Control Plan
The Certified Quality Engineer Handbook
Fourth Edition
Sarah E. Burke and Rachel T. Silvestrini
ASQ Quality Press Milwaukee, Wisconsin
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American Society for Quality, Quality Press, Milwaukee 53203 © 2017 by ASQ All rights reserved. Printed in the United States of America 22 21 20 19 18 17 5 4 3 2 1
Library of Congress Cataloging-in-Publication Data Names: Burke, Sarah E. (Sarah Ellen), 1989– editor. | Silvestrini, Rachel T.,
editor. Title: The certified quality engineer handbook / Sarah E. Burke and Rachel T.
Silvestrini, editors. Description: Fourth edition. | Milwaukee, Wisconsin : ASQ Quality Press,
[2017] | Includes bibliographical references and index. Identifiers: LCCN 2017018957 | ISBN 9780873899444 (hardcover : alk. paper) Subjects: LCSH: Production management—Quality control—Handbooks, manuals,
etc. | Reliability (Engineering)—Handbooks, manuals, etc. Classification: LCC TS156 .C423 2017 | DDC 658.4/013—dc23 LC record available at https://lccn.loc.gov/2017018957
No part of this book may be reproduced in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.
Director of Quality Press and Programs: Ray Zielke Managing Editor: Paul Daniel O’Mara Sr. Creative Services Specialist: Randy L. Benson
ASQ Mission: The American Society for Quality advances individual, organizational, and community excellence worldwide through learning, quality improvement, and knowledge exchange.
Attention Bookstores, Wholesalers, Schools, and Corporations: ASQ Quality Press books, video, audio, and software are available at quantity discounts with bulk purchases for business, educational, or instructional use. For information, please contact ASQ Quality Press at 800-248-1946, or write to ASQ Quality Press, P.O. Box 3005, Milwaukee, WI 53201–3005.
To place orders or to request a free copy of the ASQ Quality Press Publications Catalog, visit our website at http://www.asq.org/quality-press.
Printed on acid-free paper
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We dedicate this handbook to Connie Borror. Connie was a Foundation Professor at the School of Mathematical and Natural
Sciences at the west campus of Arizona State University. She was also the first female recipient of the Shewhart Medal in 2016 which speaks to her leadership within and contributions to the field of quality and applied statistics. She was an animal lover, Bette Davis aficionado, and all-around great friend. We know it was important to her to see this new edition of The Certified Quality Engineer Handbook
published. Connie, we miss you tremendously every day.
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Table of Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Preface to the Fourth Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii How to Use this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv Certified Quality Engineer (CQE) Body of Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii
Chapter 1 Management and Leadership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A. Quality Philosophies and Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
A.1. What Is Quality? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A.2. History of Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A.3. Continuous Improvement Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
B. The Quality Management System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 B.1. Strategic Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 B.2. Deployment Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 B.3. Quality Information System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
C. ASQ Code of Ethics for Professional Conduct . . . . . . . . . . . . . . . . . . . . . . . . 40 C.1. Code of Ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 C.2. Ethical Dilemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
D. Leadership Principles and Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 D.1. Developing, Building, and Organizing Teams . . . . . . . . . . . . . . . . . . . 44 D.2. Leading Quality Initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
E. Facilitation Principles and Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 E.1. Facilitator Roles and Responsibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 E.2. Facilitation Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
F. Communication Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 G. Customer Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
G.1. Customer Needs and Wants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 G.2. Quality Function Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 G.3. Customer- Driven Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
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H. Supplier Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 H.1. Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 H.2. Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 H.3. Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
I. Barriers to Quality Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Chapter 2 The Quality System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A. Elements of the Quality System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.1. Basic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.2. Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
B. Documentation of the Quality System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 B.1. Document Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 B.2. Document Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
C. Quality Standards and Other Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 C.1. The ISO 9000 Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 C.2. Other Quality Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 C.3. Malcolm Baldrige National Quality Award . . . . . . . . . . . . . . . . . . . . . . 87
D. Quality Audits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 D.1. Types of Audits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 D.2. Roles and Responsibilities in Audits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 D.3. Audit Planning and Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 D.4. Audit Reporting and Follow- Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
E. Cost of Quality (COQ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 E.1. The Economics of Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 E.2. Goal of a Quality Cost System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 E.3. Management of Quality Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 E.4. Quality Cost Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 E.5. Quality Cost Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 E.6. Quality Cost Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 E.7. Quality Cost Summary and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 E.8. Quality Cost Reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 E.9. Using Quality Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 E.10. Quality Cost Principles and Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . 101
F. Quality Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Chapter 3 Product, Process, and Service Design . . . . . . . . . . . . . . . . . . . . . . . . . 107 A. Classification of Quality Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 B. Design Inputs and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B.1. Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.2. Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
C. Technical Drawings and Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 C.1. Geometric Dimensioning and Tolerancing (GD&T) . . . . . . . . . . . . . . . 112 C.2. Positional Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
D. Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 E. Reliability and Maintainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
E.1. Predictive and Preventive Maintenance Tools . . . . . . . . . . . . . . . . . . . . 118 E.2. Reliability Systems and Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 E.3. Reliability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 E.4. Reliability, Safety, and Hazard Assessment Tools . . . . . . . . . . . . . . . . . 135
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Chapter 4 Product and Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 A. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 B. Material Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B.1. Material Identification, Status, and Traceability . . . . . . . . . . . . . . . . . . 143 B.2. Material Segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 B.3. Material Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 B.4. Material Review Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C. Acceptance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 C.1. Sampling Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 C.2. Sampling Standards and Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 C.3. Sample Integrity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
D. Measurement and Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 D.1. Measurement Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 D.2. Destructive and Nondestructive Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 191
E. Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 E.1. Standards of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 E.2. Uncertainty in Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 E.3. Traceability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 E.4. Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
F. Measurement System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 F.1. Terms and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 F.2. Gage Repeatability and Reproducibility . . . . . . . . . . . . . . . . . . . . . . . . 204 F.3. Additional Considerations for Measurement Systems . . . . . . . . . . . . 216
Chapter 5 Continuous Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 A. Quality Control Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
A.1. Flowcharts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 A.2. Cause- and-Effect Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 A.3. Check Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 A.4. Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 A.5. Pareto Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 A.6. Run Charts and Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 A.7. Scatter Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
B. Quality Management and Planning Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 B.1. Affinity Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 B.2. Force Field Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 B.3. Matrix Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 B.4. Interrelationship Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 B.5. Tree Diagrams and Prioritization Matrices . . . . . . . . . . . . . . . . . . . . . . 237 B.6. Process Decision Program Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 B.7. Activity Network Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 B.8. Process Maps and SIPOC Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 B.9. Process Value Chain Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 B.10. Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
C. Continuous Improvement Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 C.1. Total Quality Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 C.2. Kaizen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 C.3. Plan- Do-Study-Act (PDSA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
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C.4. Theory of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 C.5. Six Sigma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
D. Lean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 D.1. Continuous Flow Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 D.2. Non- Value-Added Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 D.3. Lean Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 D.4. Cycle Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
E. Corrective Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 E.1. Problem Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 E.2. Failure and Root Cause Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 E.3. Problem Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 E.4. Recurrence Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 E.5. Verification of Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
F. Preventive Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 G. Continuous Improvement Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Chapter 6 Quantitative Methods and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 A. Collecting and Summarizing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
A.1. Types of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 A.2. Measurement Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 A.3. Data Collection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 A.4. Data Accuracy and Integrity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 A.5. Graphical Methods for Depicting Relationships . . . . . . . . . . . . . . . . . 303 A.6. Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
B. Quantitative Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 B.1. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 B.2. Drawing Statistical Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 B.3. Probability Terms and Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
C. Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 C.1. Theoretical Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 C.2. General Form of Expected Value and Variance . . . . . . . . . . . . . . . . . . . 324 C.3. Common Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 C.4. Common Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 C.5. Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 C.6. Sampling Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 C.7. Probability Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
D. Statistical Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 D.1. Point Estimates and Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . 347 D.2. Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 D.3. Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 D.4. The p-Value Approach to Hypothesis Testing . . . . . . . . . . . . . . . . . . . . 380 D.5. Analysis of Variance (ANOVA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 D.6. Hypothesis Tests for Discrete Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
E. Relationships between Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 E.1. Simple Linear Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 E.2. Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 E.3. Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 E.4. Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
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F. Statistical Process Control (SPC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 F.1. Objectives and Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 F.2. Common and Special Causes of Variation . . . . . . . . . . . . . . . . . . . . . . 414 F.3. Selection of Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 F.4. Rational Subgrouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 F.5. Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 F.6. Control Chart Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 F.7. Pre- Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 F.8. Short- Run Statistical Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
G. Process and Performance Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 G.1. Process Capability Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 G.2. Process Performance versus Specifications . . . . . . . . . . . . . . . . . . . . . . 451 G.3. Process Capability Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 G.4. Process Performance Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
H. Design and Analysis of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 H.1. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 H.2. The Process of Designing and Analyzing an Experiment . . . . . . . . . . 461 H.3. Design Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 H.4. One- Factor Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 H.5. Factorial Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 H.6. Two- Level Fractional Factorial Experiments . . . . . . . . . . . . . . . . . . . . . 487 H.7. Designed Experiments and Statistical Control . . . . . . . . . . . . . . . . . . . 491
Chapter 7 Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 A. Risk Oversight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
A.1. Risk Management Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 A.2. Planning and Oversight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
B. Risk Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 B.1. Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 B.2. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 B.3. Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
C. Risk Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 C.1. Mitigation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 C.2. Risk Control and Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 C.3. Auditing and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
Appendix A Control Limit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
Appendix B Constants for Control Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
Appendix C Statistical Tolerance Factors for at Least 99% of the Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
Appendix D Standard Normal Distribution for Selected Z-Values . . . . . . . . . 528
Appendix E Areas under Standard Normal Distribution to the Left of Z-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
Appendix F F Distribution F0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
Appendix G F Distribution F0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
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Appendix H F Distribution F0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
Appendix I Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
Appendix J Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
Appendix K Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
Appendix L Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
Appendix M Median Ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
Appendix N Normal Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
Appendix O Values of t Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
Appendix P Selected National and International Quality System Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
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List of Figures
Figure 1.1 Deming’s 14 points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Figure 1.2 Ten effectiveness tests for strategic quality plans . . . . . . . . . . . . . . . . . . . . . . 13
Figure 1.3 XYZ corporation dashboard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Figure 1.4 Decision tree for production machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 1.5 Work breakdown structure (partial) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Figure 1.6 Gantt chart example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 1.7 CPM chart example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 1.8 Action plan format example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 1.9 Action plan implementation schedule example . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 1.10 Information systems strategy matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Figure 1.11 The V model for software development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 1.12 Current status report example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 1.13 ASQ Code of Ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Figure 1.14 Linking team structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 1.15 QFD house of quality diagram for a paperwork process . . . . . . . . . . . . . . . . 60
Figure 1.16 Input–output requirements matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Figure 1.17 House of quality for a car door . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Figure 2.1 Product life cycle and quality system elements . . . . . . . . . . . . . . . . . . . . . . . . 77
Figure 2.2 Tiers of the quality documentation hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . 80
Figure 3.1 Some geometric tolerancing symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Figure 3.2 Interpretation of a geometric tolerance on a drawing . . . . . . . . . . . . . . . . . . . 113
Figure 3.3 Part drawing with and without tolerances of form . . . . . . . . . . . . . . . . . . . . . 114
Figure 3.4 Two parts dimensioned with positional tolerances . . . . . . . . . . . . . . . . . . . . . 115
Figure 3.5 Reliability function versus time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Figure 3.6 Cumulative distribution function versus time . . . . . . . . . . . . . . . . . . . . . . . . . 122
Figure 3.7 Failure density versus time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Figure 3.8 Hazard rate versus time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Figure 3.9 A typical series system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Figure 3.10 A typical parallel system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Figure 3.11 A standby system with n components in standby mode . . . . . . . . . . . . . . . . 128
Figure 3.12 The general failure rate model (the bathtub curve) . . . . . . . . . . . . . . . . . . . . . 131
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Figure 3.13 Probability density function for constant failure rate . . . . . . . . . . . . . . . . . . . 132
Figure 3.14 Reliability function for constant failure rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Figure 3.15 Probability density functions for the Weibull model with different shape and scale parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Figure 3.16 Hazard rate functions for the Weibull model with different shape and scale parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Figure 4.1 Control plan example: page 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Figure 4.2 Control plan example: page 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Figure 4.3 AOQ curve for a single sampling plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Figure 4.4 An operating characteristic (OC) curve for n = 50 and c = 3 . . . . . . . . . . . . . 152
Figure 4.5 Effect on an OC curve of changing sample size (n) when accept number (c) is held constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Figure 4.6 Effect of changing accept number (c) when sample size (n) is held constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Figure 4.7 Effect of changing lot size (N) when acceptance number (c) and sample size (n) are held constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Figure 4.8 OC curves for sampling plans having the sample size equal to 10% of the lot size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Figure 4.9 OC curve for double sampling plan where n1 = 75, c1 = 0, r1 = 3, n2 = 75, c2 = 3, r2 = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Figure 4.10 Average outgoing quality curve for double sampling plan . . . . . . . . . . . . . . 161
Figure 4.11 Average sample number curve for double sampling plan . . . . . . . . . . . . . . . 163
Figure 4.12 Switching rules for normal, tightened, and reduced inspection . . . . . . . . . . 165
Figure 4.13 Structure and organization of ANSI/ASQ Z1.9-2003 (R2013) . . . . . . . . . . . . 170
Figure 4.14 Decision areas for a sequential sampling plan . . . . . . . . . . . . . . . . . . . . . . . . . 172
Figure 4.15 Go/no-go gage to check the diameter of a shaft . . . . . . . . . . . . . . . . . . . . . . . 182
Figure 4.16 ISO/R 468 surface roughness parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Figure 4.17 Other parameters of surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Figure 4.18 Two types of roundness-measuring instruments: (a) rotating table, (b) rotating workpiece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Figure 4.19 Four ways by which a center may be chosen . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Figure 4.20 Coordinate measuring machine classifications . . . . . . . . . . . . . . . . . . . . . . . . 190
Figure 4.21 Classification of standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Figure 4.22 Factors affecting the measuring process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Figure 4.23 x– and R control charts for the thermal performance example . . . . . . . . . . . . 213
Figure 5.1 Four primary flowcharting symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Figure 5.2 Flowchart for diagnostic testing process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Figure 5.3 Cause-and-effect diagram/template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Figure 5.4 Cause-and-effect diagram: product damaged after shipping . . . . . . . . . . . . 222
Figure 5.5 A simple check sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Figure 5.6 Histogram of compressive strength of concrete samples, where 3500 psi is the minimum allowed strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Figure 5.7 Typical Pareto chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
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Figure 5.8 Run chart example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Figure 5.9 Three possible relationships identified by scatter diagrams . . . . . . . . . . . . . 228
Figure 5.10 Training time versus defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Figure 5.11 Student focus group affinity diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Figure 5.12 Force field analysis for computer support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Figure 5.13 Quality function deployment matrix diagram example . . . . . . . . . . . . . . . . . 235
Figure 5.14 Line support subprocess interrelationship digraph . . . . . . . . . . . . . . . . . . . . 237
Figure 5.15 Simplified line shutdown fault tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Figure 5.16 Partial manufacturing cost improvement goal tree . . . . . . . . . . . . . . . . . . . . . 240
Figure 5.17 Receiving/storage/stocking subprocesses PDPC . . . . . . . . . . . . . . . . . . . . . . 242
Figure 5.18 Simplified CPM schedule network–line support improvement implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Figure 5.19 Enterprise-level process map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Figure 5.20 Visual alternative–improved subprocess map/PDPC . . . . . . . . . . . . . . . . . . 250
Figure 5.21 SIPOC diagram for work-related injuries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Figure 5.22 Generic production system process value chain diagram. (a) Analytical view. (b) General systems view . . . . . . . . . . . . . . . . . . . . . . . . . 252
Figure 5.23 Benchmarking and breakthrough thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Figure 5.24 Basic plan-do-study-act cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Figure 5.25 Implication of sigma-quality level. The ppm rate for part or process step considers a 1.5σ shift of the mean where only 3.4 ppm fail to meet specifications at a six sigma quality level . . . . . . . . . . . . . . . . . . . . . . 264
Figure 5.26 Normal distribution curve illustrates three sigma and six sigma parametric conformance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Figure 5.27 With a centered normal distribution between six sigma limits, only two devices per billion fail to meet the specification target . . . . . . . . . . . . . . 265
Figure 5.28 Effects of a 1.5σ shift where only 3.4 ppm fail to meet specifications . . . . . 266 Figure 5.29 Defect rates (ppm) versus sigma-quality level . . . . . . . . . . . . . . . . . . . . . . . . 267
Figure 5.30 Six Sigma metrics and implementation strategy . . . . . . . . . . . . . . . . . . . . . . . 269
Figure 5.31 A sea of inventory often hides unresolved problems . . . . . . . . . . . . . . . . . . . 273
Figure 5.32 Example of a heijunka box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Figure 5.33 C-shaped manufacturing cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Figure 5.34 Example of visual control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Figure 5.35 Example of a kanban system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Figure 5.36 A poka-yoke technique example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Figure 5.37 An example of a standard work chart for a skateboard assembly production line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Figure 5.38 Value stream map for a manufacturing process . . . . . . . . . . . . . . . . . . . . . . . 282
Figure 5.39 Value stream map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Figure 5.40 The seven phases of corrective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Figure 6.1 Example of tally or check sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
Figure 6.2 Stem-and-leaf diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Figure 6.3 Box-and-whisker diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
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Figure 6.4 Box plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Figure 6.5 Multiple box plot example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Figure 6.6 Dot plot of blood analysis turnaround times . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Figure 6.7 Histograms of variously shaped distributions . . . . . . . . . . . . . . . . . . . . . . . . . 310
Figure 6.8 A probability density function for a random variable X . . . . . . . . . . . . . . . . 322
Figure 6.9 A line graph of the pmf for random variable X . . . . . . . . . . . . . . . . . . . . . . . . 323
Figure 6.10 Probability density function for the normal distribution . . . . . . . . . . . . . . . . 325
Figure 6.11 Probability density function for a standard normal distribution . . . . . . . . . 326
Figure 6.12 Probability density function for a standard normal distribution . . . . . . . . . 327
Figure 6.13 Probability density function for an exponentially distributed random variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Figure 6.14 Probability density function for the Weibull distribution . . . . . . . . . . . . . . . 330
Figure 6.15 Continuous uniform probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . 331
Figure 6.16 Approximations to probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 338
Figure 6.17 Several chi-square distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Figure 6.18 Probability density functions for three t distributions . . . . . . . . . . . . . . . . . . 342
Figure 6.19 Normal probability plot of puncture force for toasted corn flakes . . . . . . . . 345
Figure 6.20 Weibull probability plot for life of a part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
Figure 6.21 Statistical software output of a t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
Figure 6.22 Possible decisions and errors in hypothesis testing . . . . . . . . . . . . . . . . . . . . 382
Figure 6.23 Box plots for catalyst amount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
Figure 6.24 Interaction plots of factors A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Figure 6.25 Interaction plot for temperature and catalyst . . . . . . . . . . . . . . . . . . . . . . . . . 392
Figure 6.26 Various scatter plots for two variables x and y . . . . . . . . . . . . . . . . . . . . . . . . . 399
Figure 6.27 Scatter plot of temperature and yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
Figure 6.28 Scatter plot and fitted regression line for the yield data . . . . . . . . . . . . . . . . . 404
Figure 6.29 Run chart for NMCM rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
Figure 6.30 x– and R control charts for turnaround times . . . . . . . . . . . . . . . . . . . . . . . . . . 419
Figure 6.31 x– and s control charts for complete blood count analysis turnaround times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
Figure 6.32 I and MR control charts for package weights . . . . . . . . . . . . . . . . . . . . . . . . . . 425
Figure 6.33 p chart for errors in account activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Figure 6.34 Revised p chart for errors in account activities . . . . . . . . . . . . . . . . . . . . . . . . . 429
Figure 6.35 p chart for surgical site infection rate using varying sample sizes . . . . . . . . 430
Figure 6.36 np control chart for errors in account activities . . . . . . . . . . . . . . . . . . . . . . . . 431
Figure 6.37 c chart for number of billing errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
Figure 6.38 u chart for billing statement errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Figure 6.39 CUSUM chart for package weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
Figure 6.40 EWMA control chart for package weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
Figure 6.41 Some guidelines for univariate control chart selection . . . . . . . . . . . . . . . . . . 442
Figure 6.42 Example of a pre-control chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
Figure 6.43 Short-run c chart for printed circuit boards . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
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Figure 6.44 General system process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
Figure 6.45 All possible combinations of two factors A and B, with two levels each . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Figure 6.46 Cube plot for partition length, partition height, and gap underneath . . . . . 475
Figure 6.47 Main effects plot for the air quality example . . . . . . . . . . . . . . . . . . . . . . . . . . 476
Figure 6.48 Interaction plot for partition length and partition height . . . . . . . . . . . . . . . 479
Figure 6.49 Interaction plot for partition length and gap underneath . . . . . . . . . . . . . . . 479
Figure 6.50 Interaction plot for partition height and gap underneath . . . . . . . . . . . . . . . 480
Figure 6.51 Normal probability plot of the residuals for the air quality example . . . . . 483
Figure 6.52 Residuals plotted against levels of factor A (partition length) . . . . . . . . . . . . 483
Figure 6.53 Residuals plotted against factor B (partition height) . . . . . . . . . . . . . . . . . . . 484
Figure 6.54 Contour plot for the air quality example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
Figure 6.55 Normal probability plot of the estimated effects for the air quality example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
Figure 7.1 Steps common to a risk management process . . . . . . . . . . . . . . . . . . . . . . . . . 495
Figure 7.2 AND gate for fault tree analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
Figure 7.3 OR gate for fault tree analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
Figure 7.4 Simple fault tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
Figure 7.5 Blank design FMEA form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Figure 7.6 Blank process FMEA form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
Figure 7.7 Design FMEA example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
Figure 7.8 Process FMEA example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Figure 7.9 Failure mode effects and criticality analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
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List of Tables
Table 1.1 Comparing the impact quality can have . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Table 1.2 A timeline of quality methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Table 1.3 Nominal group technique ranking table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Table 1.4 Multivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Table 1.5 Customer perspectives of value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Table 2.1 Quality cost elements by category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Table 2.2 Five different levels of evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Table 3.1 Number of failures in the time intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Table 3.2 Reliability, cdf, failure density, and hazard rate for the light bulb example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Table 4.1 Probability of acceptance for various levels of fraction nonconforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Table 4.2 AOQ levels for various levels of fraction nonconforming . . . . . . . . . . . . . . . 155
Table 4.3 Probability of acceptance for various n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Table 4.4 Fraction defective at indifference quality level . . . . . . . . . . . . . . . . . . . . . . . . 156
Table 4.5 Probability of acceptance for various lot sizes . . . . . . . . . . . . . . . . . . . . . . . . . 157
Table 4.6 Double sampling plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Table 4.7 OC curve calculations for double sampling plan . . . . . . . . . . . . . . . . . . . . . . 160
Table 4.8 Acceptance and rejection number for single, double, and multiple sampling plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Table 4.9 Percentage of acceptance sampling for previously discussed plans . . . . . . . 168
Table 4.10 Points for accept and reject lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Table 4.11 Typical standards and instrumentation for industrial length and angle measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Table 4.12 Base units of the international system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Table 4.13 Definitions of the SI base units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Table 4.14 Bias and average estimates for parts of different sizes . . . . . . . . . . . . . . . . . . 203
Table 4.15 Necessary quantities for an analysis of variance . . . . . . . . . . . . . . . . . . . . . . . 208
Table 4.16 Typical data for the gage R&R experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Table 4.17 Gage R&R estimates using the tabular method . . . . . . . . . . . . . . . . . . . . . . . . 210
Table 4.18 ANOVA for the gage R&R example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Table 4.19 Gage R&R results using the ANOVA method . . . . . . . . . . . . . . . . . . . . . . . . . 212
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Table 4.20 Variance component estimates for both methods . . . . . . . . . . . . . . . . . . . . . . 212
Table 5.1 Training data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Table 5.2 Issues, possible root causes, and general impact summary for receiving/storage/stocking PDPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Table 5.3 Line support improvement process activities, sequences, and durations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Table 5.4 Line support improvement scheduling details . . . . . . . . . . . . . . . . . . . . . . . . 247
Table 5.5 Six Sigma needs checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Table 5.6 Positrol of a wave soldering process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Table 5.7 Checklist for process certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
Table 5.8 Types of fail-safe devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Table 6.1 Frequency and cumulative frequency distributions for the ungrouped diameter data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Table 6.2 Frequency and cumulative frequency distributions for the grouped diameter data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Table 6.3 Categorical frequency distribution of manufacturing defects . . . . . . . . . . . . 311
Table 6.4 Probabilities associated with medication errors . . . . . . . . . . . . . . . . . . . . . . . 315
Table 6.5 Contingency table of part color and part size . . . . . . . . . . . . . . . . . . . . . . . . . 316
Table 6.6 pdf, mean, and variance for certain continuous distributions . . . . . . . . . . . . 333
Table 6.7 pmf, mean, and variance for certain discrete distributions . . . . . . . . . . . . . . 338
Table 6.8 Rejection regions for a single sample mean, variance known . . . . . . . . . . . . 361
Table 6.9 Rejection regions for a single sample mean, variance unknown . . . . . . . . . 363
Table 6.10 Summary of situations outlined for testing the population mean . . . . . . . . 363
Table 6.11 Rejection regions for a single sample hypothesis test on the variance . . . . . 364
Table 6.12 Rejection regions for a single sample hypothesis test on the proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Table 6.13 Rejection region for a hypothesis test on the means of two independent samples, variance known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
Table 6.14 Rejection regions for a hypothesis test on the means of two independent samples, variance equal, but unknown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
Table 6.15 Wall thickness measurements for two vendors . . . . . . . . . . . . . . . . . . . . . . . . 370
Table 6.16 Rejection regions for a hypothesis test on two independent variances . . . . 372
Table 6.17 Rejection regions for a hypothesis test on two independent proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
Table 6.18 Heart rate data for two types of exercise equipment . . . . . . . . . . . . . . . . . . . 377
Table 6.19 Heart rate data for paired observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Table 6.20 Paired heart rate data with combined differences . . . . . . . . . . . . . . . . . . . . . . 378
Table 6.21 Conversion rates for three levels of catalyst . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Table 6.22 A typical table of data for an experiment with one factor . . . . . . . . . . . . . . . 385
Table 6.23 One-way ANOVA table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Table 6.24 ANOVA table for conversion rate data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
Table 6.25 Conversion rates for experiment with two factors . . . . . . . . . . . . . . . . . . . . . 390
Table 6.26 Two-way ANOVA table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
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Table 6.27 ANOVA table for conversion rate example with two factors of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
Table 6.28 Historical percentages of nonconformities for rejected products . . . . . . . . . 393
Table 6.29 Number of nonconformities for a random week . . . . . . . . . . . . . . . . . . . . . . . 393
Table 6.30 Observed and expected frequencies for nonconformity data . . . . . . . . . . . . 394
Table 6.31 A generic contingency table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
Table 6.32 A contingency table for machine breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . 396
Table 6.33 Expected frequencies for machine breakdown . . . . . . . . . . . . . . . . . . . . . . . . 397
Table 6.34 Temperature and yield data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
Table 6.35 ANOVA table for testing significance of regression . . . . . . . . . . . . . . . . . . . . 408
Table 6.36 ANOVA table for temperature and yield regression model . . . . . . . . . . . . . . 409
Table 6.37 Peach damage data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
Table 6.38 General notation for subgroup data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
Table 6.39 Turnaround time data for x– and R charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
Table 6.40 Turnaround time data for x– and s charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
Table 6.41 Weights for dry food packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Table 6.42 Surgical site infection rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
Table 6.43 Errors in account activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
Table 6.44 Errors on hospital billing statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Table 6.45 Billing statement errors for a 24-day period . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
Table 6.46 Weights for dry food packages with custom values . . . . . . . . . . . . . . . . . . . . 437
Table 6.47 Weights for dry food packages with EWMA values . . . . . . . . . . . . . . . . . . . . 440
Table 6.48 Number of nonconformities for printed circuit boards . . . . . . . . . . . . . . . . . 448
Table 6.49 Recommended minimum values of the process capability ratio . . . . . . . . . 456
Table 6.50 A 23 full-factorial data collection table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
Table 6.51 A 23 full-factorial data collection table with data . . . . . . . . . . . . . . . . . . . . . . . 460
Table 6.52 A 23 full-factorial data collection table with run averages . . . . . . . . . . . . . . . 460
Table 6.53 Guidelines for designing an experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
Table 6.54 Randomized block general data table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
Table 6.55 ANOVA table for a randomized block design . . . . . . . . . . . . . . . . . . . . . . . . . 467
Table 6.56 Bacterial growth data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
Table 6.57 ANOVA table for the washing solution example . . . . . . . . . . . . . . . . . . . . . . 469
Table 6.58 Bacterial growth data without blocking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
Table 6.59 ANOVA table for the washing solution example without blocking . . . . . . . 470
Table 6.60 ANOVA table for two-factor factorial experiment . . . . . . . . . . . . . . . . . . . . . 471
Table 6.61 Coded factor levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Table 6.62 Combinations for terms in a two-factor interaction model . . . . . . . . . . . . . . 473
Table 6.63 Factor levels for ventilation experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
Table 6.64 Partitioning effect on ventilation effectiveness . . . . . . . . . . . . . . . . . . . . . . . . 475
Table 6.65 Main effect and interaction table for the ventilation factorial design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
Table 6.66 Estimated effects for the air quality example . . . . . . . . . . . . . . . . . . . . . . . . . . 478
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xxii List of Tables
Table 6.67 ANOVA table for the ventilation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
Table 6.68 ANOVA table for the ventilation example, with only statistically significant terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
Table 6.69 t-tests for factors and interactions for the air quality example . . . . . . . . . . . 482
Table 6.70 A single replicate of the air quality example . . . . . . . . . . . . . . . . . . . . . . . . . . 486
Table 6.71 t-test results for the air quality example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
Table 6.72 Main effects and interactions table for a 23–1 design . . . . . . . . . . . . . . . . . . . . 489
Table 6.73 Half-fraction of a 24 factorial design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
Table 7.1 HAZOP guide words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
Table 7.2 Design FMEA severity criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Table 7.3 Process FMEA severity criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Table 7.4 Design FMEA occurrence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
Table 7.5 Process FMEA occurrence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
Table 7.6 Design FMEA detection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
Table 7.7 Process FMEA detection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
Table 7.8 Risk assessment score matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
Table 7.9 Example RPNs for several potential failures . . . . . . . . . . . . . . . . . . . . . . . . . . 520
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Preface to the Fourth Edition
Like the previous editions of The Certified Quality Engineer Handbook, the fourth edition of the handbook is intended to provide the quality professional with a reference book aligned with the ASQ Certified Quality Engineer (CQE) Body of Knowledge (BoK).
The CQE BoK, updated in 2015, has for the most part minor changes from the 2006 CQE BoK. The CQE exam still contains 160 multiple- choice questions; however, with the addition of a new section in the BoK, the distribution of ques- tions from each part has changed. The BoK indicates the number of questions on the exam that cover material from each part. As before, it remains our intention to provide a handbook that is useful not only as a resource for the CQE exam but also as a general reference book for the quality professional (see the “How to Use This Book” section).
Based on the changes to the CQE BoK, as well as helpful feedback from col- leagues and reviewers, our revised fourth edition contains the following major changes:
• A new chapter on risk management
• An extensively updated glossary
• New and updated references
• A new layout of the book
We added a new chapter on risk management to reflect the addition to the BoK. An important component in risk management is the use of reliability, safety, and hazard assessment tools, many of which were covered in the Product and Pro- cess Design category of the 2006 BoK. These tools can now be found in Chap- ter 7 (“Risk Management”). While these tools are important to product design, we believe they are integral to successful risk management, particularly failure modes and effects analysis, failure mode effects and criticality analysis, and haz- ard and operability analysis.
We added many new items and definitions to the glossary; however, we also felt that some items should be removed. Our goal is that the glossary be used as a resource for those terms that are most important for the quality professional. For most glossary items, we included a reference to the chapter and section of this handbook where additional details on the defined term can be found. Those terms without specific chapter references can be found in several places throughout the handbook.
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xxiv Preface to the Fourth Edition
We included many new textbook and journal article references throughout the entire book. In Chapter 5, within the discussion of continuous improvement methods, we added descriptions of several case studies along with references to additional case studies. These case studies give the reader a broader context on how to apply many of the methods discussed in a real- life scenario. In particular, some of these case studies are related to quality in the service industry. Addition- ally, we updated discussions of and references to new technology that we feel is important to the quality professional, including emerging topics like Industry 4.0.
The BoK contains seven main categories: (I) Management and Leadership; (II) The Quality System; (III) Product, Process, and Service Design; (IV) Product and Process Control; (V) Continuous Improvement; (VI) Quantitative Methods and Tools; and (VII) Risk Management. In the third edition of the book, each chap- ter represented a subsection from these categories of the BoK. In this edition, we have aligned each chapter with a category in the BoK, so this new edition has seven chapters. We moved the BoK from its previous location in Appendix A to the beginning of the book. In some instances, we moved material in the BoK to other sections to maintain the flow of material. All content in the BoK is covered in this edition, and we refer the reader to the table of contents for specific loca- tions in the text.
We hope you find this handbook useful, and we welcome feedback for future editions.
Sarah E. Burke and Rachel T. Silvestrini
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The third edition of The Certified Quality Engineer Handbook was written to provide the quality professional with an updated resource that follows the CQE Body of Knowledge. Today it is not uncommon for the quality engineer to be involved in quality improvement activities in healthcare, finance, education, software applica- tions, and other nonmanufacturing sectors. In response, numerous new examples and illustrations that cover applications in some of these areas are included in this edition.
Key features of the third edition include:
• New and updated references
• Extensive revision of the statistical methods complete with numerous updated examples and illustrations
• Over 75 new glossary entries
Much of the material in Parts I and II of the second edition has been retained in this edition, with updated references. However, the reader will find an extensive revi- sion of the statistical methods presented throughout the book. Part V and Part VI have been significantly revised with new discussions, definitions, and examples illustrating each of the statistical techniques as they appear in the Body of Knowl- edge. Portions of Part IV have also been rewritten to reflect advances in meth- ods and applications in quality improvement activities such as conducting gage repeatability and reproducibility studies.
The goal in writing the third edition was to provide a handbook that could be used in preparation for the CQE Exam or as a reference text for professional development. When a complete description or discussion of a topic is beyond the scope of the handbook, useful references have been included for further reading. It is our hope that the reader will find the new examples, explanations, and refer- ences useful.
It is important to recognize that a handbook of this magnitude could not be completed without the dedication of many people. I would like to thank the pre- vious editors, Roger W. Berger, Donald W. Benbow, Ahmad K. Elshennawy, and H. Fred Walker, for their contributions and organization of the material in the first and second editions. In addition, gratitude goes to the authors who contributed to the first edition of the text. They wrote many of the chapters in the first edi- tion, portions of which were included in the second edition. The oversight and
Preface to the Third Edition
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xxvi Preface to the Third Edition
production of the third edition was professionally and carefully carried out by Paul O’Mara, Matthew Meinholz, William Tony, and Randy Benson at ASQ, and Leayn and Paul Tabili at New Paradigm Graphics.
Lastly, I would like to thank Dr. G. Geoffrey Vining and Dr. Douglas C. Mont- gomery for their efforts in seeing the third edition come to fruition, their careful editing of the new material, and recommendations for presentation of material. They rightfully could have been coeditors of this edition.
Connie M. Borror Editor
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In revising The Certified Quality Engineer Handbook our primary goal has been to reflect the changes in the Body of Knowledge for the Certified Quality Engineer that was published by the American Society for Quality early in 2006. We recog- nize new developments in the quality engineering profession such as:
Greater emphasis on communications
New problem- solving tools
More widespread application of Six Sigma and lean enterprise concepts
Revisions to the ISO 9000 standards
A need for more examples of how tools are applied to quality problems
As Dr. Gregory Watson said in his preface to the first edition, the American Society for Quality has been developing a more strategic perspective of the quality pro- fession and has investigated the implications that current trends across business sectors will have on our profession. The role of quality engineers has continued to shift toward being mentors and trainers for others in using the tools and concepts of quality. Mastering these tools and passing the certification exam are essential steps along the path of becoming recognized as professional quality engineers.
The revised edition provides you with both a textbook and a reference book that is completely aligned with the 2006 ASQ Body of Knowledge.
The Editors
Preface to the Second Edition
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xxviii
QuaLiTy EnginEEring—an Enduring ProFESSion
Perspective on the origins of Quality Engineering
Quality engineering was one of the disciplines that drove the American Society for Quality (ASQ, formerly known as ASQC) into its existence at the close of the Sec- ond World War. ASQ was founded to preserve and expand the expertise acquired in the war. Many improvements in production, statistical application, inspection, and management became standard practice thanks to the ASQ pioneers.
From its inception, ASQ emphasized both technical and educational aspects of the quality profession. The first certification program we developed was for quality engineering, and the body of knowledge (BoK) was prepared by a team of educators and practitioners. It was supported by the ASQ General Technical Council and soon became recognized as the core of the emerging science of quality. By creating the CQE and its body of knowledge, ASQ stabilized the meaning of “quality engineer” and also created an operational definition of quality engineer- ing. Over the years this credential has come to mean that the person who possesses it has achieved an objective standard of performance that indicates the ability to perform those tasks required of a quality engineer.
Challenge for Future Quality Engineers
Over the past 10 years, the American Society for Quality has been developing a more strategic perspective of the quality profession and has investigated the implications that current trends across business sectors will have on our profes- sion. In 1995 and again in 1999, ASQ took out crystal balls to “study the future” and determine what actions to take in supporting the quality movement and its cadre of professionals.
Several trends have been observed in these studies: some are disturbing and others serve as a beacon to warn us to take corrective action in navigating our course into the future. One major implication already observed in many compa- nies is the transference of advanced quality tools from their almost exclusive use by quality professionals into application by frontline managers and their specially trained problem solvers. This trend will challenge quality engineering profession- als in two major ways.
First, while we observe that quality tools are being disseminated to the masses, this cascade may or may not involve quality professionals. This wider application
Preface to the First Edition
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Preface to the First Edition xxix
of advanced statistical methods and quality tools requires quality professionals to accept new roles as technical mentors to the managers of our organizations. This challenge requires each of us to develop a new approach to leadership and to be the catalyst that aids in the dissemination of these methods by finding ways to encourage the proper use and application of these tools.
Second, with more and more managers knowing the same tools that we use, we cannot afford to be amateurs in the use and application of advanced quality methods. In order to earn the right to serve as the technical advisors to this next generation of more enlightened managers, all quality professionals must not only seek training in the more advanced technical methods but also must become the masters of these tools and be perceived as such by senior managers.
Call to Continuous Learning and Personal Excellence
Rather than giving up on the viability of our profession, this challenge is a call for an even higher commitment to professional performance. In the quality profes- sion, our tradition has been to use independent certifications as evidence of per- sonal mastery of a particular body of knowledge.
The achievement of certification as a quality engineer through the ASQ CQE examination is a distinction of professional achievement that represents personal mastery of the basic quality tools and analytical methods. The certified quality engineer is exposed to increased professional opportunity, promotion potential, and salary increases. Most CQEs go on to further develop skills as quality train- ers, facilitators, business managers, auditors, applied statisticians, and technical specialists. For all of these career potentials the CQE certification serves as a mark of professionalism that proclaims a readiness to meet new levels of professional challenge and extend knowledge into more complex and difficult areas to master.
Significance and Meaning of Certification
There is an old story of a young man who served as an apprentice, passed the tests and skill demonstrations as a journeyman, and was ready to be named an independent tradesman. He went to his master craftsman and told him that he was ready to go out and establish his own practice. The master said he had one more test to pass. The young man replied: “I am ready.” The master asked him to describe the true meaning of his professional credential. The young man imme- diately replied: “It means the end of my journey, a well- deserved reward for all of my hard work.” The master said that he did not have the right perspective. After a month the young man returned saying that he was ready to answer the question. Again the master asked him the true meaning of his professional cre- dential. This time the young man replied: “It is a symbol of distinction and a sign of high achievement.” Again the master was dissatisfied, and said, “Return to me next month when you understand the full meaning!” In humility, the young man returned after a third month. The master again asked his question and the young man replied: “This credential only represents the beginning. It is the start of a never- ending journey of work, discipline, and a ceaseless commitment to continu- ous learning.” The master said: “Now you are ready to work on your own!”
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xxx Preface to the First Edition
Studying for Self- improvement
As markets become more and more competitive, companies will need to enhance their agility in order to provide a flexible response to changing customer demands. This trend will require more customer intimacy as companies seek to understand the value proposition required by the market and define what customer require- ments will deliver the most value to the market that they choose to serve. Quality professionals will be asked to develop real- time quality monitoring systems and data collection and analysis methods that provide corrective feedback to minimize waste, reduce defects, and improve cost- effectiveness of inventory and capital equipment. Quality professionals also will be asked to build systems for moni- toring customer behavior, and to use the information in defining better product designs. Quality will become more and more fundamental in the management of routine business operations. Preparation for this emerging trend will call for personal dedication to developing oneself as not only a competent technician, but also as a local leader capable of influencing others to achieve quality performance results in a wide variety of applications.
Enhancing professional competence is the starting point to prepare yourself to be a force in this field. The certified quality engineer credential is a big step in the right direction toward personal development and assuring the continued viability of your set of professional skills.
This handbook will guide you through the recently updated body of knowl- edge and provide you with an exceptionally relevant textbook in your preparation for taking the CQE examination.
Gregory H. Watson Past President, American Society for Quality (2000–2001)
Fellow, American Society for Quality Certified Quality Engineer, American Society for Quality
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Quality engineering is recognized as a core technical discipline in a variety of industries and functions. The effective quality engineer must understand numer- ous concepts and techniques. To foster and recognize such achievement, the Amer- ican Society for Quality Control (as it was then called) created a Certified Quality Engineer (CQE) program in 1967 and has updated it frequently since then. This handbook and the certification program are offered by ASQ as ways to maintain and stimulate the profession of quality engineering.
This book has two main uses: as a learning tool and as a reference tool. We kept these two uses in mind as we assembled the various parts of the book.
LEarning TooL When you are in learning mode, you first need to see the big picture and then fill in the details. You seek continuity, rationale, and examples. Following the ASQ- prescribed Body of Knowledge (BoK), we have organized the subject into seven broad categories:
1. Management and Leadership
2. The Quality System
3. Product, Process, and Service Design
4. Product and Process Control
5. Continuous Improvement
6. Quantitative Methods and Tools
7. Risk Management
Each of these categories is a separate chapter in this book, which begins with a summary to give a broad picture of how the details fit together. The CQE BoK, which is printed just before Chapter 1, contains 82 elements. Keep in mind that the book was not written as a study guide to pass the certification exam, but as a com- prehensive guide to the field of quality engineering. Therefore, most of the chap- ters include material that goes well beyond the CQE exam requirements. If you are using this book to study for the exam, you must carefully examine the wording of the BoK to see which topics are of most immediate concern. For the most part, the
How to use this Book
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xxxii How to Use this Book
sections of each chapter follow the BoK, but there are some slight deviations that best match practice of use. We suggest that the reader refer to the table of contents and index to locate specific items.
Some of the more technical material must be studied intensely and repeti- tively before it is fully grasped. Examples are often essential to complete the learning process, and we have therefore provided many. We also recognize that your thirst for knowledge often cannot be satisfied by the contents of just this one book, so we have listed many sources of additional information. Also, we suggest using The ASQ CQE Study Guide for practice test questions for the CQE certifica- tion exam.
rEFErEnCE TooL When you are in reference- using mode, your thought process is quite different from that of learning mode. You want information and you want it quick. Often a single fact, procedure, or definition is required. Regardless of the kind of informa- tion you seek, the best starting point is the index. The editors and production staff have greatly extended the index of the fourth edition, and we recommend you use it regularly to look things up.
Several other features serve your reference needs. Immediately following the main text are the necessary statistical tables, all of which are cited in the text. Once you become familiar with a given statistical tool, you can often use the appropriate table without consulting the chapter. Statistical tables are listed both in the table of contents (front matter) and immediately after Chapter 7.
All figures and tables in the chapters are listed in the front matter, immedi- ately following the table of contents. Consulting these lists may lead you to a key answer in certain cases. An extensive glossary provides another reference tool. These definitions come from a variety of sources, including the fourth edition of ASQ Quality Press’s Glossary and Tables for Statistical Quality Control. For most of the glossary items, we have included a chapter and section reference where additional information can be found.
The editors believe that you will find this book a valuable learning and refer- ence tool. But you are the final judge of our success, so we welcome your com- ments and suggestions. Please e-mail, call, or write using the contact information located on the back cover.
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xxxiii
We would like to thank the people who helped us with the preparation and com- pletion of this handbook. ASQ was wonderful throughout the process, and we would like to specifically thank Paul O’Mara, managing editor for ASQ, and Matt Meinholz, former ASQ associate publisher. Thanks to Westchester Publishing Services for a thorough and excellent job in the editing and layout phases of this handbook. Several of our colleagues were instrumental in helping us with the updated edition, specifically Doug Montgomery, Dan McCarville, and Geoff Vining. All of the reviewers of this handbook provided excellent information and thought- ful comments, and we especially would like to thank Tamara Rehm and Wouter Mollers. We would also like to thank all of the first and second edition editors of the book for their contributions: Roger W. Berger, Donald W. Benbow, Ahmad K. Elshennawy, and H. Fred Walker. Finally, we would like to thank the third edition editor, Connie M. Borror, for her extensive revisions and work, especially in the Quantitative Methods section.
We would also like to acknowledge the following authors and contributors to the previous editions of this book:
Martha Atkins
Dennis Arter
Andy Barnett
Dale H. Besterfield
Connie M. Borror
Forrest W. Breyfogle III
Elsayed A. Elsayed
Hugh Jordan Harrington
Bradley Jones
William Kolarik
Kreg Kukor
Becki Meadows
Roderick A. Munro
Duke Okes
Jack B. ReVelle
Denise Robitaille
David Shores
Galal Wehaba
Russ Wescott
Chris White
acknowledgments
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xxxiv
AHP: analytical hierarchy process AIAG: Automotive Industry Action Group AND: activity network diagram ANOVA: analysis of variance AOQ: average outgoing quality AOQL: average outgoing quality limit AQL: acceptable quality limit AQP: advanced quality planning ARL: average run length ASN: average sample number ASQ: American Society for Quality ASTM: American Society for Testing and Materials ATE: automated test equipment ATI: average total inspection BoK: body of knowledge cdf: cumulative density function/cumulative distribution function CL: center line CLA: center line average CM: configuration management CMM: coordinate measuring machines COPQ: cost of poor quality COQ: cost of quality CPM: critical path method CQE: certified quality engineer CSA: Canadian Standards Association CUSUM: cumulative sum df: degrees of freedom DMAIC: define, measure, analyze, improve, control DMRCS: define, measure, reduce, combine, select DoD: Department of Defense
List of acronyms
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List of Acronyms xxxv
DOE: design of experiments DPMO: defects per million opportunities DPU: defects per unit DR: discrimination ratio EC: earliest completion time ECP: engineering change proposal EEO: equal employment opportunity 8D: eight disciplines ES: earliest start time ESS: environmental stress screening ESSEH: Environmental Stress Screening of Electronic Hardware EWMA: exponentially weighted moving average F: Fahrenheit FMEA: failure modes and effects analysis FMECA: failure modes effects and criticality analysis FS: free slack FT: fault tree FTA: fault tree analysis GD&T: geometric dimensioning and tolerancing HACCP: hazard analysis and critical control points HAZOP: hazard and operability analysis IATF: International Automotive Task Force IQ: installation qualification IQR: inter- quartile range IRR: internal rate of return ISO: International Organization for Standardization IT: information technology JIT: just- in-time K: Kelvin LC: latest completion time LCL: lower control limit LRM: linear responsibility matrix LS: latest start time LSC: least squares circle LSL: lower specification limit LSS: Lean- Six Sigma LTPD: lot tolerance percent defective MA: moving average MAP: measurement assurance protocol MBNQA: Malcolm Baldrige National Quality Award
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xxxvi List of Acronyms
MCC: minimum circumscribed circle MIC: maximum inscribed circle MMC: maximum material condition MRB: material review board MRP: manufacturing resource planning MRS: minimum radial separation MS: mean square MSA: measurement systems analysis MTBF: mean time between failures MTTF: mean time to failure MTTR: mean time to repair MZC: minimum zone circle NAVAIR: Naval Air Systems Command NAVMAT: naval material command NDT: nondestructive testing NIST: National Institute of Standards and Technology NMCM: not- mission capable equipment due to maintenance NPV: net present value OC: operating characteristic OEM: original equipment manufacturer OQ: operational qualification PC: peak count PCB: printed circuit board PDCA: plan–do–check–act pdf: probability density function PDPC: process decision program chart PDSA: plan–do–study–act PERT: program evaluation and review technique PII: personally identifiable information PLC: programmable logic controller pmf: probability mass function PPAP: part production approval process ppm: parts per million PPQ: process performance qualification PQ: process qualification PTR: precision- to-tolerance ratio PVC: process value chain QE: quality engineer QFD: quality function deployment QIS: quality information system
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List of Acronyms xxxvii
QMS: quality management system R&D: research and development R&R: repeatability and reproducibility RCDQ: reactive customer- driven quality RCI: rapid continuous improvement RFID: radio frequency identification RMS: root mean square ROI: return on investment RPN: risk priority number RQL: rejectable quality level RRM: resource requirements matrix RTY: rolled throughput yield SAE: Society of Automotive Engineers SCADA: supervisory control and data acquisition SDWT: self- directed work team s.e.: standard error SI: Systems International SIPOC: suppliers, inputs, process, outputs, customers SMED: single minute exchange of dies SNR: signal- to-noise ratio SPC: statistical process control SQA: supplier quality assurance SQM: supplier quality management SQP: strategic quality planning SS: sum of squares SWOT: strengths, weaknesses, opportunities, threats TOC: theory of constraints TPM: total productive maintenance TQM: total quality management TS: total slack UCL: upper control limit USL: upper specification limit VOC: voice of the customer VSM: value stream map WBS: work breakdown structure WIP: work in process
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xxxviii
The topics in this Body of Knowledge include subtext explanations and the cog- nitive level at which the questions will be written. This information will provide useful guidance for both the Exam Development Committee and the candidate preparing to take the exam. The subtext is not intended to limit the subject mat- ter or be all-inclusive of that material that will be covered in the exam. It is meant to clarify the type of content that will be included on the exam. The descriptor in parentheses at the end of each entry refers to the maximum cognitive level at which the topic will be tested. A complete description of cognitive levels is pro- vided at the end of this document.
I. Management and Leadership (18 Questions)
A. Quality Philosophies and Foundations
Describe continuous improvement tools, including lean, six sigma, theory of constraints, statistical process control (SPC), and total quality management, and understand how modern quality has evolved from quality control through statistical process control (SPC) to total quality management and leadership principles (including Deming’s 14 points). (Understand)
B. The Quality Management System (QMS)
1. Strategic planning
Identify and define top management’s responsibility for the QMS, including establishing policies and objectives, setting organization- wide goals, and supporting quality initiatives. (Apply)
2. Deployment techniques
Define, describe, and use various deployment tools in support of the QMS such as:
a. Benchmarking
Define the concept of benchmarking and why it may be used. (Remember)
b. Stakeholder
Define, describe and use stakeholder identification and analysis. (Apply)
Certified Quality Engineer (CQE) Body of Knowledge
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Certified Quality Engineer (CQE) Body of Knowledge xxxix
c. Performance
Define, describe and use performance measurement tools. (Apply)
d. Project management
Define, describe and use project management tools, including PERT charts, Gantt charts, critical path method (CPM), and resource allo- cation. (Apply)
3. Quality information system (QIS)
Identify and describe the basic elements of a QIS, including who will contribute data, the kind of data to be managed, who will have access to the data, the level of flexibility for future information needs, and data analysis. (Understand)
C. ASQ Code of Ethics for Professional Conduct
Determine appropriate behavior in situations requiring ethical decisions. (Evaluate)
D. Leadership Principles and Techniques
Analyze various principles and techniques for developing and organizing teams and leading quality initiatives. (Analyze)
E. Facilitation Principles and Techniques
1. Roles and responsibilities
Describe the facilitator’s roles and responsibilities on a team. (Understand)
2. Facilitation tools
Apply various tools used with teams, including brainstorming, nominal group technique, conflict resolution, and force- field analysis. (Apply)
F. Communication Skills
Identify specific communication methods that are used for delivering information and messages in a variety of situations across all levels of the organization. (Analyze)
G. Customer Relations
Define, apply, and analyze the results of customer relation tools such as quality function deployment (QFD) and customer satisfaction surveys. (Analyze)
H. Supplier Management
1. Techniques
Apply various supplier management techniques, including supplier qualification, certification, and evaluation. (Apply)
2. Improvement
Analyze supplier ratings and performance improvement results. (Analyze)
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xl Certified Quality Engineer (CQE) Body of Knowledge
3. Risk
Understand business continuity, resiliency, and contingency planning. (Understand)
I. Barriers to Quality Improvement
Identify barriers to quality improvement, analyze their causes and impact, and implement methods for improvement. (Analyze)
II. The Quality System (16 Questions)
A. Elements of the Quality System
1. Basic elements
Interpret the basic elements of a quality system, including planning, control, and improvement, from product and process design through quality cost systems and audit programs. (Evaluate)
2. Design
Analyze the design and alignment of interrelated processes to the stra- tegic plan and core processes. (Analyze)
B. Documentation of the Quality System
1. Document components
Identify and describe quality system documentation components, including quality policies and procedures to support the system. (Understand)
2. Document control
Evaluate configuration management, maintenance, and document control to manage work instructions and quality records. (Evaluate)
C. Quality Standards and Other Guidelines
Apply national and international standards and other requirements and guidelines, including the Malcolm Baldrige National Quality Award (MBNQA), and describe key points of the ISO 9000 series of standards. [Note: Industry- specific standards will not be tested.] (Apply)
D. Quality Audits
1. Types of audits
Describe and distinguish between various types of quality audits such as product, process, management (system), registration (certification), compliance (regulatory), first, second, and third party. (Apply)
2. Roles and responsibilities in audits
Identify and define roles and responsibilities for audit participants such as audit team (leader and members), client, and auditee. (Understand)
3. Audit planning and implementation
Describe and apply the stages of a quality audit, from audit planning through conducting the audit. (Apply)
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Certified Quality Engineer (CQE) Body of Knowledge xli
4. Audit reporting and follow- up
Apply the steps of audit reporting and follow up, including the need to verify corrective action. (Apply)
E. Cost of Quality (COQ)
Identify and apply COQ concepts, including cost categorization, data collection, reporting, and interpreting results. (Analyze)
F. Quality Training
Identify and apply key elements of a training program, including conducting a needs analysis, developing curricula and materials, and determining the program’s effectiveness. (Apply)
III. Product, Process, and Service Design (23 Questions)
A. Classification of Quality Characteristics
Define, interpret, and classify quality characteristics for new and existing products, processes, and services. [Note: The classification of defects is covered in IV.B.3.] (Evaluate)
B. Design Inputs and Review
1. Inputs
Translate design inputs such as customer needs, regulatory require- ments, and risk assessment into robust design using techniques such as failure mode and effects analysis (FMEA), quality function deployment (QFD), Design for X (DFX), and Design for Six Sigma (DFSS). (Analyze)
2. Review
Identify and apply common elements of the design review process, including roles and responsibilities of participants. (Apply)
C. Technical Drawings and Specifications
Interpret specification requirements in relation to product and process characteristics and technical drawings, including characteristics such as views, title blocks, dimensioning and tolerancing, and GD&T symbols. (Evaluate)
D. Verification and Validation
Interpret the results of evaluations and tests used to verify and validate the design of products, processes and services, such as installation qualification (IQ), operational qualification (OQ), and process qualification (PQ). (Evaluate)
E. Reliability and Maintainability
1. Predictive and preventive maintenance tools
Describe and apply the tools and techniques used to maintain and improve process and product reliability. (Apply)
2. Reliability and maintainability indices
Review and analyze indices such as MTTF, MTBF, MTTR, availability, and failure rate. (Analyze)
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3. Reliability models
Identify, define, and distinguish between the basic elements of reliabil- ity models such as exponential, Weibull, and bathtub curve. (Apply)
4. Reliability/safety/hazard assessment tools
Define, construct, and interpret the results of failure mode and effects analysis (FMEA), failure mode, effects, and criticality analysis (FMECA), and fault tree analysis (FTA). (Evaluate)
IV. Product and Process Control (25 Questions)
A. Methods
Implement product and process control methods such as control plan development, critical control point identification, and work instruction development and validation. (Analyze)
B. Material Control
1. Material identification, status, and traceability
Define and distinguish between these concepts, and describe methods for applying them in various situations. (Analyze)
2. Material segregation
Describe material segregation and its importance, and evaluate appro- priate methods for applying it in various situations. (Evaluate)
3. Material classification
Classify product and process defects and non- conformities. (Evaluate)
4. Material review board (MRB)
Describe the purpose and function of an MRB and evaluate noncon- forming product or material to make a disposition decision in various situations. (Evaluate)
C. Acceptance Sampling
1. Sampling concepts
Interpret the concepts of producer and consumer risk and related terms, including operating characteristic (OC) curves, acceptable quality limit (AQL), lot tolerance percent defective (LTPD), average outgoing qual- ity (AOQ), and average outgoing quality limit (AOQL). (Analyze)
2. Sampling standards and plans
Identify, interpret, and apply ANSI/ASQ Z1.4 and Z1.9 standards for attributes and variables sampling. Identify and distinguish between single, double, multiple, sequential, and continuous sampling meth- ods. Identify the characteristics of Dodge- Romig sampling tables and when they should be used. (Analyze)
3. Sample integrity
Identify and apply techniques for establishing and maintaining sam- ple integrity. (Apply)
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D. Measurement and Test
1. Measurement tools
Select and describe appropriate uses of inspection tools such as gage blocks, calipers, micrometers, and optical comparators. (Analyze)
2. Destructive and nondestructive tests
Identify when destructive and nondestructive measurement test meth- ods should be used and apply the methods appropriately. (Apply)
E. Metrology
Apply metrology techniques such as calibration, traceability to calibration standards, measurement error and its sources, and control and maintenance of measurement standards and devices. (Analyze)
F. Measurement System Analysis (MSA)
Calculate, analyze, and interpret repeatability and reproducibility (Gage R&R) studies, measurement correlation, capability, bias, linearity, precision, stability and accuracy, as well as related MSA quantitative and graphical methods. (Evaluate)
V. Continuous Improvement (27 Questions)
A. Quality Control Tools
Select, construct, apply, and interpret the following quality control tools:
1. Flowcharts
2. Pareto charts
3. Cause and effect diagrams
4. Control charts
5. Check sheets
6. Scatter diagrams
7. Histograms (Analyze)
B. Quality Management and Planning Tools
Select, construct, apply, and interpret the following quality management and planning tools:
1. Affinity diagrams and force field analysis
2. Tree diagrams
3. Process decision program charts (PDPC)
4. Matrix diagrams
5. Interrelationship digraphs
6. Prioritization matrices
7. Activity network diagrams (Analyze)
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C. Continuous Improvement Methodologies
Define, describe, and apply the following continuous improvement methodologies:
1. Total quality management (TQM)
2. Kaizen
3. Plan-do-check-act (PDCA)
4. Six sigma
5. Theory of constraints (TOC) (Evaluate)
D. Lean tools
Define, describe, and apply the following lean tools:
1. 5S
2. Value-stream mapping
3. Kanban
4. Visual control
5. Waste (Muda)
6. Standardized work
7. Takt time
8. Single minute exchange of die (SMED) (Evaluate)
E. Corrective Action
Identify, describe, and apply elements of the corrective action process, including problem identification, failure analysis, root cause analysis, problem correction, recurrence control, and verification of effectiveness. (Evaluate)
F. Preventive Action
Identify, describe and apply various preventive action tools such as error- proofing/poka-yoke, robust design and analyze their effectiveness. (Evaluate)
VI. Quantitative Methods and Tools (36 Questions)
A. Collecting and Summarizing Data
1. Types of data
Define, classify, and compare discrete (attributes) and continuous (variables) data. (Apply)
2. Measurement scales
Define and describe nominal, ordinal, interval, and ratio scales. (Understand)
3. Data collection methods
Describe various methods for collecting data, including tally or check sheets, data coding, automatic gaging, and identify the strengths and weaknesses of the methods. (Apply)
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4. Data accuracy and integrity
Apply techniques that ensure data accuracy and integrity, and iden- tify factors that can influence data accuracy such as source/resource issues, flexibility, versatility, inconsistency, inappropriate interpreta- tion of data values, and redundancy. (Apply)
5. Descriptive statistics
Describe, calculate, and interpret measures of central tendency and dis- persion (central limit theorem), and construct and interpret frequency distributions, including simple, categorical, grouped, ungrouped, and cumulative. (Evaluate)
6. Graphical methods for depicting relationships
Construct, apply, and interpret diagrams and charts such as stem- and- leaf plots, and box- and-whisker plots. [Note: Scatter diagrams are cov- ered in V.A.] (Analyze)
7. Graphical methods for depicting distributions
Construct, apply, and interpret diagrams such as normal and non- normal probability plots. [Note: Histograms are covered in V.A.] (Analyze)
B. Quantitative Concepts
1. Terminology
Define and apply quantitative terms, including population, parameter, sample, statistic, random sampling, and expected value. (Analyze)
2. Drawing statistical conclusions
Distinguish between numeric and analytical studies. Assess the valid- ity of statistical conclusions by analyzing the assumptions used and the robustness of the technique used. (Evaluate)
3. Probability terms and concepts
Describe concepts such as independence, mutually exclusive, multi- plication rules, complementary probability, and joint occurrence of events. (Understand)
C. Probability Distributions
1. Continuous distributions
Define and distinguish between these distributions such as normal, uniform, bivariate normal, exponential, lognormal, Weibull, chi square, Student’s t and F. (Analyze)
2. Discrete distributions
Define and distinguish between these distributions such as binomial, Poisson, hypergeometric, and multinomial. (Analyze)
D. Statistical Decision- Making
1. Point estimates and confidence intervals
Define, describe, and assess the efficiency and bias of estimators. Cal- culate and interpret standard error, tolerance intervals, and confidence intervals. (Evaluate)
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2. Hypothesis testing
Define, interpret, and apply hypothesis tests for means, variances, and proportions. Apply and interpret the concepts of significance level, power, type I and type II errors. Define and distinguish between statis- tical and practical significance. (Evaluate)
3. Paired-comparison tests
Define and use paired- comparison (parametric) hypothesis tests, and interpret the results. (Apply)
4. Goodness-of-fit tests
Define chi square and other goodness- of-fit tests, and understand the results. (Understand)
5. Analysis of variance (ANOVA)
Define and use ANOVAs and interpret the results. (Analyze)
6. Contingency tables
Define and use contingency tables to evaluate statistical significance. (Apply)
E. Relationships Between Variables
1. Linear regression
Calculate the regression equation for simple regressions and least squares estimates. Construct and interpret hypothesis tests for regres- sion statistics. Use linear regression models for estimation and predic- tion. (Analyze)
2. Simple linear correlation
Calculate the correlation coefficient and its confidence interval, and construct and interpret a hypothesis test for correlation statistics. (Analyze)
3. Time-series analysis
Define, describe, and use time- series analysis, including moving aver- age to identify trends and seasonal or cyclical variation. (Apply)
F. Statistical Process Control (SPC)
1. Objectives and benefits
Identify and explain the objectives and benefits of SPC. (Understand)
2. Common and special causes
Describe, identify, and distinguish between these types of causes. (Analyze)
3. Selection of variable
Identify and select characteristics for monitoring by control chart. (Analyze)
4. Rational subgrouping
Define and apply the principles of rational subgrouping. (Apply)
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5. Control charts
Identify, select, construct, and use various control charts, including x– and R, x– and s, individuals and moving range (ImR or XmR), mov- ing average and moving range (MAMR), p, np, c, and u. (Analyze)
6. Control chart analysis
Read and interpret control charts and use rules for determining statis- tical control. (Evaluate)
7. Pre-control charts
Define and describe these charts and how they differ from other con- trol charts. (Understand)
8. Short-run SPC
Identify and define short- run SPC rules. (Understand)
G. Process and Performance Capability
1. Process capability studies
Define, describe, calculate, and use process capability studies, includ- ing identifying characteristics, specifications and tolerances, develop- ing sampling plans for such studies, and establishing statistical control. (Analyze)
2. Process performance vs. specifications
Distinguish between natural process limits and specification lim- its, and calculate percent defective, defects per million opportunities (DPMO), and parts per million (ppm). (Analyze)
3. Process capability indices
Define, select, and calculate Cp , Cpk, Cpm, and Cr, and evaluate process capability. (Evaluate)
4. Process performance indices
Define, select, and calculate Pp and Ppk, and evaluate process perfor- mance. (Evaluate)
H. Design and Analysis of Experiments
1. Terminology
Define terms such as dependent and independent variables, factors, levels, response, treatment, error, and replication. (Understand)
2. Planning and organizing experiments
Identify the basic elements of designed experiments, including deter- mining the experiment objective, selecting factors, responses, and mea- surement methods, and choosing the appropriate design. (Analyze)
3. Design principles
Define and apply the principles of power and sample size, balance, replication, order, efficiency, randomization, blocking, interaction, and confounding. (Apply)
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4. One-factor experiments
Construct one- factor experiments such as completely randomized, randomized block, and Latin square designs, and use computa- tional and graphical methods to analyze the significance of results. (Analyze)
5. Full-factorial experiments
Construct full- factorial designs and use computational and graphical methods to analyze the significance of results. (Analyze)
6. Two-level fractional factorial experiments
Construct two- level fractional factorial designs and apply computa- tional and graphical methods to analyze the significance of results. (Analyze)
VII. Risk Management (15 Questions)
A. Risk Oversight
1. Planning and oversight
Understand identification, planning, prioritization, and oversight of risk. (Understand)
2. Metrics
Identify and apply evaluation metrics. (Apply)
3. Mitigation planning
Apply and interpret risk mitigation plan. (Evaluate)
B. Risk Assessment
Apply categorization methods and evaluation tools to assess risk. (Analyze)
C. Risk Control
1. Identification and documentation
Identify and document risks, gaps and controls. (Analyze)
2. Auditing and testing
Apply auditing techniques and testing of controls. (Evaluate)
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LEvELS oF CogniTion BaSEd on BLooM’S TaxonoMy—rEviSEd (2001)
In addition to content specifics, the subtext for each topic in this BOK also indi- cates the intended complexity level of the test questions for that topic. These levels are based on “Levels of Cognition” (from Bloom’s Taxonomy—Revised, 2001) and are presented below in rank order, from least complex to most complex.
remember
Recall or recognize terms, definitions, facts, ideas, materials, patterns, sequences, methods, principles.
understand
Read and understand descriptions, communications, reports, tables, diagrams, directions, regulations.
apply
Know when and how to use ideas, procedures, methods, formulas, principles, theories.
analyze
Break down information into its constituent parts and recognize their relationship to one another and how they are organized; identify sublevel factors or salient data from a complex scenario.
Evaluate
Make judgments about the value of proposed ideas, solutions, by comparing the proposal to specific criteria or standards.
Create
Put parts or elements together in such a way as to reveal a pattern or structure not clearly there before; identify which data or information from a complex set is appropriate to examine further or from which supported conclusions can be drawn.
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1
Chapter 1 Management and Leadership
Chapter One
The two main themes of Chapter 1 are a broad perspective on the quality pro- fession and the human element in quality. Areas such as strategic planning and leadership may require additional training and years of experience before full competency is achieved. In the same vein, developing communication skills and removing barriers to quality improvement could be callings of a lifetime. After a careful study of this chapter, you will have a clear idea of the elements upon which the profession of quality engineering is based.
The quality profession has both a human element and a technical element, and in Chapter 1 we examine the human element of quality from several differ- ent perspectives. First, we discuss definitions of quality, followed by a review of the history of quality. We note the contributions of the leading gurus over the past 80 years, starting with Walter Shewhart and highlighting his two greatest successors, W. Edwards Deming and Joseph M. Juran. Some major quality pro- grams we discuss are statistical process control, total quality management, lean philosophy, theory of constraints, and Six Sigma. No matter whether the quality program is one that is discussed here or something else, a successful organiza- tion will have some kind of system for managing its quality. One way to view the quality management system is to break it into three parts: strategic planning of the vision and goals, deployment techniques for converting the vision/goals into reality, and an information system to collect, analyze, and report the data. Deployment techniques used for selecting and managing projects include return on investment (ROI), program evaluation and review technique (PERT), and Gantt charts. Heavy emphasis is also given to performance measurement tools. Next, we discuss professional ethics, including the ASQ Code of Ethics and legal constraints on the quality engineer (QE).
Leadership, facilitation, and communication skills are all interrelated. For the organization to achieve its goals in a positive and efficient manner, leaders must translate vision and goals into tangible activities. Executive direction and indirect or “soft” leadership known as facilitation unleash the energy of mid- and lower- level employees. Communication skills are critical to effective leadership and facilitation, as well as to individual career success.
The final three sections of Chapter 1 address the role of quality in dealing with customers, suppliers, and barriers to improvement. Two typical techniques for addressing the role of quality are supplier surveys, which tell us what we can expect from our suppliers, and customer surveys, which tell us what our cus- tomers think of us. Finally, the section on barriers reinforces the idea that quality
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improvement is a constant struggle, and that the various ideas of this book must be applied again and again in order to maintain momentum toward that elusive but unobtainable goal of perfection.
a. QuaLiTy PHiLoSoPHiES and FoundaTionS In this section we discuss the meaning of quality and provide a brief history of quality. We highlight the pioneers of the quality engineering movement: Walter A. Shewhart, W. Edwards Deming, and Joseph M. Juran. We also introduce several continuous improvement methods.
a.1. What is Quality?
Quality means different things to different people in different situations. As Henry Ford said, “Quality means doing it right when no one is looking.” Some additional informal descriptions and results of quality include the following:
• Quality is inversely proportional to variability
• Quality is not a program; it is an approach to business
• Quality is a collection of powerful tools and concepts that are proven to work
• Quality is defined by customers through their satisfaction
• Quality includes continual improvement and breakthrough events
• Quality tools and techniques are applicable in every aspect of business
• Quality is aimed at perfection; anything less is an improvement opportunity
• Quality increases customer satisfaction, reduces cycle time and costs, and eliminates errors and rework
Additionally, typical elements used to assess quality include totality of features, essential characteristics, ability to satisfy needs, conformance to requirements, degree or grade of excellence, free of deficiencies, and meeting/exceeding customer expectations. Garvin (1987) discusses eight dimensions of quality, and Montgom- ery (2013) adds three more in regard to service and transactional organizations. These dimensions include performance, reliability, durability, serviceability, aes- thetics, features, perceived quality, conformance to standards, responsiveness, professionalism, and attentiveness.
Quality is not just for businesses; it is also for nonprofit organizations such as schools, healthcare and social services, and government agencies. Results, perfor- mance and financial, are the natural consequence of effective quality management. Table 1.1 compares the consequences and impact of quality management at two different quality levels, three sigma (99.74% good) and six sigma (99.9998% good). Six sigma quality is discussed in more detail in Chapter 5, section C.5.
The above quality descriptors show that quality is difficult to define, and no one definition can be all- inclusive. The word “quality” is highly nuanced and
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allows many interpretations. For example, a popular online dictionary defines quality as “an inherent or distinguishing characteristic” (http://dictionary. reference.com/search?q= quality); this definition is one of many distinctly differ- ent definitions from the same authority. The reader quickly comes to realize that most of the definitions are quite specialized and not really pertinent to the prac- tice of quality engineering. According to ISO 9000:2015, section 3.6.2, quality is defined as “the degree to which a set of inherent characteristics of an object fulfills requirements.”
This definition is really quite interesting, first because it is published by ISO, an international standards organization, and second because it specifically rebuts the definition that Joseph Juran used throughout his career: “quality = fitness for use.” In contrast, Philip Crosby used the definition “quality = conformance to specifica- tions.” There probably never will be an ultimate definition of this all- important word, as the definition is constantly evolving.
The views of eight well- known quality experts appeared in the July 2001 issue of Quality Progress. Although these experts differ on details and nuances, some common themes appear in all their different quality philosophies:
1. Quality improvement is a never- ending process
2. Top management commitment, knowledge, and active participation are critical
3. Management is responsible for articulating a company philosophy, goals, measurable objectives, and a change strategy
4. All employees in the organization need to be active participants
5. A common language and set of procedures are important to communicate and support the quality effort
6. A process must be established to identify the most critical problems, determine their causes, and find solutions
7. Changes in company culture, roles, and responsibilities may be required
Table 1.1 Comparing the impact quality can have.
99.74% good = 3 sigma 99.9998% good = 6 sigma
20,000 lost articles of mail per hour Seven articles lost per hour
Unsafe drinking water for almost 15 minutes each day
One unsafe minute every seven months
5000 incorrect surgical operations per week 1.7 incorrect operations per week
Two short or long landings at most major airports each day
One short or long landing every five years
200,000 wrong drug prescriptions each year 68 wrong prescriptions per year
No electricity for almost seven hours each month
One hour without electricity every 34 years
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a.2. History of Quality
The quality profession has a long history, which has greatly accelerated over the past 80 years. Joseph M. Juran (1988) traced the practice of the quality profession back to the ancient Egyptians and the building of the pyramids. For centuries, quality was intrinsically associated with craftsmanship, and each craftsman con- trolled all aspects of the final product of his craft. This changed dramatically with the Industrial Revolution.
Modern quality practices originated in two stages: mass inspection in the early 1900s and the control chart around 1930. Mass inspection became commonplace as a result of Frederick Taylor’s Scientific Management. Workers stopped check- ing the quality of their work and instead passed it on to specially trained inspec- tors. Although inspection is a vital element of quality, Walter Shewhart’s invention of the process control chart really initiated the quality profession. Awareness of worker motivation and attitudes as contributors to quality became prevalent in the early 1930s as a result of Elton Mayo’s Hawthorne studies for Western Electric.
The next big push for quality emerged during World War II when suddenly people’s lives could be destroyed by poor- quality products. At the same time, hun- dreds of American companies were called on to manufacture goods to the most exacting requirements. Many quality control techniques, such as acceptance sam- pling and process control charts, which were merely encouraged before the war, became mandatory as part of the defense effort. Two of the leading practitioners of the quality profession—W. Edwards Deming and Joseph M. Juran—established their professional credentials during this time. Both later went to Japan to teach statistical and management tools. The Japanese excelled in developing quality methods, and in the 1970s, Americans made repeated trips to Japan to explore Japanese successes and to bring home proven Japanese methods.
The American Society for Quality Control, now known as the American Soci- ety for Quality (ASQ), was established soon after World War II when Martin Brum- baugh saw that great benefits would be attained if he could unify various local quality control societies into one national organization. As he struggled with this task, he recognized the superb skills of George Edwards, who was then head of inspection engineering at Bell Telephone Laboratories. Edwards became the first president of the society and helped establish policies that guide its operation to this day.
The first three awards the society created to recognize these three pioneers of quality were the Brumbaugh Award, the Shewhart Medal, and the Edwards Medal. In time, the society created numerous other awards, each honoring a spe- cific hero of the profession and recognizing outstanding achievement in a particu- lar area of the profession. These awards include the Crosby Medal, Feigenbaum Medal, Juran Medal, Deming Medal, and Ishikawa Medal, among others.
Table 1.2 provides a detailed timeline that shows the development and pro- gression of formal methods and practice in quality engineering. Management and statistics are the most critical aspects of the quality control and engineering move- ment. In the remainder of this section, we discuss in more detail the influences of the three people who have arguably had the biggest impact on the quality move- ment: Shewhart, Deming, and Juran. Shewhart, Edwards, Juran, and Deming all worked for and learned from the Bell System in one way or another. Edwards and Shewhart retired as Bell System employees. Both Juran and Deming went on from the Bell System to become world- famous consultants and authors.
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Table 1.2 A timeline of quality methods. (Continued)
1700–1900 Quality is largely determined by the efforts of an individual craftsman. Eli Whitney introduces standardized, interchangeable parts to simplify assembly.
1875 Frederick W. Taylor introduces “Scientific Management” principles to divide work into smaller, more easily accomplished units—the first approach to dealing with more complex products and processes. The focus was on productivity. Later contributors were Frank Gilbreth and Henry Gantt.
1900–1930 Henry Ford—the assembly line—further refinement of work methods to improve productivity and quality; Ford developed mistake-proof assembly concepts, self- checking, and in-process inspection.
1901 First standards laboratories established in Great Britain.
1907–1908 AT&T begins systematic inspection and testing of products and materials.
1908 W. S. Gosset (writing as “Student”) introduces the t-distribution—results from his work on quality control at Guinness Brewery.
1915–1919 WWI—British government begins a supplier certification program.
1919 Technical Inspection Association is formed in England; this later becomes the Institute of Quality Assurance.
1920s AT&T Bell Laboratories forms a quality department—emphasizing quality, inspection and test, and product reliability. B. P. Dudding at General Electric in England uses statistical methods to control the quality of electric lamps.
1922 Henry Ford writes (with Samuel Crowtha) and publishes My Life and Work, which focused on elimination of waste and improving process efficiency. Many Ford concepts and ideas are the basis of lean principles used today.
1922–1923 R. A. Fisher publishes series of fundamental papers on designed experiments and their application to the agricultural sciences.
1924 W. A. Shewhart introduces the control chart concept in a Bell Laboratories technical memorandum.
1928 Acceptance sampling methodology is developed and refined by H. F. Dodge and H. G. Romig at Bell Labs.
1931 W. A. Shewhart publishes Economic Control of Quality of Manufactured Product— outlining statistical methods for use in production and control chart methods.
1932 W. A. Shewhart gives lectures on statistical methods in production and control charts at the University of London.
1932–1933 British textile and woolen industry and German chemical industry begin use of designed experiments for product/process development.
1933 The Royal Statistical Society forms the Industrial and Agricultural Research Section.
1938 W. E. Deming invites Shewhart to present seminars on control charts at the U.S. Department of Agriculture Graduate School.
1940 The U.S. War Department publishes a guide for using control charts to analyze process data.
(continued)
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Table 1.2 A timeline of quality methods. (Continued)
1940–1943 Bell Labs develops the forerunners of the military standard sampling plans for the U.S. Army.
1942 In Great Britain, the Ministry of Supply Advising Service on Statistical Methods and Quality Control is formed.
1942–1946 Training courses on statistical quality control are given to industry; more than 15 quality societies are formed in North America.
1944 Industrial Quality Control begins publication.
1946 The American Society for Quality Control (ASQC) is formed as the merger of various quality societies. The International Standards Organization (ISO) is founded. Deming is invited to Japan by the Economic and Scientific Services Section of the U.S. War Department to help occupation forces in rebuilding Japanese industry. The Japanese Union of Scientists and Engineers (JUSE) is formed.
1946–1949 Deming is invited to give statistical quality control seminars to Japanese industry.
1948 G. Taguchi begins study and application of experimental design.
1950 Deming begins education of Japanese industrial managers; statistical quality control methods begin to be widely taught in Japan.
1950–1975 Taiichi Ohno, Shigeo Shingo, and Eiji Toyoda develop the Toyota Production System, an integrated technical/social system that defined and developed many lean principles such as just-in-time production and rapid setup of tools and equipment. K. Ishikawa introduces the cause-and-effect diagram.
1950s Classic texts on statistical quality control by Eugene Grant and A. J. Duncan appear.
1951 A. V. Feigenbaum publishes the first edition of his book Total Quality Control. JUSE establishes the Deming Prize for significant achievement in quality control and quality methodology.
1951+ G. E. P. Box and K. B. Wilson publish fundamental work on using designed experiments and response surface methodology for process optimization; focus is on chemical industry. Applications of designed experiments in the chemical industry grow steadily after this.
1954 Joseph M. Juran is invited by the Japanese to lecture on quality management and improvement. British statistician E. S. Page introduces the cumulative sum (CUSUM) control chart.
1957 J. M. Juran and F. M. Gryna’s Quality Control Handbook is first published.
1959 Technometrics (a journal of statistics for the physical, chemical, and engineering sciences) is established; J. Stuart Hunter is the founding editor. S. Roberts introduces the exponentially weighted moving average (EWMA) control chart. The U.S. manned spaceflight program makes industry aware of the need for reliable products; the field of reliability engineering grows from this starting point.
1960 G. E. P. Box and J. S. Hunter write fundamental papers on 2k–p factorial designs. The quality control circle concept is introduced in Japan by K. Ishikawa.
1961 National Council for Quality and Productivity is formed in Great Britain as part of the British Productivity Council.
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Table 1.2 A timeline of quality methods. (Continued)
1960s Courses in statistical quality control become widespread in industrial engineering academic programs. Zero defects (ZD) programs are introduced in certain U.S. industries.
1969 Industrial Quality Control ceases publication, replaced by Quality Progress and the Journal of Quality Technology (Lloyd S. Nelson is the founding editor of JQT).
1970s In Great Britain, the NCQP and the Institute of Quality Assurance merge to form the British Quality Association.
1975–1978 Books on designed experiments oriented toward engineers and scientists begin to appear. Interest in quality circles begins in North America—this grows into the total quality management (TQM) movement.
1980s Experimental design methods are introduced to and adopted by a wider group of organizations, including the electronics, aerospace, semiconductor, and automotive industries. The works of Taguchi on designed experiments first appear in the United States.
1984 The American Statistical Association (ASA) establishes the Ad Hoc Committee on Quality and Productivity; this later becomes a full section of the ASA. The journal Quality and Reliability Engineering International appears.
1986 Box and others visit Japan, noting the extensive use of designed experiments and other statistical methods.
1987 ISO publishes the first quality systems standard. Motorola’s Six Sigma initiative begins.
1988 The Malcolm Baldrige National Quality Award is established by the U.S. Congress. The European Foundation for Quality Management is founded; this organization administers the European Quality Award.
1989 The journal Quality Engineering appears.
1990s ISO 9000 certification activities increase in U.S. industry; applicants for the Baldrige award grow steadily; many states sponsor quality awards based on the Baldrige criteria.
1995 Many undergraduate engineering programs require formal courses in statistical techniques, focusing on basic methods for process characterization and improvement.
1997 Motorola’s Six Sigma approach spreads to other industries.
1998 The American Society for Quality Control becomes the American Society for Quality (see www.asq.org), attempting to indicate the broader aspects of the quality improvement field.
2000s ISO 9000:2000 standard is issued. Supply-chain management and supplier quality become even more critical factors in business success. Quality improvement activities expand beyond the traditional industrial setting into many other areas, including financial services, health care, insurance, and utilities. Organizations begin to integrate lean principles into their Six Sigma initiatives, and lean Six Sigma becomes a widespread approach to business development.
Source: Reprinted from D. C. Montgomery, Introduction to Statistical Quality Control, 7th ed. (Hoboken, NJ: John Wiley & Sons, 2013).
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A.2.a. Walter A. Shewhart
The industrial age was approaching its second century when a young engineer named Walter A. Shewhart altered the course of industrial history by bringing together the disciplines of statistics, engineering, and economics. He referred to his greatest achievement, the invention of the process control chart, as “the for- mulation of a scientific basis for securing economic control.” The Shewhart control chart is sometimes referred to as a process behavior chart.
Shewhart wanted statistical theory to serve the needs of industry. He exhibited the restlessness of one looking for a better way. A man of science who patiently developed his and others’ ideas, he was an astute observer of the world of science and technology. While the literature of the day discussed the stochastic nature of both biological and technical systems, and spoke of the possibility of applying statistical methodology to these systems, Shewhart actually showed how it could be done. In that respect, the field of quality control can claim a genuine pioneer in Shewhart. His book Economic Control of Quality of Manufactured Product, published in 1931, is regarded as a complete and thorough exposition of the basic principles of quality control.
Called on frequently as a consultant, Shewhart served the War Department, the United Nations, the government of India, and others. He was active with the National Research Council and the International Statistical Institute. He was a fel- low of numerous societies and in 1947 became the first honorary member of ASQ. Many consider the Shewhart Medal, given for outstanding technical contributions to the quality profession, to be the most prestigious award ASQ offers. As of 2016, 67 people have been awarded the Shewhart Medal in recognition of their contribu- tions to the quality profession.
A.2.b. W. Edwards Deming
Deming became the best- known quality expert in the United States. He delivered his message on quality not only throughout the United States but also around the world. In recognition of his valuable contribution to Japan’s postwar recovery, the Union of Japanese Scientists and Engineers established an annual award for qual- ity achievement called the Deming Prize.
Deming (1982) emphasized that the keys to quality are in management’s hands—85% of quality problems are due to the system and only 15% are due to employees. The heart of his quality strategy is the use of statistical quality control to identify special (erratic, unpredictable) causes and common (systemic) causes of variation. Statistical tools provide a common language for employees throughout a company and permit quality control efforts to be widely diffused. Each employee assumes considerable responsibility for the quality of his or her own work. Those in traditional quality control functions are then able to take more proactive roles in the quality improvement effort.
Deming introduced statistical quality control to the Japanese in the early 1950s when Japan was recovering from World War II and trying to overcome a reputa- tion for shoddy workmanship. Deming’s guidance was instrumental in transform- ing “made in Japan” from a liability to an asset. Deming asserted that there was no point in exhorting employees to produce higher- quality work because the changes needed to improve quality were almost always outside the workers’ control, such
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as having the right tools, training, and materials. Instead, management had to accept responsibility for quality. Based on his experience, Deming developed a 14-point set of requirements called Deming’s 14 points, shown in Figure 1.1. He also described seven deadly diseases of the workplace, including emphasis on short- term profits, use of personnel performance evaluations (which he labeled “management by fear”), and mobility of management (management as a profession independent of the product/service or commitment to the organization).
A.2.c. Joseph M. Juran
Juran, like Deming, built his quality reputation in America and then took his expertise to Japan in the 1950s. The two complemented each other well in Japan, as Deming showed the use of statistical tools and Juran taught the techniques of managing for quality. Juran originated the concept of “the vital few” and the
1. Create consistency of purpose toward improvement of products and services, with a plan to become competitive and to stay in business. Decide to whom top management is responsible.
2. Adopt the new philosophy. We are in a new economic age. We can no longer live with commonly accepted levels of delays, mistakes, defective materials, and defective workmanship.
3. Cease dependence on mass inspection. Require instead statistical evidence that quality is built-in to eliminate need for inspection. Purchasing managers have a new job and must learn it.
4. End the practice of awarding business on the basis of price tag. Instead, depend on meaningful measures of quality, along with price. Eliminate suppliers who cannot qualify with statistical evidence of quality.
5. Find problems. It is management’s job to work continually on the system (design, incoming materials, composition of material, maintenance, improvement of machines, training, supervision, retraining).
6. Institute modern methods of training on the job.
7. Institute modern methods of supervision of production workers. The responsibility of foremen must be changed from sheer numbers to quality. Improvement of quality will automatically improve productivity. Management must prepare to take immediate actions on reports from foremen concerning barriers such as inherited defects, machines not maintained, poor tools, fuzzy operation definitions.
8. Drive out fear, so that everyone may work effectively for the company.
9. Break down barriers between departments. People in research, design, sales, and production must work as a team, to foresee problems of production that may be encountered with various materials and specifications.
10. Eliminate numerical goals, posters, and slogans for the workforce, asking for new levels of productivity without providing methods.
11. Eliminate work standards that prescribe numerical quotas.
12. Remove barriers that stand between the hourly worker and his right to pride of workmanship.
13. Institute a vigorous program of education and retraining.
14. Create a structure in top management that will push every day on the above 13 points.
Figure 1.1 Deming’s 14 points.
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“useful many” (originally “trivial many”). He called this concept the Pareto princi- ple, which is now implemented in the well- known Pareto diagram. An economist, Vilfredo Pareto, had noticed the phenomenon, but it was Juran who applied it to quality improvement.
Juran recognized that improving quality requires a completely different approach from what is needed to maintain existing quality. He demonstrated this idea in his book Managerial Breakthrough, first published in 1964, and later con- densed his ideas into the Juran trilogy:
1. Quality control: monitoring techniques to correct sporadic problems (analogous to special causes)
2. Quality improvement: a breakthrough sequence to solve chronic problems (analogous to common causes)
3. Quality planning: an annual quality program to institutionalize managerial control and review
Juran served the quality profession well when in 1951 he created the monumental Juran’s Quality Handbook, now in its seventh edition (Defeo 2016). Juran’s contribu- tions are extensive and varied. He defined quality as “fitness for use by the cus- tomer.” He emphasized the need for top managers to become personally involved in order for a quality effort to be successful and for middle and lower- level managers to learn the language and thinking of top management—money, for example—in order to secure their involvement. Juran’s universal process for qual- ity improvement requires studying symptoms, diagnosing causes, and applying remedies. He repeatedly emphasized that major improvement could be achieved only on a project- by-project basis. The basis for selecting projects was the ROI, now a major component of Six Sigma.
a.3. Continuous improvement Tools
In the seven decades since World War II ended, great quality leaders have emerged. Besides those mentioned previously, the following individuals have become famous for their contributions. Philip Crosby popularized the concept of zero defects and established the Crosby Quality College. Kaoru Ishikawa, who helped sponsor Deming’s seminars in Japan, created quality circles and invented the cause- and-effect diagram, also called the Ishikawa diagram. Armand Feigen- baum coined the term “total quality control” and tirelessly preached its funda- mentals around the world. Genichi Taguchi, a Japanese engineer, developed a unique system for designing industrial experiments to establish robust systems. Eliyahu Goldratt created an improvement system built around the theory of con- straints. Other notable contributors to the profession include George Box, Eugene Grant, Jack Lancaster, Frank Gryna, Richard Freund, and Dorian Shainan.
The most notable continuous improvement methodologies (all of which include the use of various quality tools) in quality engineering are the following:
• Statistical process control
• Total quality management
• Theory of constraints
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• Lean
• Six Sigma
Statistical process control (SPC) is considered one of the major areas of statistical technology useful in quality improvement. The main tool in SPC is the control chart, which has been widely adapted and is utilized by all practitioners of quality engineering. Process control and improvement is discussed in Chapter 4, while Chapter 6, section F, covers SPC in detail.
Total quality management (TQM) is based on the principles of Feigenbaum, Deming, and Juran. The exact origin of TQM has been debated, but TQM as a pro- cess improvement methodology received the most use and attention in the mid- 1980s and early 1990s before being mostly replaced by lean and Six Sigma efforts. TQM is a structured approach to managing quality improvement methods within an organization. While it is important as a continuous improvement tool, its lack of recent success is attributed to insufficient effort on the technical aspects associ- ated with improving and maintaining quality in an organization. TQM is further discussed in Chapter 5, section C.1.
Theory of constraints (TOC) has become a popular catchphrase for a system improvement program. It is based on the principle that one—and often more than one—specific factor or element constrains or prevents the system from reaching a more desirable state of existence. Goldratt had an insight: managing a complex system or organization can be made both simpler and more effective by providing managers with a few specific areas on which to focus, maximizing performance in the areas of key constraints, or elevating the constraints, making them less restric- tive. This leads to a view of the company where the constraint guides all strategic decisions. Goldratt’s clients and students have claimed numerous major successes in applying his concepts. He coauthored The Goal, the first famous business novel that informed and entertained many thousands of managers and engineers as it showed the path to success by applying his concepts. TOC, sometimes referred to as constraint management, has been actively developed by a loosely coupled community of practitioners around the world. TOC is discussed in more detail in Chapter 5, section C.4.
Lean philosophy, discussed in detail in Chapter 5, section D, is exemplified by its terse name: get the job done as simply as possible. It was originally called lean manufacturing but has migrated into many different service industries. A good example of lean philosophy is just- in-time (JIT), where a process is managed so that parts arrive just prior to their actual insertion into the assembly.
Six Sigma, a widely used quality philosophy, combines and exploits the strengths of other approaches to the extent that it now dominates all others. There are journals, conferences, study groups, and consulting firms devoted solely to Six Sigma. Six Sigma combines effective communication, organization of effort, financial accountability, and strong techniques to enable organizations to make sustained improvements over a period of time. Improvements such as cost reduction, quality improvement, cycle time reduction, improved morale, greater profits, and so forth, are all attainable through Six Sigma, but these improvements require a great deal of dedicated work, dedication to the process, and continuous training and learning. See Chapter 5, section C.5, for more about Six Sigma.
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B. THE QuaLiTy ManagEMEnT SySTEM In this section, we discuss aspects of the quality management system (QMS), which include strategic planning, deployment techniques, and the quality infor- mation system.
B.1. Strategic Planning
Strategic planning usually begins with an analysis phase. The strengths and weak- nesses of the organization are assessed and forecasts are generated to predict how market opportunities and competitive threats will change during the time period covered by the study. This analysis is sometimes called a SWOT (strengths, weak- nesses, opportunities, and threats) analysis. Ideally, strategic planning for quality will address each aspect of the SWOT analysis.
The strengths of the organization can be leveraged to create or sustain a com- petitive advantage. The weaknesses of the organization should be addressed through appropriate measures such as training initiatives to develop strategic skills or process improvement efforts. The opportunities available to the organiza- tion can be identified through various marketing research techniques. Key outputs of the marketing research may include estimates of the size and growth rate of the market and clearly articulated customer expectations, desires, and perceptions. This information should drive new product development efforts.
Finally, the business environment should be assessed, with particular empha- sis on potential threats to the success of the organization. Threats can come from direct competitors offering similar products, indirect competitors offering sub- stitute products or services (e.g., butter vs. margarine), suppliers of critical pro- prietary components, and even from distributors that can influence the purchase decisions of the final customers.
After the SWOT analysis is complete, the organization can develop strategic quality plans. As the strategy is being formulated, management should evaluate whether the plans will ensure the success of the organization. To discern the effec- tiveness of strategic quality plans, management should employ a series of sequen- tially ordered effectiveness tests, shown in Figure 1.2 and discussed in more detail below:
1. Does the strategy adequately address all four SWOT elements? Leverage the organization’s strengths; remedy the weaknesses. Exploit the opportunities in the market; minimize the potential impact of external threats. It also may be prudent to prepare contingency plans that can be implemented quickly in response to threatening actions from competitors. It is crucial for this stage of the planning process to be data- driven. The analysis should be comprehensive, including product quality, finance, purchasing, human resources, marketing and sales, delivery, customer service, and the internal processes that drive these activities. The notion that quality improvement is limited to the factory floor is obsolete. When management begins to apply quality disciplines and statistical methods to assess advertising campaigns and HR initiatives, the transformation is under way. The organization is poised to establish strategic quality plans.
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2. Will the strategic plan result in a significant competitive advantage in the marketplace? Incremental improvements in quality may not be sufficient to ensure success. Furthermore, the advantage must be recognized and valued by the customer. Engineering and manufacturing can create superior products, but that may not help the organization succeed if the customers do not know about the products. Other activities must be involved in the strategic planning process. For example, marketing is responsible for raising customer awareness of product enhancements and influencing purchase decisions through advertising or promotions. Keep in mind that the current strengths of an organization may only generate passing interest among customers. For example, a product may have best- in-class durability, but customers may be more interested in appearance, availability, or ease of use. In such cases, consider strategic initiatives that will strengthen the organization’s ability to maximize customer satisfaction throughout the purchase and ownership experience. Such market research tools as conjoint analysis and the Kano quality model can measure how product or service features influence customer purchase behavior. Companies that use market research to help select targets for creating a competitive advantage are more likely to thrive in the marketplace.
3. Is the competitive advantage sustainable? Can your competitors quickly and easily imitate your strategy? Will they respond with counteroffensives that weaken your position? Will your competitors’ strategic efforts pay off a year from now and undermine your leadership in the market? Some consultants recommend avoiding cost reduction as a primary strategy because price is one of the easiest things to imitate in the market. Both you and your competitors will lose if a price war erupts. Anyone can reduce costs by using cheaper components or
Figure 1.2 Ten effectiveness tests for strategic quality plans.
Strategic Planning Effectiveness Tests
1. Does the plan adequately address strengths, weaknesses, opportunities, and threats (SWOT)?
2. Will the plan result in a significant competitive advantage in the marketplace?
3. Is this advantage sustainable?
4. Does the vision statement inspire a sense of mission and purpose among employees?
5. Are the goals and objectives SMART (specific, measurable, achievable, realistic, and time-based)?
6. Are the goals and objectives aligned throughout the organization?
7. Have adequate resources been allocated to achieve the plan?
8. Are organizational structures, systems, and processes appropriate to execute the plan?
9. Is a review/reporting system in place to monitor the execution of the plan?
10. Does the strategic planning team include representatives from all key stakeholders?
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reducing staff in service or support activities. The risk of this approach is that customers may perceive deteriorating quality, which damages the organization’s reputation and results in lost sales. Insisting on a strategy that will deliver outstanding quality through continuous improvement is much more likely to generate a sustainable competitive advantage. The popularity of the Six Sigma movement and its impressive success stories demonstrate that it is possible to embark on a major strategic quality improvement initiative and reap substantial benefits on the bottom line.
4. Does the vision inspire and motivate your employees? The vision should be customer focused and provide a clear, succinct view of the desired future state of the organization. A major strategic effort will require dedication and commitment. Resources may be stretched to achieve the vision. If the vision is too difficult to achieve, employees may become discouraged and give up. If the vision is too easy to achieve, your competitors may implement something better, and you will be playing catch- up.
5. Goals and objectives are established to direct the efforts of the organization and measure whether the vision is being achieved. Are the goals and objectives SMART?
— Specific. State what is expected in precise terms.
— Measurable. Demonstrate progress through quantitative rather than qualitative or subjective measures.
— Achievable. The goal can be achieved with available resources if appropriate actions are taken.
— Realistic. A reasonable, sensible person would accept the goal after considering the degree of difficulty and the probability of success.
— Time-based. Deadlines serve a useful purpose. Companies that are first to market with new innovations frequently enjoy a significant, sustainable advantage over their competitors. For more information on innovation and competitive advantage, see Porter and Stern (2001), Hockman and Jensen (2016), and Box and Woodall (2012).
6. Are the goals and objectives aligned throughout the organization? Goals and objectives must be in harmony with each other. As goals are cascaded through an organization and broken down into manageable tasks to be performed by various departments or individuals, unity of purpose and alignment of priorities must be maintained to avoid conflicts.
7. Are resources (staffing, equipment, financing, etc.) adequate to achieve the plan? Can the additional workload be absorbed? Are the skill levels of the employees sufficient? Has the time line been reviewed by affected participants to ensure that there are no scheduling conflicts? Project management techniques such as critical path method (CPM) may be helpful. CPM will identify the critical paths in the program and provide documentation as to when the resources will be required. CPM is discussed in more detail in section B.2.d of this chapter.
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8. Are organizational structures, systems, and processes suitable for executing the plan? Is a departmental reorganization necessary to streamline the flow of work and facilitate concurrent activities? Is a research and development (R&D) effort necessary to upgrade designs or manufacturing equipment capability?
9. Is a review and reporting system in place to periodically assess progress? These reviews should be conducted by management at a high enough level within the organization to marshal additional resources as needed when the program is in danger of falling behind schedule. Key program milestones should have clearly defined expectations to ensure consistency and excellence in the execution of the activities. Checklists are a simple yet effective means of communicating the expectations.
10. Does the strategic planning team include the participation of experienced professionals from all affected work groups? Do the team members fully understand the strategy, and have they bought into it? The benefits of a cross- functional planning effort cannot be overemphasized. Consider an analogy to the product development process: manufacturing personnel contribute expert advice during the early stages of product design and thereby avoid costly, time- consuming delays and redesigns. Ford Motor Company’s advanced quality planning process lists the use of a cross- functional team as the number one expectation for executing many of the quality disciplines within a product development effort.
The importance of establishing the right strategy is critical to the success of an organization. Countless years of sincere toil have been wasted by implementing poorly developed strategies. Excellent execution will not ensure success unless the plan is also excellent. Juran argues that a structured planning process results in products that perform better and have a shorter development cycle from concept to customer (Juran and Godfrey 1999).
Management must explore strategic quality initiatives that go beyond mere incremental improvement: Drive the philosophy of continuous improvement throughout the organization and create a culture of innovation. Look beyond the factory floor for breakthroughs in all systems, such as R&D, product development, marketing, human resources, and purchasing. Strive for quality initiatives that add value for the customer and establish a sustainable competitive advantage.
B.2. deployment Techniques
Quality improvement does not just happen; it must be planned, supported, and monitored just like any other process. Planning requires ways to identify the spe- cific initiatives to be undertaken, while support and monitoring require methods for tracking and communicating progress. Establishing goals is not enough. Goals must be supported by measurable objectives that are in turn supported by action plans that delineate how and when the objectives are to be achieved and by whom. There must be measurable objectives in order to know what the projected results should be. In addition, a means for measuring the attainment of these objectives must be established. Similarly, action plans provide more specific information
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about attaining objectives. An example of the hierarchical relationships between strategies, goals, objectives, and action plans is as follows:
Organizational strategy: Continually build and retain a loyal customer base.
Organizational goal: Deliver all products to all customers 100% on time.
Organizational objective: Given current capacity, improve delivery dates of all future customer orders from 35% to 75% on- time delivery by February of the current year and to 100% by August of the current year.
Functional objectives: The quality department will assign a QE to convene a cross- functional process improvement team by November 1 of the current year. The team will utilize lean manufacturing techniques to reduce cycle time and will continue its efforts until the production process has achieved 100% on- time delivery performance.
Action plans: Detailed plans state how and when the objective will be achieved and by whom. Action plans may resemble mini project plans or may be more complex project planning documents as needs dictate. In either case, action plans influence planning and scheduling.
Deployment techniques in support of the QMS include benchmarking against competitors, feedback from stakeholders, performance assessment via metrics, and project management. These are all discussed in this section as well as addi- tional information regarding deployment such as policy considerations and use- ful tools.
B.2.a. Benchmarking
Benchmarking is a process in which organizations compare their performance with that of their competition or with best practices found internally or in outside orga- nizations. It was pioneered by Xerox in the late 1970s in response to growing pres- sure in the photocopy industry. Benchmarking is now recognized as an important input to strategic planning. It can be applied to any business process or function, such as optimizing inventory levels or improving service delivery.
Benchmarking can help an organization identify new ideas and methods to improve operational effectiveness. It can help break through institutional barri- ers and resistance to change because some other organization has already dem- onstrated that the new methods are more effective. Once these best practices are identified, the organization can develop plans to adopt them. In this way, bench- marking can become an integral part of the continuous improvement process.
Internal benchmarking is used to compare performance between plants or divisions. Competitive benchmarking is used to assess performance relative to that of direct competitors within an industry. Internal and competitive bench- marks are useful in identifying gaps in performance. For example, automotive manufacturers use customer surveys to compare quality and customer satisfac- tion. Poor performance must be addressed to ensure survival in the marketplace. However, competitive benchmarking may not identify the best practices needed to close the gap in performance. Furthermore, although benchmarking internally or among competitors may identify incremental improvement opportunities, it is not likely to identify breakthroughs leading to world- class performance.
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Collaborative benchmarking requires cooperation between two or more orga- nizations. Each organization freely shares information about its best practices in exchange for information about other best practices from a partner. Suppose, for example, Wal- Mart wishes to team with Dell Corporation. Wal- Mart offers to share information on forecasting consumer demand, and Dell reciprocates by sharing insights on how it minimizes order- to-delivery times. With collaborative bench- marking, the key is to identify the very best performer. Use trade associations, publications, financial analysis, market research, or other tools to find the leader.
External benchmarking may identify the best opportunities, but it requires a significant investment of time and effort. It may be useful to employ internal benchmarking first because it will generate quicker results. Internal successes should receive recognition, which can help convince skeptics that the process works. The benchmarking team also will gain valuable experience and be better prepared for pursuing external benchmarking partners. A typical benchmarking project includes the following:
• Planning. Identify what is to be benchmarked. Establish the objectives for the study. If the scope is too narrow, the benefits will be limited. If the scope is too broad, the task may become unmanageable and the probability of successfully implementing the best practices will diminish. Select the team members and search for target organizations to benchmark.
• Data collection. Develop a mutually acceptable protocol with the partner, including a code of conduct, confidentiality agreements, and performance measures to be analyzed. Data sharing may include information about procedures, standards, software, training, and other supporting systems. The key is to gain enough understanding and direction to replicate the best practice within your organization.
• Analysis. Assess the data for accuracy and credibility. Determine current performance levels and identify gaps. Explore the feasibility of implementing the best practice. Some practices are not readily transferable—is adaptation necessary? Forecast the expected improvement.
• Implementation. Obtain the support of key stakeholders. Use project management techniques or action plans to initiate the change. Monitor performance. Document activities and communicate progress.
Benchmarking is not a precise discipline, and common pitfalls include lack of commitment, insufficient planning, comparing processes that are not sufficiently similar to generate useful insights, and measuring processes that have little poten- tial for significant gains. A well- executed benchmarking project will help in both deploying strategic plans and suggesting modifications to future strategic plans. But real leadership means not just catching up with other industry leaders but surpassing them. Benchmarking can never accomplish that.
B.2.b. Stakeholder Analysis
Congruence between policy and results is evaluated through audits that periodi- cally check for conformance. The stakeholders need to be clearly identified and their differing needs must be met. If adaptation of a policy must occur, it must
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remain within the original intent if the policy is to remain credible to the stake- holders. Frequent feedback from all stakeholders helps to quickly identify and correct any disparity. Performance measures, discussed below, must take into account the differing needs and perceptions of each stakeholder group. Stakehold- ers include the following:
• Stockholders, the owners of the company. Their role is often passive and their needs are primarily of a financial nature. They expect the company to maintain its credibility in the financial markets and hope for growth in earnings and share price.
• The executive group, including the board of directors and the top tier of managers. They must acknowledge and serve the other stakeholders. Conversely, the health of any organization is critically dependent on its decision making and deployment.
• Employees other than top management. This critical group of stakeholders has little direct impact on policy but all other groups depend on them to carry out the policy efficiently and promptly. The quality of any organization’s end product depends on how well the employees are recruited, trained, and supervised.
• Suppliers and customers. These two groups are concerned with external inputs and outputs. Suppliers must adhere to contractual requirements and therefore can insist on fair and prompt payment for their goods and services. Customers are paramount stakeholders; if customers do not want the organization’s products, this organization will eventually cease to exist. Sections G and H later in the chapter deal with customer relations and supplier management, respectively.
• The community at large. Communities, neighbors, environmental regulators, law enforcement agencies, chambers of commerce, legislatures, and similar bodies often are indirect stakeholders. Individually their impact is relatively slight, but if a major issue arises, the concerns of a community can have an overwhelming influence. This stakeholder group is especially critical when plant openings or closings are being planned. The community often is concerned about treatment of minorities, public service (or the absence thereof), and environmental abuses.
B.2.c. Performance
The strategic plan is a vision with broad goals and objectives for the organization to achieve. Management at all levels is charged with implementing the strategic plan. Metrics must be developed to monitor activities and track progress toward achieving the goals and objectives. But before discussing numbers and types of metrics, it is important to emphasize that the metrics should reflect the strategic vision. Some authors use the word “linkage” to describe the connection between strategic goals and performance metrics. We are on the right path if people two or three levels down from top management in the organization can articulate how their activities support a strategic objective.
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Once the strategic plan is finalized, management must cascade the goals and objectives down through the organization and identify specific tasks with time lines, methods, and responsibilities. This is not a trivial task. Considerable care should be taken to select appropriate measures. Stakeholders and subject matter experts within the organization should be involved in the selection process. Team participation is more likely to result in performance measures that are aligned with strategic objectives. Participation also fosters ownership of the metrics. Some managers go a step further and link the objectives to annual employee perfor- mance evaluation programs or to bonus programs.
For a clear example of how to cascade performance measures, we can look to the field of reliability engineering. When designing a system, we establish reli- ability targets for the system as a whole. When designing the components of the system, we must establish more stringent reliability targets for each component so that the system as a whole continues to meet the overall performance target. This process, called reliability allocation, is a highly technical process that should be performed by someone with expertise in reliability. Unfortunately, management science has not progressed to the same level of discipline as the reliability field. Nevertheless, the basic concepts still apply. When cascading a high- level objective down to operations, we must allocate tasks and apportion the targets to ensure that the organization as a whole will meet the objectives. For more details on reli- ability concepts, see Chapter 3, section E.
Performance measures should be:
• Linked to strategic objectives
• Rigorous, objective, quantifiable, and standardized
• Achievable, realistic, and time- based
• Assigned to appropriate personnel who are held accountable and who are empowered with some level of control to influence outcomes
In general, performance measures should focus on the vital few. Use your judg- ment and avoid using too many metrics, which may dilute the results. Automate data collection and calculations if possible. Spend more time making decisions than generating reports. Select measures that are resistant to problematic behav- ior. In the following subsections we discuss performance metrics as well as two tools for assessing performance: the balanced scorecard and the dashboard.
B.2.c.i. Performance Metrics
Most of the guidelines for performance measures are self-evident, but the recom- mendation to select measures that are resistant to problematic behavior warrants explanation. Suppose an organization faces stiff competition in a commodity mar- ket. Cost reduction is a key strategic initiative. When the objective is cascaded to plant operations, the maintenance department decides to support the objective by postponing costly equipment overhauls. This “problematic” behavior may help in the short run but could cause a catastrophe in the future. How can this be
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avoided? One solution is to use combined metrics. For example, we could create a maintenance productivity metric:
Maintenance productivity = MTTFRΔ
MTTR)Maintenance budget)(( R RΔΔ
In this metric, bigger is better. The symbol ∆R is applied to each variable and refers to the ratio of the variable in period t divided by the variable in period t – 1. This little math trick results in a dimensionless equation that is “normalized” to a value of 1.0 when there is no change in the variable from one period to the next. If the productivity value is greater than 1, performance is improving; if it is less than 1, performance is deteriorating. Since maintenance spending is in the denominator, less spending is encouraged because it will increase the produc- tivity metric. But we can also increase the productivity metric by increasing the equipment mean time to failure (MTTF) or by decreasing the mean time to repair (MTTR) (for more details see Chapter 3, section E). If the maintenance depart- ment starts scrimping on the budget, breakdowns will probably occur more frequently and repair times may increase. Declining performance will offset the benefit of reduced spending in the metric. Thus, this combined metric encour- ages appropriate behavior.
The point of this illustration is not to advocate specifically for a maintenance productivity metric but to suggest that a little creativity can overcome inherent weaknesses in traditional performance measures.
B.2.c.ii. Balanced Scorecard
Robert Kaplan and David Norton introduced the balanced scorecard in 1992. Kaplan and Norton argued that most strategic plans were unbalanced because one stakeholder group—the stockholders—was overemphasized (see the list of stakeholders in section B.2.b). They proposed a “balanced” scorecard with four perspectives:
1. Financial fundamentals
2. Business processes
3. Customer
4. Learning and growth
Financial measures include traditional indicators such as cash flow, sales, and ROI. Business processes include manufacturing measures such as yield and rework, along with support activities such as order processing. Customer measures may include trends in customer satisfaction or average customer service wait times. The learning and growth perspective recognizes the human element in an organi- zation and looks at softer measures such as participation in employee suggestion programs and training.
The balanced scorecard provides a framework to translate the strategic plan into specific tasks that can be managed by frontline employees. In a typical score- card, the objective is listed along with associated measures, targets for perfor- mance, and initiatives that will drive the organization to achieve the objective.
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B.2.c.iii. Dashboard
A dashboard provides a visual, at- a-glance display of key business indicators (see Figure 1.3). Dashboards provide a compact view of the current organizational state. Dashboards may include trend charts and bar charts, and green/yellow/red lights to indicate performance relative to a target. Some dashboards include “drill down” features so that managers can dig into lower- level data. Digital dashboards must be customized for various activities throughout the organization. High- level dashboards are appropriate for executives, but frontline employees need to access low- level data appropriate for their sphere of influence.
The elements in a dashboard should be linked to the strategic objectives. Sales for the example company shown in Figure 1.3 are targeted to grow at 3.75% per year. To avoid revealing confidential information, the dashboard shows only dif- ferences from target. Sales below target are negative. In Figure 1.3, although sales in the recent past have fallen short of the goal, the trend is favorable. Inventory turns (annual sales divided by current inventory) have met or exceeded the target in two of the past three quarters. The milestone review for new product develop- ment shows two tasks behind schedule. The year- to-date (YTD) performance to target chart includes several elements that were selected in the balanced scorecard process. Calculating the ratio between actual performance and the target allows us to combine various metrics on a single chart with a common scale. In this example, management should be concerned that employee suggestions are not being closed
Figure 1.3 XYZ corporation dashboard.
1 –100
–75 –50 –25
0 25 50 75
100
0%
Validation testing
Capability studies
Packaging specifications
Control plan
Process FMEA
Prototype testing
Design reviews
Design FMEAs
Customer requirements
Sourcing decision
20% 40% 60% 80% 100%
8765432
Deviation to quarterly sales target ($ 1000s)
New product development—milestone review
Completion status
Inventory turnover
0
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TargetInventory turns
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promptly and that customer calls are still not being processed fast enough through the call center.
Remember that what you measure will determine to a great extent the activi- ties of your organization. Therefore, carefully select the metrics for your dash- boards, scorecards, and other performance measurement tools. More information on performance measures such as process capability indices is presented in Chap- ter 6, section G.
B.2.d. Project Management
QEs often become involved in project activity, either as a project team member or as a project leader. A number of proven techniques and tools are available to assist in cost- effective project management. The first is proper project selection. In the following subsections, we discuss project tools, project planning techniques, how to monitor and measure project activities, documentation, and strategies for policy deployment.
B.2.d.i. Project Tools
Projects must be prioritized in order to select those having the most merit. Projects should be evaluated for their fit with overall business needs, financial payoff, and potential risks. Exceptions will be made for legal mandates, consumer safety, and customer demands. Only projects that are optional should be prioritized.
Major projects involve risk of loss. Risk assessment involves identifying poten- tial problems that could occur, their impact, and what, if any, actions should be taken to offset them (e.g., taking countermeasures, purchasing risk insurance, or developing contingency plans). For complex projects, it may be prudent to apply a formal risk assessment tool such as a failure modes and effects analysis (FMEA) or simulation. (See Chapter 7, section B.1.d, for more details on FMEA.)
If the benefits of a project are uncertain and multiple outcomes are possible, then a decision tree can help estimate the expected value of gain or loss (see Exam- ple 1.1). A decision tree lists the potential outcomes and assigns a probability to each branch. The financial payout for each outcome is shown at the end of the branch. A few simple rules apply to the creation of a decision tree:
• At each branch point, the probabilities must sum to 1.0
• The expected value for each branch is calculated by multiplying all the probabilities along the branch by the financial payout
• Add the expected payout for all the outcomes within a decision branch
• Choose the decision with the highest payout
There are many other financial methods for justifying projects. Three very com- mon methods are the following:
• Payback period. The number of years it will take to recover the investment from net cash flows.
• Net present value (NPV). NPV takes the time value of money into account. NPV involves finding the present value of each cash flow (yearly) discounted at the cost of capital percentage used by the organization,
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summing the discounted cash flows, and determining if the project is a candidate for approval based on positive NPV (see Example 1.2).
• Internal rate of return (IRR). A discount rate that causes the NPV to equal zero. If the IRR is greater than the minimum required by the organization for typical capital investments, the project is a candidate for acceptance.
The payback period is widely used because it is easy to calculate and simple to under- stand. In the decision tree example (Example 1.1), the payback period for installing a new machine is less than one year, which implies a very high ROI. But a major weak- ness of the payback period is that it does not give any insight into the magnitude of future savings, that is, savings after the initial investment has been recovered.
ExaMpLE 1.1
A QE is considering several options to fix a problem with a production machine. The machine is starting to wear out, so it has excessive variation and approximately 1% of production must be scrapped. He can replace the machine with a prototype machine. There is an 80% chance the new machine will eliminate the variability problem and it will probably increase capacity by 2%. The second choice is to overhaul the machine, with a 60% chance of improving the yield. The third choice is to perform selected repairs. This choice has the lowest initial investment but also is least likely to solve the variability problem. This problem is summarized in the decision tree in Figure 1.4. The probabili- ties associated with the choices are shown in brackets.
Currently, the variation problem generates scrap worth $50,000 per year. A 2% increase in capacity would be worth an additional $100,000 profit per year. There- fore, the financial payout changes depending on whether the scrap is eliminated and the capacity is increased.
Choice
$110,000
New machine
Yes [0.8] Yes [0.85] $150,000
Reduce variation?
Increase capacity?
Financial payout
No [0.15]
Yes [0.85]
No [0.15]
$ 50,000
$100,000
$0
$ 50,000
$0
$0
$ 50,000
No [0.2]
Yes [0.6]
No [0.4]
Yes [0.3]
No [0.7]
$ 35,000
Overhaul
$ 15,000
Selective repair
Figure 1.4 Decision tree for production machine.
Continued
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NPV and IRR rectify this deficiency. Both methods give more accurate infor- mation, provided that suitable estimates of future cash flows can be obtained. The major difference between the two methods is that IRR generates an interest rate that balances all future cash flows against the present outlay, while NPV generates a dollar amount of present and future cash flows. With both calculation methods, bigger is better. Many companies have an internal hurdle rate, such as an IRR greater than 10% or 20%, that projects must achieve to be considered. The com- pany probably could not consistently earn such a high return on stocks or bonds, yet it requires projects to clear this hurdle. One reason for this conservatism is the difficulty of getting accurate estimates of future cash flows.
A final cautionary word about project estimating: sometimes things do not work out as planned. Assumptions may be misleading, probabilities may be opti- mistic, and factors beyond your control may come into play, such as unexpected changes in the market, new legislative policies, or changes in regulatory require- ments. If you enter the calculations in a spreadsheet, it is easy to make adjustments and perform a sensitivity analysis. Sensitivity analysis allows you to evaluate how the projected results would be affected by changes in the estimated inputs (e.g., probabilities of success or potential risks). For example, how much would the NPV change if the probability of success decreased by 10%? For more details and exam- ples see Park (2007).
The expected value (EV) for a decision is given by the equation:
EV = Σx p(x) where x is the financial payout, and p(x) is the associated probability of the outcome. Expected value calculations are discussed in more detail in Chapter 6, section C. We sum all the values within the decision branch. Therefore, the expected value of the new machine is:
EV = (0.8)(0.85)$150,000 + (0.8)(0.15)$50,000 + (0.2)(0.85)$100,000 + (0.2)(0.15)$0 EV = $125,000
Note that the expected value of the new machine is less than the maximum payout because there is a chance the new machine will not work perfectly. We can calculate the expected value for the other options using the same approach.
For the overhaul: EV = (0.6)$50,000 + (0.4)$0 = $30,000 For the repairs: EV = (0.3)$50,000 + (0.7)$0 = $15,000
Finally, we must subtract the initial investment, shown in Figure 1.4, from the expected value to get the net return.
New machine = $125,000 – $110,000 = $15,000 Overhaul machine = $ 30,000 – $ 35,000 = ($ 5,000) Selective repairs = $ 15,000 – $ 15,000 = $ 0
In the first year, we will make money on the new machine, we will break even using repairs, but we will lose money if we select the overhaul. (Note: when evaluating proj- ects, you should always consider the savings in future years, not just the first year.) At the end of the first year, we will gain experience with the option that we implemented. We can update the probability assumptions and repeat the decision tree exercise in sub- sequent years.
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ExaMpLE 1.2
The NPV method converts all future cash flows to today’s dollars at a specified interest rate. It is easy to calculate using a spreadsheet. From the decision tree example above, we enter the initial investment and the expected values of the payouts for year 1, year 2, and so on. In year 3, the warranty expires and we start performing repairs. After year 5, the machine is starting to wear out, and by year 7, we are ready to overhaul or replace the machine. Note: the NPV example shown here can be understood without reference to the decision tree in Example 1.1.
a B C D
1 Cash flow Cash flow New Year machine Overhaul Comments
2 0.10 0.10 Interest rate
3 1 ($110,000) ($35,000) Initial investment
4 2 $125,000 $30,000 First year, expected value
5 3 $125,000 $30,000 Second year
6 4 $110,000 $15,000 Offset savings, paying for repairs
7 5 $110,000 $15,000 $15,000 in repairs
8 6 $105,000 $10,000 Machine is starting to wear out
9 7 $ 98,000 $8,500 Variability increasing, yield decreasing
10 8 $ 62,000 $2,400 Time to replace machine?
To calculate the NPV, use the formula: N
t=1
NPV = Rt
(1 + i)tΣ where Rt is the cash flow in year t, and i is the interest rate.
The interest rate should be the prevailing rate for raising cash in capital markets (that is, a bank loan). Ten percent is typical. The NPV function assumes that the initial investment is made at year 0 and the first payout is at the end of year 1. The results are surprising:
NPV, new machine = $379,136 NPV, overhaul = $ 46,200
The net return for the new machine option in the first year was $15,000. But when you consider the life of the investment, the return is enormous. The overhaul option loses money in the first year but proves to generate positive cash flows in subsequent years. The selective repair option has zero NPV—it is a basic maintenance strategy.
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B.2.d.ii. Project Planning and Estimation
The success of a project is significantly impacted by the effectiveness of project planning. A typical project- planning sequence for a larger project, often identified in the project charter, is as follows:
1. Statement. This first step is where the kernel of an idea or the basic concept visualized is translated into a clear statement of the problem, deficiency, or opportunity to be realized. Careful definition at this point helps later to clarify the scope of the project.
2. Project justification. Risk analyses and assessment (payback period, NPV, IRR, ROI, return on assets, and benefits/cost) are performed, and a go/no-go decision is made.
3. Drafts of mission statement, project scope, and project objectives. These documents clarify the overall direction of the project, what it is to accomplish, the breadth and depth of the project, and the measurable objectives by which progress and completion are to be measured.
4. Stakeholder requirements. Stakeholders consist of two groups: (1) those with a direct commitment to the project team, for example, a process manager who provides a skilled person to serve on a process improvement team working to reduce machine downtime, and (2) those without involvement but who can influence project results, for example, the purchasing department that selects the vendor for a new machine. A macro- level process map may be used to identify areas from which potential team members should be selected.
5. Project team formation. Team members should be selected based on the need to represent a stakeholder group and/or specific skill sets required. Stakeholder groups not represented on the project team should have opportunities to provide input. Some members may be required on an as- needed basis only. Whenever possible, the interests, values, and personality profiles of individuals nominated should be considered. The Myers- Briggs Type Indicator (http://www.myersbriggs.org) can be a useful tool for building a team with complementary interpersonal skills and interests.
6. Finalized mission statement, project scope, project objectives, and project charter. Team members refine the original drafts. A benchmarking study may be appropriate to better define target outcomes. Items that are out of scope of the current project are also identified in order to prevent “scope creep.”
7. Contractual requirements and deliverables. All requirements and outputs of the project are identified, defined, and documented.
8. Work breakdown structure (WBS). Project work is further defined by breaking the work down into a hierarchy of work categories (families of like work clusters) down to the task level. Boxes on a WBS may be annotated with “person/work unit responsible,” “resources required,” “cost estimates,” various other cross- references, and so on. We discuss WBS in more detail below.
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9. Gantt chart. Major project steps or task clusters are listed vertically on a time line chart with each item’s estimated start- to-finish time depicted as a bar across the chosen time intervals (weeks, months, quarters). As the project progresses, the same chart may be used to plot the actual time expended next to the estimated time. Major milestones are shown as points along the time bar. See Figure 1.5 and the discussion below for a more detailed example.
10. Time-dependent task diagram (AND, CPM, PERT charts). Depending on the size, complexity, and duration of the project, it may be necessary to plot the time dependencies of each task to each other task. An activity network diagram (AND) depicts the interrelationships of each task or task cluster in the project. A critical path method (CPM) chart adds the dimension of normal (most likely) time to complete tasks and allows for computing the critical path (longest timeline) through the project. A program evaluation and review technique (PERT) chart adds two additional time estimates for each task (optimistic, pessimistic), allowing further what- if planning. Typically, AND is used for shorter- term, simpler projects, CPM is used where data are available for reasonably accurate time estimates, and PERT is most often used for projects for which there may be no prior precedent. See the discussion below and Figure 1.6 for additional information on CPM charts.
11. Resource requirements matrix (RRM). An RRM delineates the various types of resources needed (e.g., personnel, facilities, equipment, materials, consultants, etc.), quantity, when needed, and cost.
12. Linear responsibility matrix (LRM). An LRM, used for larger projects, defines the interfaces: who has what responsibility for what tasks, and to what degree (e.g., primary, secondary, resource only, need to know).
13. Project budget. A detailed itemized budget is prepared based on the time and cost estimate prepared by the team.
14. Measurements. Define quantifiable measurements to track both project progress and achievement of project objectives. Determine and document the progress- monitoring process, methods for analyzing data gathered, reporting protocols, and checkpoints for initiating corrective action.
15. Approved. Final approval of the project and authorization for implementation project plan is given.
WBS, Gantt charts, and CPM are useful project management tools from a project- planning standpoint. Figure 1.5 shows a three- level WBS under development. A WBS allows determination of the many activities that must occur during the proj- ect. The numbering scheme in Figure 1.5 may seem unduly complex at first, but the consistent use of multiple decimal points allows nesting of levels and facilitates changes to dynamic projects. The project budget details the anticipated expendi- tures over time for each category of expense. Depending on the size of the project, budgets may be prepared for successive levels of the project (usually paralleling the WBS hierarchy).
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Figure 1.6 is a Gantt, or milestone, chart showing the major implementation phases and their relative timing for an ISO 9001 implementation. The Gantt chart is one of the earliest planning tools, dating back to the early years of the twentieth century. Solid bars indicate activities that require an elapsed period of time, while triangles denote events that occur at specific points in time. The figure is fairly simple; computerized Gantt charts created in programs such as Visio can involve multiple layers and interactions of activities.
The CPM chart illustrated in Figure 1.7 is useful for showing every activity in the project. The starting point of each activity depends on the completion of other activities. The sequence that takes the longest total time constitutes the critical path and determines the minimum time to completion of the project.
RRMs are essentially spreadsheets that lay out the requirements over time against the activities in the project. RRMs may be compiled for facilities, equip- ment, materials, contract/consulting services, personnel, and so on. Understand- ing the project life cycle can also help in estimating the resources required. The five
Figure 1.5 Work breakdown structure (partial).
1.0 ISO 9001 Quality management system implementation project 1.1 Quality system documentation 1.1.1 Quality policy and objectives 1.1.2 Quality system manual (QSM) 1.1.3 Quality system procedures (QSP) 1.1.4 Quality system work instructions (WI) 1.2 Training 1.2.1 ISO 9001 briefing 1.2.2 Steering committee meetings 1.2.3 Management representative training 1.2.4 Internal auditor training 1.2.5 Audit behavior training 1.2.6 Statistical process control training 1.3 Quality system implementation 1.3.1 Calibration system 1.3.2 QSPs and WIs 1.3.3 Supplier qualification process 1.3.4 Document control system 1.3.5 Auditing schedule 1.3.6 Customer information system 1.3.7 Corrective/preventive action process 1.4 Controls 1.4.1 Document control 1.4.1.1 QSM, QSP, WI 1.4.1.2 Forms 1.4.1.3 External documents 1.4.2 Audits 1.4.2.1 Internal audits 1.4.2.2 Preassessment 1.4.2.3 Certification assessment 1.4.2.4 Surveillance audits 1.4.3 Corrective/preventive actions 1.4.4 Supplier evaluations 1.4.5 Management reviews
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Figure 1.6 Gantt chart example.
18-month ISO 9001 quality management system implementation project
Task
Select consultant
Conduct ISO 9000 briefing
Conduct gap analysis
Form steering committee
Prepare quality system procedures (QSP)
Prepare quality policy, objectives
Prepare work instructions
Employee kickoff meeting
Evaluate registrars
Train internal auditors
Implement QSPs
Select, schedule registrar
Conduct internal audits
Prepare quality system manual
Conduct audit behavior meeting
Conduct preassessment
Take corrective action
Conduct final assessment
Registration—celebrate
Weeks 66–78Weeks 53–65Weeks 40–52Weeks 27–39Weeks 14–26Weeks 1–13
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stages of a project are (1) concept, (2) planning, (3) design, (4) implementation, and (5) evaluation and closeout.
Follow the old adage: plan your work, work your plan. Work planning requires a clear understanding of the overall goal and objectives, also referred to as outcomes to be achieved (the “what”). The planning process must also take into account how the initiative relates to other projects (e.g., sharing of resources) and therefore often requires input from, or participation by, multiple stakeholders. Fig- ure 1.8 shows an action plan format that can be used to document the plan.
After the plan has been documented, activities can be associated with a sched- ule. Figure 1.9 shows a format for an implementation schedule. A Gantt chart may be added to show the timing of each step in the schedule and would allow for activities to overlap one another.
Periodic work review meetings are held to provide the following:
• An opportunity for the project leader and the sponsor of the project to discuss progress
• A summary of performance (presuming day- to-day feedback was given), evaluation of progress, determination of actions to correct/ improve performance, and renegotiation of such activities as may be necessary
• An assessment documentation relative to the work objectives
• An effective time for the manager to reinforce work done well, assuming the work climate is conducive to frank, open, two- way discussion and problem solving
• An effective time for the sponsor to provide input on the project and assist with elevating to upper management any issues/concerns regarding the project
Figure 1.7 CPM chart example.
1 3 1 1 1
1 4
1
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Critical path: 1, 2, 3, 4, 5, 16, 17, 18, 19, E = 78 weeks
39
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Figure 1.8 Action plan format example. Source: © 2000 R. T. Westcott & Associates (Reprinted with permission of R. T. Westcott & Associates).
Objective/plan title:
Description
Major outcomes desired/required:
Scope (Where will the solution/implementation be applied? What limitations exist?):
By what criteria/measures will completion and success of project be measured?
Assumptions made that may impact project (resources, circumstances outside the project):
Describe the overall approach to be taken:
When should the project be started in order to meet the date needed/wanted?:
Estimate the resources required (time and money):
Outline the tentative major steps to be taken, a projected start and complete date for each step, and the person to be responsible for each step. (Use the back of this sheet to sketch your time line.)
Plan no.:
Date initiated:
Date needed:
Approval:
Team (L):
Team (M):
Team (M):
Team (M):
Team (M):
Action plan
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The specific time to review progress is a matter of preference. Different objectives or projects may be reviewed at different time intervals depending on complexity, time span of work, competency level of performer, criticality of work outcomes, disruptions in due dates, resource shortages, and so on. As a rule of thumb, work reviews should be scheduled at least once a month for objectives spanning more than a three- month period, in addition to project milestones. It is never appropri- ate to wait until just before the planned achievement date to review progress on work objectives. Also, a review should be conducted any time the project deviates from the plan.
B.2.d.iii. Monitoring and Measuring Project Activity and Results
Critical project performance measures include timeliness, budget variance, qual- ity, scope, and resource usage. Project measurements must then be determined and a system for tracking, monitoring, and reporting progress is established.
In medium to large projects, milestones (critical checkpoints) are established in the planning stage and the project is monitored against these milestones. The CPM discussed earlier and illustrated in Figure 1.7 can be built into the quality information system for projects of any size. Thorough periodic project reviews are conducted, including assessment of schedules against the critical path, expendi- tures against budgets, resource utilization against plans, implementation results achieved, a possible reevaluation of risks, and any major issues impacting project continuance. Based on these reviews, the project may be continued as planned, modified, put on hold, or canceled. A similar review is conducted to evaluate the results when the project is completed.
Figure 1.9 Action plan implementation schedule example. Source: © 2000 R. T. Westcott & Associates (Reprinted with permission of R. T. Westcott & Associates).
Step no. Activity/event description
Depends on step
Start date
Finish date
Person responsible
Action plan implementation schedule
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B.2.d.iv. Project Documentation
Documenting the project throughout the process will make it easier to complete the project and potentially any associated paperwork required to close out the project. If the team has not documented every aspect of the project, begin to docu- ment as soon as you can in order to capture details such as the following while they are still available:
• Assumptions, risks, and rationale for selecting the project
• Decisions made to initiate project and approvals
• Detailed plans for design and implementation
• Design and/or implementation changes
• Major obstacles encountered and how they were resolved
• Details of implementation (e.g., measurements established)
• Progress reports and resulting decisions
• Risk log
• Budget information
• Scope changes
• Final evaluation of project results
• Results of post- project audits
All documentation is valuable in planning and estimating new projects and in avoiding previous mistakes. Likewise, the documented knowledge base is a tool for training those new to project management. Documentation can also be use- ful in the development of new policies or employment of old policies to support future and ongoing projects.
B.2.d.v. Policy Deployment
Policies provide direction to guide and determine present and future decisions. They indicate the principles to be followed or what is to be done, but not spe- cifically how it is to occur. For example, a quality policy should summarize the organization’s view on the meaning and importance of quality as it relates to com- petitiveness, customers, suppliers, employees, and continual improvement.
To ensure consistency and understanding throughout the organization, poli- cies need to be integrated with the strategic plan, then deployed through appropri- ate initiatives and performance checks. Projects must be justified and scheduled. Performance must be measured and reported. An organization’s policies should be actionable. Some situations may call for temporary adaptation of the policy to meet unanticipated needs. A documented and deployed quality policy provides the following:
• A written guide to managerial action, lending stability to the organization
• Consideration of quality problems and their ramifications
• A basis for auditing practices against policy
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Deployed policies cascade throughout the organization, directly impacting each functional area and indirectly affecting events, activities, and outcomes depending on those functions. If policies do not have this effect, they are not fulfilling their purpose. Each function and person impacted by the organization’s policy must align their objectives and procedures to support the policy.
B.3. Quality information System
A quality information system (QIS) is a collection of data, rules, and equipment that creates information about quality in a systematic way. A QIS will collect, store, analyze, and manage quality- related data from customers, suppliers, and internal processes. It will generate information in the form of printed reports, screen dis- plays, and signals sent to mechanical devices. Depending on the degree of auto- mation, it may give answers to questions posed by humans, or it may have built- in action rules. Above all, if it is well done it will enhance profit and productivity. In some industries, a QIS is required by law; for example, the Federal Drug Adminis- tration requires pharmaceutical companies to maintain a QIS.
The first requirement in studying QISs is to understand what, exactly, a “sys- tem” is. The word is used in many different contexts. For example, this book dis- cusses management systems, information systems, strategic planning systems, and quality systems, for starters. The essence of a system is this: it ties a number of components together that act in common with each other. Systems that QEs are interested in are dynamic and goal oriented. They have inputs, outputs, operating rules (procedures or transformational processes), data storage, and boundaries. They are designed by people to achieve specified goals. Computerized informa- tion systems are explicitly designed, usually by cross- functional teams.
A QIS is both a quality system and an information system. It is naïve to speak of the QIS, because an effective organization will have numerous quality systems, which may be manual, computerized, or a hybrid of the two, with both manual and computer elements. A well- designed information system allows information generated at one level or in one part of the organization to be used for many dif- ferent purposes.
QISs may be used to:
• Initiate action (e.g., generating a shop order from a customer’s order)
• Control a process (e.g., controlling the operation of a laser cutting machine within given specification limits)
• Monitor a process (e.g., real- time production machine interface with control charting)
• Record critical data (e.g., measurement tool calibration)
• Create and deploy operating procedures (e.g., an ISO 9001–based quality management system)
• Manage a knowledge base (e.g., capturing, storing, and retrieving needed knowledge)
• Schedule resource usage (e.g., personnel assignments)
• Archive data (e.g., customer order fulfillment)
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• Store quality processes and procedures
• Document required training
The importance of information systems becomes apparent when looking at their impact on various aspects of quality management. Both process management and problem solving require accurate and timely information. Contrast the following two cases. One information system in a plant might be hardwired into manufac- turing and testing equipment, with monitors displaying real- time information complete with alarms and action signals; it could have options for graphic display of statistical and trend analysis for quick intervention. Another system in the same plant could tie executives, project teams, and off- site employees together through an intranet; organizational objectives and milestones appropriate for each level and function could be displayed as both text and graphics, along with actual per- formance and gaps. These two QISs are quite different.
Good information systems are critical to cross- functional collaboration, since access to distributed information is required in order for groups and employees to make quicker and better decisions. For example, some projects can be carried out largely through computerized discussions and transmission of documents. Often this enables highly skilled team members to participate regardless of their physi- cal location and can also reduce the amount of time required for the project.
The modern QE must be competent in the selection, application, and use of hardware and software technology appropriate to the tasks and responsibilities assigned. Consideration should be given not only to the functionality of the sys- tem for the task, but also issues such as required user skills, compatibility with other systems, and information security. Furthermore, if the quality system is of any magnitude, the QE must understand project and data management techniques and must be a good team member.
Industry 4.0 refers to the fourth industrial revolution of manufacturing and partially focuses on data management and analysis in a system. This revolution is characterized by four trends: big data, advanced analytics, human- machine inter- faces, and digital- to-physical transfer (3D printing) (Baur and Wee 2015). Some associate Industry 4.0 with some of its large- scale data exchange capability. Organi- zations, particularly those in the manufacturing technology sector, make use of the data exchange system (see B. Lydon, “Industry 4.0—Only One- Tenth of Germany’s High- Tech Strategy,” April 4, 2014, http://www.automation.com/automation- news/article/industry-40-only-one-tenth-of-germanys-high-tech-strategy). Refer to Brettel et al. (2014) for additional information on Industry 4.0.
B.3.a. PLC and SCADA Systems
The widespread use of microcomputers and programmable logic controllers (PLCs) has transformed the factory floor. There is a growing trend toward distrib- uted measurement and control, where PLCs have built- in programs and logic to control machines and processes. Fewer and fewer technicians are turning dials or opening and closing valves to control processes. These tasks are now controlled by PLCs; however, many PLCs do not have a human interface such as a monitor or keyboard. The PLCs are widely distributed throughout the plant, making manual data collection time- consuming and cumbersome. Furthermore, PLC language is not user- friendly. These drawbacks have given rise to large- scale supervisory
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control and data acquisition (SCADA) systems (e.g., Systems, Applications, and Products [SAP], a system used in many industries). Other automatic data extrac- tion and data exchange programs have been built for industry- specific needs.
The SCADA system interfaces with all of the PLCs through a network. The SCADA system periodically polls the PLC memory registers to collect data. The system includes a human interface, usually in a central location such as a con- trol room, to monitor the processes, generate alarms, and allow the operator to intervene or issue an override as necessary. The SCADA system typically includes real- time trend charts and graphic displays of the current status of the equipment. The system also provides for data storage in a database program, which allows for rapid retrieval of data for subsequent analysis and reporting.
What is the role of a QE in the creation of a large- scale SCADA system? The information system should be viewed as no different from a manufacturing sys- tem. The QE should be involved from the earliest planning stages to ensure that user and system requirements are thoroughly documented. It may be appropriate and beneficial to apply some of the advanced quality planning disciplines dis- cussed in Chapter 3, section B, even though the “product” is a software system. For example, customer requirements should be fully understood, even if the “cus- tomer” is an hourly employee who will use the system to monitor and adjust the process. The QE should participate in creating the user requirements; after all, the QE is typically considered the local expert in data analysis and reporting. We dis- cuss the required reports and methods to display and summarize the data in the next subsection.
B.3.b. QIS Tools
Although there are many ways to design information systems, it is clear that the larger they get, the more fraught with risk of failure they become. The QE can render a real service to the employer by studying strategy and tactics of systems development. Two useful tools are the information systems strategy matrix and the V model. The need for a strategy was emphasized by Pearlson and Saun- ders (2004), who produced an information systems strategy matrix, as shown in Figure 1.10.
Figure 1.10 Information systems strategy matrix.
What
Hardware List of physical components of the system
Individuals who use it, individuals who manage it
Physical location
Software List of programs, applications, and utilities
Individuals who use it, individuals who manage it
What hardware it resides on and where that hardware is located
Networking Diagram of how hardware and software components are connected
Individuals who use it, individuals who manage it, and the company from which service is obtained
Where the nodes are located, where the wires and other transport media are located
Data Bits of information stored in the system
Individuals who own it, individuals who manage it
Where the information resides
Who Where
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The matrix shown in Figure 1.10 displays the following four categories: hard- ware, software, networking, and data. Other categorizations could also be used. This basic example demonstrates the kinds of analysis required. Another tool to consider is the V model. The V model starts on the left side at the top of the V (see Figure 1.11) with high- level user requirements and cascades down through func- tional specifications and detailed design requirements. On the right side of the V, test protocols are developed, executed, and documented to verify that the design specifications have been met. The QE should be involved in this process to ensure quality and data integrity during the execution of the project.
Tasks that seem trivial, such as naming conventions, can have a huge impact down the road. Large, real- time control systems may have hundreds of PLCs and thousands of sensors. Imagine the complexity of creating a downtime report for the packaging area of the plant. Every machine and sensor in the area must be included in the database query. A good naming convention will allow a group of variables to be captured within a single query statement that includes a “wild card.” If a nam- ing standard is not used or is poorly executed, then the user has no choice but to individually specify each sensor and PLC when the database query is created.
Similar care and consideration should be given when creating the test pro- tocols. How much data should be collected? How often will the samples be col- lected? If the sampling duration is too short, or the elapsed time between samples is too long, then it may not be possible to detect variation that is directly caused by the PLC control system. Is there a difference between the process target and the actual steady- state process average? What about including process upsets in the test protocol? Does the controller overshoot the target during initial recovery?
Further ideas to improve the success of system development projects are reported by Long and Gryna (1999), who drew the following conclusions:
• Carefully define the scope of the QIS and what it is expected to accomplish. From the very beginning emphasize operational benefits, not technical specifications. It may be wise to develop a pilot project that can be used to show what really does work and what does not. Getting some benefits in a short period of time builds confidence not only in the system itself, but also in the competence of the system developers.
Figure 1.11 The V model for software development.
Verify
User requirements specification
Functional specifications
System test protocol and user acceptance
Hardware test protocol
Detailed design specifications
Software module test protocol
System development
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• Be sure that the goal of the QIS supports the goal of the business. (This point was discussed earlier in this chapter when we discussed strategic planning.) Once the goal is set, use well- proven project management techniques.
• Get advance agreement on who will do what and when. Get buy- in to clearly understood milestones. Do not simply delegate the project to the information technology (IT) folks; keep QEs and sponsors fully engaged in the development.
• Concentrate on user expectations and how they are being realized. Focus attention on the overall performance of the system rather than specific metrics. Ongoing discussion and comparison between the users and the developers is an important key to success.
• Publish regular progress reports and keep the language in user terms. A common trap in large- scale information systems projects is to get bogged down in technical metrics and jargon; the user may cross their fingers and hope for the best without really understanding what is going on. A corollary of this is to be sure that the end user has the technical competency to understand what is being said. Reports cannot be watered down simply to avoid confusing the uneducated.
Repeatedly stress the anticipated benefits that were specified at the outset and do not abandon original goals under pressure. The exception to this guideline occurs if it becomes evident that the original specifications cannot be met. The top- level sponsors must then be fully briefed and participate in the revised benefit state- ment. This action should be viewed as a last resort and is in a sense a salvage operation.
Productivity improvement is perhaps the most frequently cited reason for investing in an information system. The investment can be considerable because the infrastructure requires hardware, networks, sensors, customized software, and information systems support personnel. Estimating the payback can be a challenge. The payback estimates may include optimistic forecasts and tenuous assumptions. Some people focus on the human benefits such as automating periodic reports. Relief from mundane tasks will free up personnel to pursue other important tasks. But much larger gains usually can be achieved by using the information system to improve production processes. A well- designed information system can identify opportunities that probably would be missed by even the most conscientious and determined analyst using a manual or paper- based data system. At many facili- ties, a 1% gain in production yield is a realistic assumption and will generate a much larger return than a few hours saved per month generating manual reports. Electronic QISs can be used effectively for traceability and ensuring quality of pro- duction. We discuss traceability in more detail in Chapter 4, section B.
B.3.c. QIS Example
To illustrate the tremendous value of a QIS, consider this case study regarding a highly automated packaging plant in Texas. Equipment breakdowns plagued the facility for the first year of production. Downtime was so excessive that the plant was operating below the break- even point. Management decided to make a major investment in a new information system. Over the course of the next year, nearly
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every machine in the facility was linked through a network to a database. Sensors were added to monitor key production processes. Automatic feedback systems were installed and gradually tuned to achieve stability in the most complex pro- cesses. Customized reports were created to distill vast amounts of data into usable information. The reports summarized and prioritized the current status so that management could quickly allocate resources where they were most needed. One such report is shown in Figure 1.12. The report executes automatically at the end of each production shift. It analyzes data from nearly 700 machines, identifies the top three concerns in each functional area, and prints a one- page summary.
The quality department and the maintenance department worked together to develop the format. The general manager participated in establishing the equip- ment performance standards needed to support the balanced scorecard objectives. If performance does not meet the objectives, then the report highlights the total with a large, bold font. Management and maintenance employees can quickly identify concerns and focus their process improvement efforts accordingly.
Figure 1.12 Current status report example.
Equipment Exception Report 7/2/16 3:00 PM to 7/2/16 10:00 PM
Concern
Critical / misc machine alarms 4
No. 2 compressor, low oil pressure 3
No. 2 compressor, oil temperature 1
Cooling hood jams 23
Shop 1: 19
This shift
Shop 3: 2
Shop 2: 2
14
Prior shift
7
3
Inspection conveyor jams 34
Shop 3, loop A 23
Shop 2, loop C 4
Shop 3, loop A 3
Check detector
Leak test
Carton forming faults 12
CF 2 11
CF 2 1
Case not at madrel
In flight jam
Downtime summary (minutes) 285
Shop 3 146
Shop 1 83
Total downtime
Total downtime
Shop 2 56Total downtime
Leak test
Number of faults Concern
Annealing oven faults 1
Shop 1, zone 5, high temperature 1
Coating sprayer alarms 260
Shop 1: 157
Shop 3: 89
Shop 2: 14
No spray alarm
No spray alarm
No spray alarm
Discharge conveyor jams 11
Shop 2, loop A 6
Shop 3, loop A 2
Shop 3, loop A 1
Check detector
Leak test
Carton packer faults 18
Shop 1, north packer 13
Shop 1, north packer 2
Missing jars
Elevator jam
Shop 2, west packer 1No glue
Throughput % of budget 0
94.6%
98.1%
Shop 3
Shop 2
98.4%Shop 1
Scanner
Number of faults
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The team designing this system took several months and gave a great deal of thought to balancing the automatic collection and processing of data with the human interpretation of information. It would have been easier to design a com- pletely closed- loop control system, but this would have precluded human inter- vention and thoughtful study of what the processes were saying. At the same time, the data on which the daily and weekly reports were based were massive and it was essential that the data be condensed and summarized before being presented to management.
The plant achieved a dramatic improvement in throughput in less than six months after implementing this QIS. The report shown in Figure 1.12 (and others like it) helped drive a transformation in quality and productivity. By the end of the second year, the plant achieved best- in-class quality and its profit margin was over 10%, exceeding the original performance target.
This example is only one type of the tremendous number and variety of QISs now being implemented. Bar codes, voice entry, optical character recognition, and local area and wide area networks are among the host of new technologies available for cost- effective automation of quality systems. Other technologies include knowledge management, audiovisual presentations, individual learn- ing programs, decision support systems, computerized conferencing, systems modeling, automated online reference services, and so on. QEs should carefully study computerized information systems techniques and possibilities. QIS is an area that will continue to revolutionize all aspects of life, both organizational and personal.
C. aSQ CodE oF ETHiCS For ProFESSionaL ConduCT
C.1. Code of Ethics
All professions are bound by specific codes of ethics, and one mark of any profes- sion is publishing and upholding standards of conduct. ASQ has adopted the code of ethics shown in Figure 1.13.
QEs must be aware of legal issues, such as equal employment opportunity (EEO) laws and other guidelines. Another example of how the legal system impinges on QEs is the Sarbanes- Oxley legislation. Because of several instances of large- scale corporate fraud at the turn of the last century, the US Congress passed this law, sometimes called Sarbox, which mandates a number of stringent requirements for corporate financial reporting that can be understood as quality assurance techniques applied to the corporate financial system. Sarbox actually enhances the role of quality engineering because it carries the same concept from the quality arena to the financial arena.
Whether your work is governed by EEO laws, Sarbox, or other relevant stat- utes, the point to remember is that your behavior must at all times be such that no embarrassment comes to the supplier, your employer (subordinates, peers, or management), the customer, or yourself. You must be polite and diplomatic and show respect to all persons. In the final analysis, you must be honest with your- self that you have acted fairly and legally, and you should have a good feeling in your gut about the things you have been involved with, including resolving ethi- cal dilemmas.
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C.2. Ethical dilemmas
Ethical dilemmas arise often when dealing with technology, in federal and state legislation, and in management activities. How technology is applied and the con- sequences of the application often call for ethical decisions. Some have equated the definition of quality and ethics with “do the right thing.”
Another area of concern is managing personally identifiable information (PII), which includes phone numbers, addresses, social security numbers, credit card numbers, and so on. There should be industry and organization policies along with government regulations to ensure this type of information is protected from abuse. These policies should include the need to encrypt, safeguard, and remove these data from computer databases as required. For example, in the medical industry
Figure 1.13 ASQ Code of Ethics.
Code of Ethics
Fundamental Principles
ASQ requires its members and certification holders to conduct themselves ethically by:
I. Being honest and impartial in serving the public, their employers, customers, and clients
II. Striving to increase the competence and prestige of the quality profession, and
III. Using their knowledge and skill for the enhancement of human welfare
Members and certification holders are required to observe the tenets set forth below:
Relations with the Public
Article 1—Hold paramount the safety, health, and welfare of the public in the performance of their professional duties.
Relations with Employers and Clients
Article 2—Perform services only in their areas of competence.
Article 3—Continue their professional development throughout their careers and provide opportunities for the professional and ethical development of others.
Article 4—Act in a professional manner in dealings with ASQ staff and each employer, customer, or client.
Article 5—Act as faithful agents or trustees and avoid conflict of interest and the appearance of conflicts of interest.
Relations with Peers
Article 6—Build their professional reputation on the merit of their services and not compete unfairly with others.
Article 7—Assure that credit for the work of others is given to those to whom it is due.
ASQ’s code of ethics will help you decide how to treat your subordinates, peers, and managers, but numerous laws, as well as company policies, are applicable. Knowledge of these may be mandatory. For example, if you are interviewing someone for a position, the law requires you to follow certain rules for asking questions. Likewise, your company may have internal rules for dealing with peers, subordinates, and suppliers.
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these policies are known as the Health Insurance Portability and Accountability Act (HIPAA), and in academia, the Family Educational Rights and Privacy Act (FERPA).
A case in point is the ongoing need for guidelines governing ethical behavior in the application of computers, e-commerce, e-business, and other new technolo- gies. Some of the issues demanding critical attention are the following:
1. Misusing employers’ computers for personal gain or pleasure
2. Destroying others’ property (e.g., injecting a virus or wiping out files)
3. Using or condoning the use of computers for fraudulent activities
4. Violating individual and company rights to privacy and confidentiality
5. Omitting safeguards that protect users
6. Infringing on copyrights and trademarks
7. Failing to maintain a sufficient level of accuracy and completeness implied when data are collected and stored in computer databases
8. Failing to make critical information known to appropriate decision makers in time to prevent a negative outcome
9. Failing to capture, manage, and make available critical knowledge to those who need it
10. Failing to upgrade computer technology
11. Managing retrieval of data files from old or different software programs/versions
12. Dealing with global employees, businesses, and markets
13. Dealing with legal requirements (including safety and environmental regulations) of different governmental groups across geographic boundaries
14. Ensuring the usage quality of the new technology itself, and ensuring that people are trained to use the new technology
Another area of concern to the engineer is the Occupational Safety and Health Administration (OSHA). Both federal- level agencies and state- level agencies mon- itor organizations to ensure compliance with the respective rules and regulations. Some of the more common sets of rules and regulations are the following:
OSHA, Labor (Randall’s Practical Guide to ISO 9000 provides a more comprehensive list of regulations)
29 CFR 1910.95, Occupational Noise Exposure (Ear Protection)
29 CFR 1910.120, Hazardous Waste Operations and Emergency Response
29 CFR 1910.132, Personnel Protective Equipment
29 CFR 1910.133, Eye and Face Protection
29 CFR 1910.147, The Control of Hazardous Energy (Lockout/Tagout)
29 CFR 1910.1200, Hazard Communication
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Engineers also find themselves involved with ethical issues usually handled by management, such as interviewing potential new employees for the organization. Without the proper training, engineers could put themselves and their employers at great risk for lawsuits by asking inappropriate questions. Engineers conducting interviews should keep the following guidelines in mind:
• Ask only job- related questions
• Do not ask about age, race, national origin, marital status, or religion
• Focus on the competencies and skills for the job in question
• Avoid any small talk that is not related to the job
The ASQ Code of Ethics emphasizes that we are professionals and must act accord- ingly. Federal law and employer rules create additional requirements for compli- ance. You must understand all of the above and more as it is presented to you.
d. LEadErSHiP PrinCiPLES and TECHniQuES Leadership is an essential part of any quality initiative. Robbins and Judge (2012) define leadership as “the ability to influence a group toward the achievement of goals.” While there are many different leadership approaches involving personal- ity and situation, the literature has indicated several theories of leadership that may help in understanding these approaches. Robbins and Judge (2012) detail two main theories of leadership:
• Trait theory of leadership: personality and social, physical, and intellectual traits may play a role in differentiating leaders from nonleaders
• Behavioral theory of leadership: specific behaviors may differentiate some leaders from nonleaders
We point the reader to the text by Robbins and Judge (2012) for further details on leadership theory and traits and discuss some important principles for leaders in the quality field to keep in mind.
The leader’s role is to establish and communicate a vision and to provide the tools, knowledge, and motivation necessary for those individuals or teams that will collaborate to bring the vision to life. This can apply to an entire organization as well as to a specific department or work group. For example, the leader of the quality engineering function is responsible for helping shape the policies for the quality technologies that will be deployed throughout the organization and for ensuring that department personnel are sufficiently qualified to support the use of the technologies.
A leader may or may not hold an officially designated leadership position. Often, someone in a work group will emerge as a leader because of their knowl- edge, skills, experience, and/or abilities. Further, teams often include facilitators, another leadership role. The facilitator’s purpose is to provide support to the team’s effort while at the same time allowing the team to maintain ownership of its decisions.
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A good leader always tries to understand where the other person is coming from, what makes them act the way they do—in other words, understanding what motivates others. Good leaders recognize and apply Maslow’s hierarchy of needs. This principle asserts that people are driven by their needs and wants and that all human needs can be roughly placed in a hierarchy. Higher- level needs are not rel- evant until lower- level needs are satisfied; however, once a need is met, it no lon- ger motivates behavior. The five levels are (1) physiological (hunger, thirst, sleep), (2) safety and security (protection from the elements and predators), (3) socializa- tion, (4) ego, and (5) self- actualization. Many people never get their ego needs fully satisfied and thus do not experience self- actualization needs, but all the great thinkers and leaders of the ages are in fact self- actualized. When trying to lead less cooperative followers, it often helps to think about what need level they are working on.
Leadership of the quality engineering function involves defining and carrying out projects that support the organization’s strategic plan, as well as providing the resources for and overseeing day- to-day quality engineering activities. While some of these activities may be performed by an individual, in today’s complex environment they are often conducted in a team setting. Examples include work- ing with an advanced quality planning team to analyze repeatability and repro- ducibility (R&R) of a new measurement system, and working with a software engineer to implement a new automated SPC online package.
d.1. developing, Building, and organizing Teams
Since around 1980, quality concepts and team concepts have moved in tandem through the economy. Teamwork is now vital in government, space exploration, healthcare, education, and most profit- oriented businesses. Compared to a genera- tion or two ago, control of a project or process has shifted from a single individual to the team level.
D.1.a. The Need for Teams
The drive for excellence includes better deployment of people at all levels. Workers at all levels now expect to have some say in designing and implementing change, and only through change can quality improve. Managing an organization through teams has become recognized as a core component of business.
There are many types and purposes of teams, each requiring different struc- tures, skills, resources, and support. Leaders of an organization must therefore be clear about what they are trying to accomplish to ensure that the appropriate team processes are utilized for their situation.
A team- based environment might be initiated as part of the strategic plan or as a response to a specific problem encountered by the organization. Regardless of the reason, there should be a process for planning and carrying out the team- based initiative. This process is often done through a steering committee that focuses on driving business improvement. A member of management—called the sponsor— also typically is identified and takes responsibility for initiating and guiding a team. The sponsor usually is the individual with ownership of the process or area where the team’s actions are focused.
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D.1.b. Types of Teams
Three major types of teams are widely used:
• Process improvement team. These are temporary teams whose mission is to develop a new process or improve an existing process. These teams are often cross- functional, consisting of representatives from multiple departments involved in the process under study. The management sponsor typically selects the team leader and will negotiate with other area managers to identify other team members appropriate for the project mission. Figure 1.14 shows how teams should be integrated within the organizational hierarchy.
• Work group. These teams consist of the personnel who work in a particular department or process area. Their mission is the ongoing monitoring and improvement of process performance, and they typically meet on a regular basis (e.g., weekly) to review indicators and identify any actions required. The work group leader is usually the individual with supervisory responsibility for the process area. The team also may initiate a process improvement team, especially when the improvement requires interfacing with other departments that are suppliers or customers of the work group. Organizations committed to applying work group–based improvement from top to bottom can use an interlocking team structure that includes all members of the organization.
Figure 1.14 Linking team structure.
Department A leader
Department B leader
Four interlocking teams, each at a
different level of the organization
Unit C4 leader
Unit C3 leader
Unit C2 leader
Unit C1 leader
Process 2 leader
Process 3 leader
Process 1 leader
Associate Associate Associate Associate Associate
Facility leader
Department C leader
Department D leader
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• Self-directed work team (SDWT). An SDWT is a group of individuals who have broader and deeper day- to-day responsibility for management and improvement of their process area. SDWT members are highly trained in subjects such as quality, safety, maintenance, and scheduling, and in some cases also carry out human resource functions. These teams are highly empowered to make their own decisions, although of course there are still limits, such as spending authority.
Whether and to what extent an organization utilizes teams is usually dependent on factors such as the rate of change in its industry, the culture of the organization, the predominant management style, employee education levels, and where the company’s product or service is in the maturity cycle.
Some teams are less formally structured, such as an ad hoc group organized to address a customer complaint or a virtual team that wants to compare the process used for design reviews by several different facilities. Regardless, many of the fol- lowing considerations will influence the success of the team and the satisfaction of its members.
D.1.c. Selecting Team Members
The primary determination of who will participate in a team effort is whether the person is involved in the process to be improved. However, when selecting team members, other issues often are considered. For example, a process improve- ment team might not be very effective if all team members have the same per- sonality style (e.g., as measured by a personality evaluation instrument such as the Myers- Briggs Type Indicator/MBTI; see http://www.myersbriggs.org). Some teams intentionally include someone from outside the process area who can pro- vide a more objective or different viewpoint. Supplier or customer personnel often are invited to participate when their input is deemed especially valuable.
Selection of team members for organizational management and improvement is vital, just as it is for a sports team. The many different activities to be carried out call for certain roles and responsibilities, which then require a certain set of skills and/or mind- set. For example, a team needs to analyze process data, minimize disruptive conflict, monitor meeting time effectiveness, and maintain records of activities. Specific roles, such as a timekeeper and a scribe, are usually defined for individuals who will carry out the latter two of these responsibilities.
D.1.d. Support Mechanisms Required for Team Success
Team-based improvement requires more than creating teams; it requires provid- ing them with adequate support to ensure success. Examples of support include:
• Equipment. Face- to-face teams need meeting space, equipment (such as tables, chairs, projectors, and whiteboards), and access to computer hardware and software in order to document meeting minutes, analyze process data, and prepare presentation materials. Virtual teams can utilize software such as BlueJeans, Google Hangouts, MyMeetings, Sharepoint, and WebEx.
• Training. Unless an organization is extremely lucky, most employees who become involved in teams will not have all of the necessary skills. Such skills may include how to plan and effectively manage meetings,
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how to analyze processes and data, and how to make group decisions based on consensus. The organization must therefore determine the specific skills required and the current skill levels of employees, and provide opportunities to close the gap.
• Management sponsor. The sponsor role is a vital leadership function that goes beyond simply launching a team. It also includes staying in contact with the team leader to ensure sufficient progress and resolving any conflicting issues with other parts of the organization. The sponsor typically has authority to cross organizational boundaries that team members would need to negotiate and can therefore resolve some types of issues quicker. The sponsor is also ultimately responsible for effective implementation of the team’s recommendations.
• Systems change. Setting up a new team in an organization that is not adequately designed for this way of working is a prescription for failure. An organization is a system, meaning that if one part is changed, other parts will be affected. If the primary management style is autocratic and people are rewarded for competition versus cooperation, teams are unlikely to be an effective mechanism. Before beginning the team process, leadership must consider what other systemic changes will be necessary to align the various parts of the organization. How team success will be recognized and rewarded is an especially vital component.
D.1.e. Team Development
Each new team is a new mini- organization. The team will therefore progress (and often regress) through the traditional stages of group development (Tuckman 1965), which are described briefly here:
• Stage 1: Forming. When team members first begin to meet, each member brings their individual identity and the perspective of their own environment (e.g., functional process area). Even for members who have participated in other teams, each team is a unique experience and individuals often approach it cautiously, uncertain of how they will perform in the new situation. During the forming stage, a team usually clarifies its mission, specifies roles that need to be carried out and who is to perform them, and defines rules of acceptable behavior, often called norms.
• Stage 2: Storming. During this phase, team members realize the size of the task before them. They still think primarily as individuals and often attempt to shape decisions to their own advantage rather than considering the impact on other team members. Arguments, testing the leader’s authority, and attempts to change the team’s mission are typical behaviors during the storming stage.
• Stage 3: Norming. In this phase, the individuals begin to shift their focus from personal concerns to that of helping the team meet the challenge at hand. Interpersonal conflicts and the tug of external loyalties have less of an impact as team members realize their interdependence. They are more willing to discuss differences of opinion in order to understand them and how they might impact team success.
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• Stage 4: Performing. At this stage, the team has matured to the point where it is working as a smooth cohesive unit. Team members have a good understanding of each other’s strengths and weaknesses and how they support the mission and are now able to work through group conflict. There is a greater appreciation of the importance of the team’s processes and members are more satisfied with being part of the team. During this phase, the team typically makes significant progress toward achieving its goals.
Although these stages indicate a logical sequence that occurs over time, actual progress by a particular team will vary greatly. For example, a team that has pro- gressed to stage 3 or 4 may fall back to stage 1 or 2 if team members find that some previous assumptions about one another are not true or if team membership changes as a result of a job transfer. Some teams may not progress beyond the earlier stages due to a short project duration or if they are unable to successfully resolve group dynamics issues.
A fifth stage, adjourning, was added to the original small- group development process in 1977 (Tuckman and Jensen 1977). This final stage represents team dis- bandment. While team members often feel happy due to the success of the team’s accomplishments, there can also be a sense of sadness due to the “death of the team.”
Team development can be enhanced by making sure that team members have a basic understanding of how to (1) interact in positive ways, (2) deal with dif- ficult people or situations, (3) contribute to accomplishing the team’s goals, and (4) give or receive constructive feedback. A facilitator can help ensure that the team is aware of its progress by commenting during meetings. Special interven- tions are also sometimes useful. Examples include simulations or outdoor adven- tures that allow the group members to become more familiar with one another’s styles, strengths, and weaknesses, and to become more effective at working with and through their differences.
d.2. Leading Quality initiatives
A QE is frequently called on to lead particular quality initiatives. Such projects might involve improving an existing product or service, working to resolve supplier per- formance issues, addressing product field performance failures, implementing new measurement technology, or obtaining ISO 9001 quality system registration.
The following list offers several recommendations for leadership of such ini- tiatives. Most are appropriate whether or not the project is a team- based initiative, because, by definition, most initiatives will influence others in the organization (and/or the supply chain), and the roles of others should therefore be taken into account throughout the project.
• Ensure that the project mission is clear, including expected results, timing, limitations, and reporting structure and methods. Obtain supporting data used to indicate the value of the project and determine how the project is related to the bigger picture (e.g., strategic plan, other projects, and/or day- to-day operations).
• Determine who the other players in the project will be and make contact with them individually. Learn of their interest in and commitment to the project.
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• Define the technical process and the time schedule to be used to carry out the initiative. For example, a problem- solving project might use a seven- step problem- solving process, while a Six Sigma project might use the DMAIC process (see Chapter 5, section C.5.c).
• Execute the project according to the process defined in the previous step, involving others as appropriate and keeping management informed.
• Evaluate outcomes of the project against the original mission. Ensure that all people involved receive appropriate recognition for their contributions.
Most of these steps are basic to effective project management. However, a sig- nificant portion of the impact of such initiatives also will be related to the qual- ity of leadership demonstrated throughout the project. Following are some useful guidelines:
• Ensure that all involved understand the mission, the goals, and the project objectives and how the team fits with the bigger picture.
• Understand that all people and organizations involved will have their own priorities, perspectives, and skills. Learn what they are, recognize the validity of the differences, and find ways to integrate them effectively.
• Be aware of your own strengths and weaknesses and how they can affect project success. Find ways to learn from and utilize the skills of others to compensate. Also, provide as many opportunities as possible for other project personnel to utilize their full capability and to develop new skills.
• Understand all of the team members’ strengths, weaknesses, and personalities to successfully lead the team.
• Communicate, communicate, communicate. People tend to fill gaps in their understanding with their own bias or fears, so keep the gaps to a minimum.
• Be a role model by emphasizing and demonstrating the importance of high- quality work.
Additionally, a QE will frequently be called on for technical advice regarding particular methods for process analysis, such as conducting a process FMEA (see Chapter 7, section B). Although the QE may not be in a leadership role, they must still understand these principles. For more information, see Snee and Hoerl (2012) and the many references within their article on leadership, leadership skills, and statistical leadership.
E. FaCiLiTaTion PrinCiPLES and TECHniQuES Concurrent with the development of teams was the emergence of the facilitator as a key organizational player. Whereas the old- fashioned boss would simply tell workers what was to be done, the facilitator must understand the objectives and constraints of the team, as well as its needs.
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In an ideal world there would be no need for facilitators. Everyone would have the skills necessary for their roles and would work effectively with everyone else. However, it is not an ideal world, since all of us are continually learning. The role of facilitator is therefore a valuable one since it allows special additional skills to be readily available to the team.
E.1. Facilitator roles and responsibilities
A facilitator’s primary mission is to ensure that a team is successful, but this must be done in a way that ensures that the team, not the facilitator, is responsible for the outcome. A successful facilitator is one that is continually working himself or herself out of the role by helping the team develop higher and higher levels of competency.
The facilitator is termed a marginal role, since facilitators are not actually members of the team with which they are working. However, facilitators are usually present at most or all of the team’s meetings, and their role is to provide support that helps the team work better. Simple examples of this support include notifying the team when it has veered from the meeting agenda, jumped to a con- clusion without any supporting data, or not allowed all team members to voice their opinions.
Facilitators usually take one of two types of roles within a team. One is that of meeting manager, where the facilitator guides the team through the agenda. The other role is that of observer, where the facilitator sits quietly to the side and simply comments when it seems necessary or useful to further team progress. The observer role also provides the opportunity to gain information that can be used to coach the team leader in team process skills.
An important distinction, though, is that facilitators do not discuss content issues, only process issues. For example, if a team were trying to reduce the amount of time patients spend in the waiting room of a healthcare clinic, the facilitator would not interject comments such as, “Should we change the patient scheduling process?” since it is relative to technical content of the subject matter. However, at the appropriate time the facilitator might ask, “What are some additional ways that the time could be reduced?” since it only involves ensuring that the team has taken a broad view of potential opportunities.
It is not necessary that facilitators be someone from outside the team. The team leader or a specific team member who has sufficient skills and experience may also take on the role of facilitator. In this case, the facilitator is allowed to contribute content, because the person is in fact a bona fide member of the team. The ultimate objective, of course, is for all teams to be fully capable of working without the need for anyone in a designated facilitation role. Each member simply pays attention to both content and process issues and ensures that the team works effectively.
Because a facilitator tries to help the team be more effective, there is a wide range of issues to consider. Here is a list of just a few of the items that facilitators must pay attention to:
• Meeting agenda. Is there an agenda for each meeting, and does the team follow it?
• Communication. Do team members listen to and discuss each other’s opinions, or does each simply state his or her own? Are discussions on
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a positive note or does negativity sap people’s energy? Does everyone have the opportunity to speak, does the team leader appear to give more attention to some members than others, or do some individuals dominate?
• Technical process model. Have the team members engaged in procedural conflict: negotiating the where, when, how, and why issues, such as defining the steps they will take to carry out the project (e.g., a seven- step problem- solving model, if appropriate), or are they simply wandering around with no defined direction?
• Conflict. Is there interpersonal or procedural conflict between group members that prevents them from working together effectively? Is all conflict being suppressed, causing ideas to be withheld? Is substantive conflict—for example, deferring consensus when discussing ideas to get to the best ideas—encouraged?
• Decision making. Does the team make decisions based on data, or does it jump to conclusions? Is consensus used when the decision is one that requires everyone’s commitment?
• Follow-up. Does the group identify action items, then ensure that they are carried out?
An effective facilitator must have a broad range of capabilities. Three of the most important are the following:
• Meeting management skills. A facilitator should know how to run meetings in a manner that effectively uses the time available. In many ways, meetings are like mini- projects, with a mission (purpose of the meeting), technical process (meeting agenda), and boundaries (meeting duration). In addition, since meetings consist primarily of discussion, the ability to communicate effectively is vital.
• People skills. Since each person brings his or her own background, skills, and priorities to meetings, the ability to understand and work with different perspectives is critical for a facilitator. An understanding of psychology (both individual and social) and methods for change (e.g., from the field of organization development) is therefore valuable.
• Technical process analysis skills. Improvement of processes involves analysis of processes. An understanding of the seven basic quality control tools, the seven management tools, SPC, and design of experiments gives a facilitator a wide range of tools that can be introduced at an appropriate time. (These tools are all discussed in Chapters 5 and 6.) Perhaps the most important knowledge for facilitators is also the most difficult to obtain: understanding themselves. It is difficult to understand others if you do not understand yourself, because you may make interpretations using filters of which you are unaware. An effective facilitator must be able to determine whether a particular intervention is being done because of the needs of the team or because of the needs of the facilitator.
When facilitators believe that the team should change the way it is working, they can select from several different ways of bringing this matter to the attention of the
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team. The particular method the facilitator chooses will often depend on a com- bination of the facilitator’s personal style and level of comfort with the team, and the team’s responses to previous interventions. Following are some of the different ways to intervene:
• Tell the team. The easiest way is simply to tell the team what it is doing wrong or what it needs to do differently. For minor issues, this is a quick and probably safe intervention; however, it may cause more resistance with some teams since it can be interpreted as being authoritative.
• State observations. A slightly more discreet way of intervening is for the facilitator to simply state what he or she is seeing that the team may want to do differently. This puts the information in front of the team, allowing them to decide whether to pay attention to it.
• Have the team explore. Another choice is to ask the team members to think about what they are doing at the moment (and perhaps frame the context of the issue, for example, whether it is relative to communications or agenda issues). Although this method takes more time, it causes the team to take more ownership of the intervention, so that learning is more likely to be internalized.
Perhaps it is clear from some of the above discussion, but it is worth emphasizing again: it is vital for the team to have ownership of decisions that are made regard- ing content and, when possible, those regarding team processes. A facilitator who gets glory from making such decisions for the team simply reduces the likelihood of the team learning from and being committed to the team process.
There are, however, situations when facilitators have a higher level of involve- ment than what has been presented here. For example, with kaizen blitz teams, which typically last three to five days, acceleration of the improvement process comes about partially due to reducing concerns over how decisions are made. The facilitator in such projects usually has much more authority to specify the direc- tion the team will take.
E.2. Facilitation Tools
Facilitation tools are useful for idea generation. One of the main purposes of idea generation is to encourage creative thinking. Robbins and Judge (2012) discuss the three components of creativity: expertise, creative- thinking skills, and intrinsic task motivation. The tools outlined in this section can help facilitate creative think- ing and improve creative thinking skills.
Most people are familiar with brainstorming as a means of generating many ideas in a short period of time to identify solutions to problems. Other facilitation tools include nominal group technique, conflict resolution, and force field analy- sis. All of these tools are detailed below, with the exception of force field analysis, which is discussed in Chapter 5, section B.2.
E.2.a. Brainstorming
Groups and teams can use both structured and unstructured brainstorming meth- ods. For unstructured brainstorming, a topic is agreed on and written in front of the
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group. The leader/facilitator then asks for ideas to be randomly called out and all are recorded without any discussion. When the flow of ideas stops, the list is reviewed and discussed, which may result in the elimination or combination of ideas.
A structured brainstorming approach involves a round- robin process whereby each person in the group is asked to state one idea. If a person has none, he or she passes and the next person is asked, and so on. When everyone has passed on a round, the brainstorming is complete. A similar process can be used by posting several sheets of paper around the room with a topic or problem written at the top of each. Each team member goes to a sheet and writes down ideas that come to mind, then the members rotate repeatedly until all have con- tributed to each sheet. Another alternative is to simply circulate sheets of paper among the group.
Another method of brainstorming, called Crawford slip, is especially use- ful when the team is working on a particularly sensitive topic or when the team does not yet have a high level of trust. All the team members are asked to record their ideas on pieces of paper that are then given to a trusted individual (e.g., the facilitator) who compiles all the items into a single list (e.g., on a whiteboard). The anonymous nature of this method helps people feel freer to include their ideas, and the team often finds that several members had the same idea, which begins to build cohesiveness.
E.2.b. Nominal Group Technique
Nominal group technique is one way of processing lists of brainstormed items. It involves using the following steps to reduce a large list to a shorter one:
1. Ask each participant to rank the items in numeric order (e.g., in a list of eight items, 1 is best and 8 is worst).
2. Record the ranks of all participants beside each item.
3. Total the rankings for each item. Those with the lowest totals are the preferred options.
Table 1.3 shows an example applied by a group of course participants who were trying to decide where to go for lunch. Of the four choices, Marlow’s received the lowest total (therefore the highest priority), making it the group’s first choice.
Table 1.3 Nominal group technique ranking table.
Individuals and rankings
Restaurant Tom Joe Mary Sue Terry Total
Marlow’s 1 2 3 1 2 9
Grunge Café 3 1 1 2 3 10
Stew & Brew 2 4 2 4 4 16
Fancaé 4 3 4 3 1 15
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Another way to narrow down a list of items is to have the group select from the list only those that they prefer. The number they are to select is usually approx- imately one- half of the total number. After all participants have made their selec- tion, the facilitator asks how many participants voted for each option and records this. The Pareto principle will usually work, with some of the options getting very few votes; they are then dropped from the list. The voting process is then repeated until the desired number of items remains. Table 1.4 shows multivoting on a larger version of the lunch selection problem. Five people are voting, and in the third round of voting Grunge Café finally emerges as the winner by a 4:1 margin.
E.2.c. Conflict Resolution
Most people identify conflict as a problem to be solved, as something that is inevi- table, and as something that is undesirable in teams. Further, many believe conflict only comes about when two or more people have ideas that appear to be totally different and where it is perceived that a choice must be made between them. In reality, however, two kinds of conflict, substantive conflict and procedural conflict, can actually enhance teamwork. A third kind, affective or interpersonal conflict, results when team members “allow personal feelings to negatively affect group interaction” (Burnett 2005), such as when hidden biases surface, normally inconse- quential behaviors become irritants, or past slights or unresolved issues spill over into team interaction.
Kilmann and Thomas (1977) detail several styles of conflict resolution, includ- ing competing, avoiding, accommodating, collaborating, and compromising. Dealing directly with conflict means that a facilitator or other team member either reminds the team of its common goal or puts the grievances on the table in as neu- tral a fashion as possible to defuse the situation or negotiate a compromise that will allow the team to function. Not all conflict can be avoided, but one way to minimize conflicts is to clearly identify roles and responsibilities in the team early in the process.
By engaging in substantive conflict, teams actively work at avoiding hasty consensus (such as jumping on the first idea instead of waiting for possibly better
Table 1.4 Multivoting.
First vote (select 4)
Second vote (select 3)
Third vote (select 1)
Pizzas R Us 2
Marlow’s 4 3
Alice’s Restaurant 1
Grunge Café 5 5 4
Mom’s Diner 0
Stew & Brew 3 2
Fancaé 5 5 1
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alternatives or making a decision before everyone has had a chance to give input). Teams can use three strategies to defer consensus:
• Elaborate key ideas by adding details, examples, or explanations. Remember that one good idea can spark several other good ideas, which means the team has more choices.
• Consider alternatives by adding to an idea or exploring an idea that has not been previously considered. One team member might add details to help another explain a suggestion or might restate the idea so that everyone understands.
• Voice disagreements to strengthen the product or process. Remember that disagreeing does not mean you do not like someone; in fact, disagreeing about ideas can mean that you are sufficiently engaged to notice strengths and weaknesses (Gillette et al. 1993).
Negotiation is key to resolving procedural conflict, especially when a team is first convened, at specific points in reaching an objective or goal, and at the begin- nings of project meetings. As the name suggests, procedural conflict has to do with how the group runs, and requires participants to be very clear; participants should write down and maintain group memory documents that keep track of where and when the team will meet, who will take on certain roles (such as team leader, recorder, time manager, devil’s advocate), what procedures and tools the team will use (such as consensus versus voting), and the anticipated time line for meeting the team’s objectives. All these issues are important and some may need to be renegotiated on an ongoing basis to keep the group running smoothly.
The following guidelines incorporate each of the three kinds of conflict:
• Encourage people to exchange ideas freely before coming to a decision
• Treat the discussion as a problem to be solved instead of an attack on a person
• Take the time to attend to housekeeping issues such as regular breaks, room temperature, and sufficient supplies of necessary items (paper, pens, tissues)
• Consider, and keep records of, the benefits and drawbacks of each option
• Keep the team’s goals, objectives, and common interests on the front burner, especially when tempers run high
One difficulty is getting everyone on the team to understand what the other team members are saying and why it is important to them. When everyone understands and is willing to share their values and the assumptions underlying their posi- tions, asking team members to restate in their own words what has been said helps ensure true understanding.
Time is also an ally for conflict resolution. If the issue is over a decision that can be delayed, the time between subsequent discussions may allow the players to not only cool off but also think over both their own positions and those of other team members. When all is said and done, many of the skills related to conflict are also communication skills.
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F. CoMMuniCaTion SKiLLS The term “communicate” comes from the Latin communicat(us), meaning to impart or make common. When we communicate, we try to establish a common ground. Every communication interaction is unique in terms of purpose, context, mode of communication, and people involved. Communication skills are critical for suc- cess, whether measured by promotion or by higher- quality processes and prod- ucts. Robbins and Judge (2012) point out that “communication can be thought of as a process” and “must include both the transference and the understanding of meaning.” When thinking about communication from both a communicator standpoint and a receiver standpoint, it may be helpful to keep in mind best prac- tices associated with both parties. Some of these best practices are detailed below.
In the quality field, effective communication is essential in order for every- one to understand and have a sense of ownership of the common vision. Every employee must be aware of objectives and necessary actions that are required for successful quality initiatives within the organization. Common goals are a unify- ing factor in virtually all successful teams. The communication skills needed to accomplish complex goals and objectives should involve some understanding of communication theory and practice. They also require the communicator to con- sider the various tools and methods available for communication as well as the audience receiving the communication.
Communication can be achieved on an individual level (one-on-one), from an individual to a group, or in a group or team environment. Methods of communi- cation include verbal, visual, and written. Regardless of the setting or the method used, it is important to try to be as transparent and as clear as possible.
Verbal or oral communication can be achieved by using a variety of methods or conveying the information in a variety of ways. Types of oral communication include, but are not limited to, interviews, formal speeches, conversation, debate, directives, briefings (in person or via telecommunication), and public announce- ments. Telecommunication using Skype or BlueJeans has become popular. In the absence of attendees, sometimes the telecommunications are recorded and saved for viewing asynchronously. While video recordings or even YouTube videos can be considered a form of visual communication, we can also think of graphics, pho- tographs, and images as visual communication methods. Written communication ranges from the very informal text, instant message, or tweet to the formal techni- cal report or peer- reviewed publication.
All forms of communication convey a message to the receiver. QEs rely heav- ily on the discovery and organization of data, facts, and evidence—systematically collecting, analyzing, and organizing the material. Descriptive statistical methods involving data visualization are an important part of the data analysis process. There are many visual displays of information presented throughout this hand- book. Some of the displays involve a combination of visual and written communi- cation, such as the FMEA diagrams. Careful thought and consideration should be involved when choosing an appropriate graphical technique to study, understand, and present data. For example, pie charts can be very informative when a popu- lation or sample can be grouped into proportions, but they are not appropriate graphical choices for use in every analysis.
Technical communication of results often includes all forms of communication. PowerPoint presentations are regularly used when one person is relaying technical
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information to a group of listeners. When creating PowerPoint presentations, we suggest paying attention to details such as numbering the slides, using large and legible fonts (especially for any graph or axes titles), clearly detailing and explain- ing the main message of the presentation, and including an outline of the talk.
To be successful, QEs should master both discussion skills and presentation skills. The ability to analyze and organize information and present it orally will consistently reap rewards. Clear, concise, and unambiguous writing is also an essential skill. Practicing presentations ahead of time as well as seeking out con- structive feedback from peers and superiors is essential.
Feedback is an important component of the communication interaction. It pro- vides the opportunity for clarification and in- depth understanding. There are five main categories of feedback that occur in communication exchanges:
1. Evaluation. Making judgment about the worth, goodness, or appropriateness of the statement
2. Interpretation. Paraphrasing or perception checking as a means of clarification
3. Support. Confirming behavior that encourages the sender to continue to communicate
4. Probing. Attempting to gain additional information, continue the discussion, or clarify a point
5. Understanding. Trying to discover what the sender of the message intends or means by the message
Being an active listener and supplying adequate feedback, including asking rele- vant questions, are important. This is especially true when communicating as part of a team. Group communication skills require some additional considerations, such as what your role is in the group, when to talk, when to listen, and how to resolve issues in the event of a conflict or confrontation. Successful quality engi- neering implementation requires the development of effective teams, and effective communication is an important aspect.
Leaders of the quality engineering teams and leaders of an organization should pay careful attention to communication details. Leaders must establish a vision, communicate that vision to those in the organization, and provide the tools and knowledge necessary to accomplish the vision. Therefore, good leaders understand and employ efficient and effective communication in order to achieve this goal. Remember that leadership is needed at all levels of the organization.
In order to accomplish a stated goal, all members involved in reaching that goal must understand and be committed to achieving that goal. One way to achieve understanding and commitment is to include all members in the complete process. Members of effective teams feel some ownership of programs and projects when they understand goals, objectives, and/or mutually well- understood expectations and are given access to needed information and resources.
To create understanding and commitment, leaders employ skills such as clear formulation of a concept, emphasis of key points, repetition, and summarization. Multiple channels are absolutely vital to convey our message in the intricate infor- mation world we inhabit. Every listener/reader is bombarded with communica- tion from myriad sources all day long (and most of the night).
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g. CuSToMEr rELaTionS Customers can be found both internally and externally to the organization, and you must find some way of communicating with your customers on a regular basis. In studies conducted over a number of years, Collins and Porras (1997) point out that the best- of-the-best companies (visionaries) in their respective industries have developed systems that transcend dependence on any single leader or great idea to build an enduring, great human institution that has lasted and will last for decades. Many of these companies stumbled along the way but somehow found a way to come back, providing the customer or client the products or services that are wanted and/or needed. The secret seems to be to try a lot of things, keeping those that work and stopping those that do not, and continually check back with the customer to see if anything has changed, thus starting the process over.
g.1. Customer needs and Wants
Your organizational objective should be to ensure that customers want and need your products or services. As Perry (1998) states, “Staying in direct, face- to-face contact with customers, in their world, is the surest way to combat organizational myopia.” Far too often a system is developed and people in that system “expect” customers to conform to the way things are done by the supplier organization. This occurs everywhere from the corner grocery store to other retail outlets, from schools to manufacturing organizations. How often have you seen cartoons with the central theme of “if it wasn’t for the unrealistic customers, this would be a great place to work”?
The QE’s job (either manufacturing or service- based) is to help the organi- zation see that customers are the reason for its existence, versus the other way around. Hayes (2008) provides details on effective methods to measure customer satisfaction and loyalty through surveys or questionnaires. These methods now include online surveys as well. Particular attention should be paid to potential measurement error in these types of surveys as response rates tend to be low. Regardless, the goal should be to receive measurements that accurately represent customers’ attitudes.
This work, however, should go beyond just collecting a sample of informa- tion (surveys, focus group meetings, plant visits, and so on). Everyone has seen the customer survey cards at hotels and restaurants that ask about customer sat- isfaction. But what is the validity of such an effort when considering issues such as response rate and nonrandomness of response? A four- stage model for evalu- ating training events devised by Kirkpatrick (2006; discussed in more detail in Chapter 2, section F) would categorize this kind of data- gathering effort—and its validity—as reaction, or level- one evaluation. Some consider these tools to be “smiley sheets,” a pejorative term referencing the halo effect, which results from the glow of the moment of the event or because the participant wants the researcher to feel good. The real question for the QE should be, “What do my customers think after using the product or service for some period of time in actual real- world settings, and what are they telling other people about my organization?”
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g.2. Quality Function deployment
Quality function deployment (QFD) is a powerful planning technique, perhaps the most comprehensive ever invented for quality planning. QFD is an overall concept that provides a means of translating customer requirements into the appropriate technical requirements for each stage of product development and production (i.e., marketing strategies, planning, product design and engineering, prototype evalu- ation, production process development, production, and sales). QFD is especially suited to large- scale products such as airplanes, automobiles, and major appli- ances because these products have heavy tooling, high design costs, and many optional features that must be selected and then produced or procured. QFD was introduced into American industry in the 1980s by the American Supplier Institute of Livonia, Michigan, which remains one of the organizations that actively pro- mote its usage. QFD continues to be employed across a variety of organizations. Recently, Camgoz- Akdag, Pinar, and Nazli (2016) discussed the use of QFD for increasing customer satisfaction in the textile industry in Turkey.
G.2.a. Definitions and Concepts of QFD
Five key terms are associated with QFD (Sullivan 1986):
1. Voice of the customer (VOC). The customer’s requirements expressed in their own terms.
2. Counterpart characteristics. An expression of the customer’s voice in technical language that specifies customer- required quality.
3. Product quality deployment. Activities needed to translate the voice of the customer into counterpart characteristics.
4. Deployment of the quality function. Activities needed to ensure that customer- required quality is achieved; the assignment of specific quality responsibilities to specific departments. (Any activity needed to ensure that quality is achieved is a quality function, no matter which department performs it.)
5. Quality tables. A series of matrices used to translate the voice of the customer into final product control characteristics.
Sometimes it is possible to incorporate all the key relationships into a simple dia- gram known as the house of quality, so- called because it resembles a house with a pitched roof. Figure 1.15 shows such a diagram.
For comprehensive coverage of more than 30 planning tools grouped under QFD, see King (1987). A typical project will require only a few of those tools. The following QFD documents are most common:
1. Customer requirements planning matrix
2. Design matrix
3. Final product characteristic deployment matrix
4. Manufacturing/purchasing matrix
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5. Process plan and quality control charts
6. Operating instructions
7. Human factor studies or surveys for capturing VOC
G.2.b. Application of QFD: The Basics
By applying QFD, customers’ expectations are translated into directly related job requirements. The objective is improved customer satisfaction at acceptable cost. The basic relationship is displayed in the input–output matrix shown in Figure 1.16. This matrix, one of many in QFD, organizes the process of determining relationships between what the customer wants (usually described in nontechnical terms) and how the supplier satisfies these wants. Wants fall into three categories: must have, expected to have, and would like to have. Numerical measures are highly desirable. The wants must be specified in sufficient detail to ensure they are clearly under- stood. Customers may or may not be involved in setting the requirements; however, their satisfaction depends on identifying and meeting their wants.
Figure 1.15 QFD house of quality diagram for a paperwork process.
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The “hows” are the technical details of each job. The strength of each rela- tionship may be strong, medium, or small, as shown in Figures 1.15 and 1.17. These symbols can be converted to weights, such as strong = 5, medium = 3, and small = 1. The weights will convert to scores indicating the importance of each job requirement. At the top of the requirements matrix, a correlation matrix is added to show the strength of the relationships among the different job requirements. A basic example is shown in Figure 1.15 for a paperwork improvement project, and a more complex example for a car door design is depicted in Figure 1.17.
QFD as a planning technique has significant benefits:
1. Product objectives based on customer requirements are not misinterpreted at subsequent stages
2. Particular marketing strategies or sales points do not become lost or blurred during the translation process from marketing through planning and on to execution
3. Important production control points are not overlooked
4. Efficiency is increased because misinterpretations are minimized
G.2.c. Customer Value Analysis
Gale and Wood (1994) describe seven tools of customer value analysis:
1. The market- perceived quality profile (an “indicator of how well you are performing overall for customers in your targeted market”)
2. The market- perceived price profile (a weighted indicator of how customers perceive different competitors’ performance on given price attributes)
3. The customer value map (a “map that reveals a sizable cluster of business units receiving premium prices that are not fully supported by superior perceived quality”)
Figure 1.16 Input–output requirements matrix.
Relationship matrix
“Whats” (Customer requirements)
“H o w
s” (J
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)
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Figure 1.17 House of quality for a car door.
Relationships:
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4. The won/lost analysis (an analysis of those factors that won or lost the sale)
5. The head- to-head area chart of customer value (a “chart of customer value displaying where you do well and where you do worse against a single competitor”)
6. The key events timeline (a chronological list of the events that changed the market’s perception of performance on each quality attribute, yours and your competitor’s)
7. A what/who matrix (“a method for tracking who is responsible for the actions that will make success in customer value possible”)
Using these tools will “enable an organization to navigate strategically even in confusing times.” Numerous factors represent value to different customers under a variety of situations. The characteristics shown in Table 1.5 illustrate different perspectives of what the customer considers important.
g.3. Customer- driven Quality
A growing number of approaches focus on greater understanding of and inter- action with customers. The two types of customer- driven quality, reactive and planned, are proving to be successful in improving quality but still do not guar- antee customer satisfaction (Foster 1998). Reactive customer- driven quality (RCDQ) responds to customer requirements after the fact. Planned customer- driven quality, on the other hand, is anticipatory and proactive in that it assesses customer needs and seeks methods for satisfying those needs before the fact. Any organization wanting to meet customer expectations is pursuing a moving target. The reactive nature of the RCDQ approach will cause the supplier to fall behind the moving target.
Planned customer- driven quality is best accomplished using some form of strategic quality planning (SQP). This is not necessarily the same as the strategic planning process, however, and is one reason that the Malcolm Baldrige National
Table 1.5 Customer perspectives of value.
Characteristics—product (examples)
Performance Reasonable price Durability Safety
Serviceability Ease/flexibility of use Simplicity of design, aesthetics Ease of disposal
Characteristics—service (examples)
Responsiveness Reliability Competence Access Courtesy Communication (sensitivity, genuine interest/concern)
Credibility/image Confidentiality/security Understanding the customer Accuracy/completeness Timeliness
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Quality Award changed the name of the SQP category to strategic planning, to counter the sense that some quality professionals had too narrow a focus on com- pany competitiveness in the marketplace.
With any given effort to become a customer- driven company, an organization should study how it is perceived by its customers, which is related to organiza- tional practices. One list of top 10 key characteristics of customer- focused compa- nies includes:
1. Total consumer experience. The ability to look at the customer from all angles of how the organization’s products and services are experienced in the real world. Look for every possible point of contact with the customer to collect information on what is happening in the field.
2. Product hits. Use of the Kano model to continually delight the customer with new products and services, some of which the customer may not even have known that they wanted.
3. Consumer loyalty. Building a sustained momentum over time to the point where the customer will use only your product or service—even waiting, if necessary, to get the “real thing.”
4. Retailing and distribution. Creating a win–win–win for your organization, distributors, and customers. Your distribution system is a customer as well.
5. Brand process. The creation of recognized products or services that are sought after in the marketplace.
6. Logistics. Providing JIT and just what is needed/wanted in the marketplace at point of usage.
7. Build to demand. Creating a lean process that is capable of rapid changeovers to give the customers the needed products and services as they want them (JIT). This process has to be built into the entire system, from suppliers, through production, to the ultimate customer.
8. Consumer knowledge system. Continuous information gathering of customers’ expectations and wants that feed into the system; used to look for continual improvement opportunities.
9. E-commerce. Becoming interactive and offering distribution, selling, and constant communication with customers online.
10. Growth. Continually improving with faster service, better value, and higher quality to create a culture that uses creativity and innovations to improve customer satisfaction.
To summarize, there is no sure way to always satisfy or delight customers, because we cannot talk to every individual customer and because customers are constantly changing their minds about what they need or expect. So we must find ways to continually talk with many customers using the techniques we can. With today’s technology this should become easier, but will the QE be able to ensure that the information received is good enough to make sound predictions? The challenge is to keep the process both simple and informative. (See section B.3 earlier in the chapter for more details on QISs.)
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H. SuPPLiEr ManagEMEnT Many years ago, companies worked under the assumption that engineers designed products and specified requirements, suppliers provided materials, manufactur- ing built the products, and quality control inspected the product after it was made to ensure quality. This approach was inherently wasteful. Since the 1940s, the use of quality standards for suppliers has gradually evolved into a system that ensures quality products that meet requirements with only a limited amount of inspec- tion by quality control personnel. MIL- Q-9858, BS 5750, industry- specific (starting in the early 1960s), and ISO 9000 standards (see Chapter 2, section C) have each made their contribution. For example, in the automotive industry, the Automotive Industry Action Group (AIAG) manages the Production Part Approval Process Man- ual (2006), a standard that supplies a list of several elements of the manufactur- ing process to ensure suppliers meet their customer’s requirements (e.g., design records, change documents, FMEA, and control plans). In the aerospace industry, AS9102B (2014), managed by the Society of Automotive Engineers (SAE), details requirements for first article inspection, a method to document requirements of aerospace parts between customers and suppliers.
Quality assurance personnel now spend greater effort ensuring that quality is built into products and that conformance is achieved during production. The lines are becoming more blurred as Six Sigma programs help everyone in the organization become concerned about quality and defect prevention. The same team cooperation and close communication used internally are now being applied to supplier rela- tions. The goal is to ensure that purchased items and materials conform to require- ments without the need for extensive inspection upon receipt by the purchaser and that continual improvement is being practiced (Johnson and Webber 1985).
Suppliers also can be found both internally and externally to the organization. It is important to find an effective way of communicating with all your suppliers on a regular basis and to put together an effective process and practice for supplier quality management (SQM). SQM practices may differ by industry. AlMaian et al. (2016) discuss the importance of SQM in the construction industry and discuss methods for quantitative analysis when faced with the problem of too little data due to time and budget constraints. In this section we present high- level tech- niques, improvement, and risk for SQM.
H.1. Techniques
At the superficial level, surveys, audits, and inspection all have the same goal: they provide internal or external customers with a degree of confidence (but not absolute assurance) that the quality of the product or process is what it should be. However, each of these tools has its own distinctive characteristics.
H.1.a. Audit
As defined by ISO 9000:2015, an audit is a “systematic, independent and docu- mented process for obtaining objective evidence and evaluating it objectively to determine the extent to which the audit criteria are fulfilled. The fundamental ele- ments of an audit include the determination (3.11.1) of the conformity of an object according to a procedure carried out by personnel not being responsible for the object audited.” Audits of a supplier’s systems or processes can only be performed
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at the supplier’s facility. Audits of a supplier’s product may be performed at either the supplier’s facility or the customer’s facility.
A system audit is a documented activity performed on a management system to verify, by examination and evaluation of objective evidence, that applicable ele- ments of the quality system are suitable and have been developed, documented, and effectively implemented in accordance with specified requirements (Russell 2013). A system audit examines everything within the system: the processes, prod- ucts, services, and supporting groups (e.g., purchasing or customer service).
A process audit is an analysis of elements of a process and appraisal of complete- ness, correctness, conditions, and probable effectiveness. It is done to ensure that the processes are working within established limits. Unlike a system audit, a pro- cess audit covers only a small portion of the total system and therefore often takes less time to complete than a system audit. Furthermore, “a process audit checks the adequacy and effectiveness of the process controls established by procedures, work instructions, flowcharts, and training and process specifications” (Russell 2013).
A product audit is “an examination of a particular product or service (hardware, processed material, software) to evaluate whether it conforms to requirements (that is, specifications, performance standards, and customer requirements)” (Russell 2013). The product audit verifies that the system and processes used to produce the product are capable of producing a product that conforms to the established specifications/requirements. Product audits are performed after the product has been completed and has passed final inspection; therefore, this method should not be confused with the term “inspection,” which concerns the acceptance or rejec- tion of the product or lot. In a product audit, the product or service is examined in terms of form, fit, and function after it has passed final inspection.
Audits can also be used to assess change control of products and configuration management programs. A product audit is used to ensure configured products or services meet specifications and perform as required. A process audit can also be used on the configuration process to verify the process is appropriate and main- tained (Russell 2013).
H.1.b. Sampling Inspection
Inspection is a process of measuring, examining, testing, gauging, or otherwise comparing a unit with the applicable requirements. Sampling inspection is some- what comparable to surveys and audits, while 100% inspection is somewhat com- parable to production line operation because each and every item is subjected to it. (See Chapter 4, section C, for inspection and sampling.)
One hundred percent inspection is required in certain highly critical processes and in processes that produce unavoidable defects, such as semiconductor fabri- cation. Generally, 100% inspections done by people are not completely effective. Thus, in today’s industrial environment, 100% inspections are nearly always auto- mated. Several types of sampling inspection include the following:
Acceptance sampling is sampling where decisions are made to accept or reject a product or service based on the results of inspected samples.
Skip-lot inspection is an acceptance sampling plan in which some lots in a series are accepted without inspection because the sampling results for a stated number of immediately preceding lots met stated criteria. Explanation of this methodology is found in American National Standard, ANSI/ASQ S1-2012.
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Incoming inspection is the inspection of purchased parts at the customer’s facility, after the shipment of parts from the supplier, to ensure supplier compliance with specifications and contractual agreements.
Source inspection is the inspection of purchased parts at the supplier’s facility by a customer representative to ensure supplier compliance with specifications and contractual agreements.
H.1.c. Survey
The survey can be defined as a broad overview of a supplier ’s system and/or processes that is used to evaluate the adequacy of that system or processes to produce quality products (Laford 1986). A system survey is used to assess whether the supplier has appropriately controlled systems that will adequately prevent the manufacture of nonconforming products. A process survey is used to evaluate whether a supplier has controls in place to ensure that the process will manufac- ture quality products. Process controls include proper tooling, equipment, inspec- tion, and so on.
The primary purpose of a survey of a supplier or potential supplier is to ascertain whether the supplier has adequate financial resources (evaluated by purchasing), adequate manufacturing capabilities (evaluated by manufacturing engineering), and adequate quality systems (evaluated by the quality assurance group).
In preparing for the survey, the team leader should obtain as much informa- tion about the supplier as possible. The purchasing agent can provide copies of the supplier’s annual reports, credit investigation, Dun & Bradstreet reports, inter- net searches, certifications, and so on. A facilities and equipment list should be obtained for review by manufacturing engineering, and a copy of the supplier’s quality manual must also be reviewed prior to the survey.
The survey team may be made up of members from purchasing, manufactur- ing, and quality control, plus various specialists in the areas of nondestructive testing, product design, or other special processes. At times, the team may consist of only the quality professional. In the latter case, the purchasing agent usually has previously evaluated the supplier’s financial status.
It is important that the team meet prior to arriving at the supplier’s facility. Based on the premise that the team has reviewed all pertinent materials, the pre- survey meeting is held to (1) ensure that all of the team members agree on the theme and purpose of the survey, (2) ensure that the roles and responsibilities of each team member are understood by the others, (3) draft a preliminary survey agenda, and (4) select the team leader.
The team leader must not overlook the obvious, such as the supplier’s current address, name of host individual to contact, correct time and date for the survey, and so on. It is important that the team leader verify that the supplier is ready for the survey. Often it is appropriate to advise the supplier of the proposed agenda, allowing supplier representatives to prepare for the visit.
In order to quantify the results of a survey, there must be a formalized approach for collecting data and evaluating the systems observed. The primary method of quantification is for the survey team to use a checklist to record survey results. Checklists commonly used cover both procurement and manufacturing/quality aspects of a supplier’s organization.
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The manufacturing/quality checklists often are broken into the following categories:
1. Drawing and specification control
2. Purchased material control
3. Measuring and test equipment control
4. Process control and product acceptance
5. Material storage area, packing, shipping, and record retention control
6. Quality program management
7. Statistical process control
8. Strength summary of system survey
9. Corrective action summary of system survey
10. Summary report
The manufacturing/quality categories may be expanded as needed. An amplifica- tion of the listed categories can be found in Laford (1986).
The supplier procurement checklist often is broken down into the following categories:
1. General information
2. Product information
3. Facilities and equipment information
4. Sales, shipping, and payment information
The supplier procurement checklist categories may be expanded as needed. An amplification of the list can be found in Laford (1986).
The use of scoring (numerical, alphabetical, or other regularly sequenced scores) in a checklist further enhances quantification and validity of judgments. Many pro- fessional evaluators prefer to have the supplier also score a copy of the checklist in order to better compare the customer’s viewpoint with that of the supplier’s.
The opening conference is get- acquainted time. The survey team members should explain why they are there, what they are going to attempt to do, and, in a general way, the sort of results they expect. Each team member should explain his or her role in the survey and in the customer’s organization. The team leader also should briefly explain the nature of the customer’s products or services. It is essen- tial that all levels of supplier management understand the scope and purpose of the survey (Vendor-Vendee Technical Committee 1977).
Each supplier representative present should explain his or her role in the sup- plier organization. At this time, the supplier representatives also should briefly describe the nature of the products manufactured and present an overview of the company and systems used. The opening conference also is a good time for the survey team to brief the supplier on the intended products to be purchased.
A brief plant tour will acquaint the survey team with the supplier’s over- all operations. Following the plant tour, the team members can proceed to their respective areas for evaluation. Each area should be evaluated in detail in accor- dance with the checklist and point scores recorded. It is imperative that each area
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be evaluated in the actual area and not in the conference room or manager’s office. Furthermore, by being in the appropriate area, verbal statements of compliance and quality procedures can be verified by witnessing the action being performed. The survey team should discuss any negative findings with the supplier escort who was present during the finding to reconfirm the facts prior to the closing con- ference with supplier top management.
Prior to the closing conference, the survey team must meet to compile the report for that conference (this is not the final report). During the closing confer- ence, the team leader should review each category, expressing the strengths and weaknesses observed. At this time it may be possible to estimate corrective actions required for deficiencies found if they have not already been addressed.
The closing conference must be kept on a positive note, with a win–win attitude on both sides, which requires careful attention to communication strat- egies and can challenge the team leader ’s communication skills. In the closing conference, the team leader should focus on the major deficiencies found, if any, and detail appropriate corrective actions. This should be followed by a brief mention of any minor deficiencies observed. All can be lost if the survey team presents an extensive list of minor observations with a few major deficiencies intertwined.
If at all possible, the survey team should leave a draft copy of the survey report with the supplier. By doing so, any questions can be cleared up immediately. It is much more difficult to clarify misunderstandings when a copy of the final report is received a month or more later.
The end product of the survey or quality program evaluation should be an understandable final report. A good report effectively communicates the findings, using the original observations to support the conclusions. The report must be an honest, objective summation of the team’s efforts.
The report should detail the following:
1. All individuals present and their correct titles
2. The areas evaluated
3. Any major deficiencies requiring written corrective action
4. Any minor deficiencies
5. A summary that states the final conclusion, for example, approval, conditional approval (corrective actions that should be addressed), or disapproval
6. A closing statement expressing appreciation for the supplier’s assistance and cooperation
Survey follow- up is carried out to ensure that a supplier that did not qualify at the time of the survey visit has taken satisfactory corrective action. The customer may have to judge whether a follow- up visit is warranted. A report from the sup- plier, accompanied by suitable documentation of corrective actions taken, may be adequate.
H.1.d. Evaluation
Rating a supplier’s capabilities involves rating or evaluating (1) the supplier’s system (financial, manufacturing, and quality) and (2) the supplier’s delivered
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product. Other considerations to take into account include the amount of effort, time, and cost necessary to replace a supplier, the ability of the supplier to deliver material on time, the location in the world of the supplier’s manufacturing facility, and the importance of the material to the producer’s product.
The rating of a supplier’s system usually begins with the initial supplier sur- vey (as discussed above). Often, the initial survey is followed up with a periodic supplier resurvey, called a systems audit. The audit provides the customer with an opportunity to evaluate the supplier’s systems over time so that any deterioration is noticed immediately.
The rating of a supplier’s delivered product basically takes the form of record- ing, in some predetermined manner, the results of incoming inspections. It also can include failures caused by the supplier’s delivered products that appeared during the customer’s manufacturing cycle or while the product was in service.
Supplier rating elements and formulas are as diverse as companies are. The common aspects are quality, price, and delivery.
The quality factor usually includes quality lot rating, quality part rating, com- parison with competition, complexity analysis, and economic conditions, where
Quality lot rating =
and
Number of lots rejected Number of lots inspected
Quality part rating = Number of parts rejected
Number of parts inspected
The delivery factor usually includes a timeliness rating and completeness rat- ing. The timeliness rating is based on the due date of the lot minus some demerit (e.g., 10%) for each day the lot is early or late beyond some specified grace period or window (e.g., due date ± two working days). It is important to note that if the supplier chooses the freight carrier, the system can base the due date on the date the lot is received on the customer’s dock. If the customer chooses the freight carrier, however, the due date should be measured by the date shipped from the supplier.
The completeness rating = Number of parts actually received
Number of parts scheduled to be received
An overall rating, for example, can be derived by assigning percentages to the aforementioned aspects of quality, price, and delivery and taking a weighted average.
Quality rating = 0.40(Quality lot rating) + 0.60(Quality part rating) Price rating = 0.40(Comparison level) + 0.30(Complexity level) + 0.30(Economic condition) Delivery rating = 0.50(Timeliness rating) + 0.50(Completeness rating)
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The next step is to assign weights to the three main factors. For example:
Overall supplier rating = 0.40(Quality rating) + 0.30(Price rating) + 0.30(Delivery rating)
This generic example can be expanded into an elaborate computerized sys- tem. It also can be tailored for use by smaller businesses that may still have manual systems.
H.2. improvement
The purchasing organization usually tracks and monitors suppliers. A special sup- plier quality assurance (SQA) group may be formed to work with the buyer to look at suppliers’ performance. Some common supplier information includes:
• Defective parts per million (ppm)
• Cost adjustment requests
• Delivery date slippages
• Performance improvement
• Adherence to quality system requirements
Using metrics such as the above, a QIS can generate reports such as supplier pro- files by select criteria. Suppliers can be ranked by defective ppm, improvement, or similar metrics. Preferred suppliers can then be selected using quantitative data instead of guesswork and politics.
Ideally, suppliers are treated like partners in satisfying customers. This requires a mature organization with objective information. Communication skills, careful fact gathering, and a good QIS are all needed to achieve this goal. You and your suppliers should keep constant communication open on many fronts to ensure that everything is working well to delight the ultimate customer.
H.3. risk
Standards and specifications are documents containing criteria that must be met, and these documents become legally binding by reference on the purchase order. They define what is being purchased from the supplier. They can be in the form of engineering drawings, catalog descriptions, or other documentation. The pur- chaser need not always develop original specifications. Commercial quality speci- fications are available and range from detailed engineering drawings (which may include references to process specifications, such as reliability verifications and inspection requirements) to off- the-shelf items (which are defined by the charac- teristics on the manufacturer’s data sheet or catalog). Such commercial specifica- tions help simplify the procurement process.
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It is important that the applicable standard or specification document be incor- porated into the purchase order so that there is no doubt that the requirements are to be met. If they are not incorporated, there is no basis for enforcing compliance. This helps to ensure business continuity and minimize risk associated with the supplier materials or items.
It is important to include contingency plans related to supplier purchases. See Chapter 7 for detailed information regarding risk management, which can be applied to suppliers as well as internally.
i. BarriErS To QuaLiTy iMProvEMEnT An organization with a properly implemented total QMS will have fewer noncon- formities, reduced rework and scrap, lower inventory levels, reduced cycle times, greater employee satisfaction, and increased customer satisfaction. Organizations that are not able to overcome the barriers or obstacles to quality improvement will not experience these benefits. In a study by Salegna and Fazel (2000), managers of TQM companies ranked 12 obstacles to implementing quality. These barriers or obstacles follow in order of importance:
1. Lack of time to devote to quality initiatives. Frequently, managers are too busy with their regular activities to take on an additional activity such as quality. Initially, senior management must provide time for employees to devote to the quality initiative. Once a program is well established, the quality activity will become part of the employees’ activities.
2. Poor intra- organizational communication. All organizations communicate with their employees in one manner or another. Communications deliver the organization’s values, expectations, and directions, provide information about developments, and allow feedback from all levels. The organization must encourage and provide the means for two- way communication so that information flows up as well as down the ladder.
3. Lack of real employee empowerment. Too often, empowerment is merely lip service. Individuals should be empowered to make decisions that affect the efficiency of their process or the satisfaction of their customers. Teams need to have the proper training and, at least in the beginning, a facilitator.
4. Lack of employee trust in senior management. In many organizations, this obstacle will not be a problem because senior management has created an atmosphere of trust in its relationship with the employees. In other organizations, this atmosphere will have to be developed by management being honest with the employees.
5. Politics and turf issues. Differences between departments and between individuals create problems. The use of multifunctional teams will help break down long- standing barriers. Restructuring to make the organization more responsive to customer needs may be needed. An
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example of restructuring is the use of product or customer support teams whose members are permanently reassigned from the areas of quality, production, design, and marketing.
6. Lack of a formalized strategic plan for change. A formalized plan for change is necessary because individuals resist change. They become accustomed to performing a process in a particular way and it becomes the preferred way. Management must understand and utilize these basic concepts of change:
— People change when they want to and to meet their own needs
— Never expect anyone to engage in behavior that serves the organization’s values unless an adequate reason (why) has been given
— For change to be accepted, people must be moved from a state of fear to one of trust
It is difficult for individuals to change their own behavior, and it is much more difficult for an organization. Honest two- way communication with respectful feedback increases the chances of success.
7. Lack of strong motivation. The building of a motivated workforce is, for the most part, an indirect process. Management at all levels cannot cause an employee to become motivated; they must create a conducive environment for individuals to become motivated.
8. View of quality program as a quick fix. Frequently, the quality program is viewed as a quick fix. Quality improvement is a race that does not have a finish. Management must constantly and forever improve the system so that quality and productivity are continually and permanently improved and costs reduced.
9. Drive for short- term financial results. Too often, organizations focus their efforts on the quarterly financial results. Quality improvement requires an organization to have a strong future orientation and a willingness to make long- term commitments.
10. Lack of leadership. In order for any organizational effort to succeed, there must be leadership. Leadership requires a substantial commitment in terms of both management time and organizational resources.
11. Lack of customer focus. Organizations need to understand the changing needs and expectations of their internal and external customers. Effective feedback mechanisms are necessary for this understanding.
12. Lack of a company- wide definition of quality. This obstacle is the least of the 12 and is easy to correct. Experienced quality professionals recommend that all areas of the organization be involved in writing the definition.
Overcoming these barriers to change can be difficult. Some ways of overcoming the barriers outlined here are self- explanatory. For example, if there is a lack of
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leadership in an organization, especially leadership interested in quality, then change will not occur unless new leadership is established. Other ways of deal- ing with barriers can be accomplished with change management. A change agent is someone responsible for managing change activities (Robbins and Judge 2012). Change agents can be successful when making changes regarding structure, technology, physical setting, and people. A change agent who is interested in improving quality in an organization can be very effective. To overcome a lack of time, a strategic process for selecting quality projects should be implemented and supported by leadership. This type of strategic planning is a fundamental principle in many continuous improvement methodologies, including Six Sigma.
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A quality system is the enabling mechanism behind the quality assurance and improvement functions of any organization. It is a statement of commitment to quality and tells how quality is to be achieved. The term “system” implies func- tional elements, attributes, and relationships. Chapter 1 explained quality manage- ment and leadership. In this chapter, we discuss the quality system, its elements, and how it is documented. We also introduce recognized standards that define or recommend quality systems and discuss quality audits. Quality is intrinsically related to cost, and we explain the most prominent quality cost systems. Finally, we suggest that training is a quality system, and provide an overview of the role of the quality engineer (QE) in quality training.
a. ELEMEnTS oF THE QuaLiTy SySTEM The basic elements of a quality system are planning, control, and improvement. These elements span the entire process and are present regardless of the type of organization or industry implementing quality initiatives. In this section, we briefly introduce these elements and the basic design of the strategic plan used to ensure that implementation of these elements is successful.
The elements of the quality system pertain to all functions in the organization. We describe four functions below and briefly discuss how quality applies to each.
1. Quality in marketing. The marketing function is an important source of information regarding the implied and stated needs of the customer, actual field performance, and the degree of customers’ satisfaction with the product. Such information will help identify product problems relative to expectations and initiate corrective measures. Consequently, the marketing function is required to define and document the requirements for a quality product, provide the organization with a formal statement or outline of product requirements, and establish an information feedback system for monitoring field performance on a continuous basis.
2. Quality in specification and design. With the customers’ needs clearly identified, the design function provides the translation of these needs into technical specifications. Formal plans should be prepared and documented for identifying critical stages of the design process and assigning responsibility for each. Design reviews should be conducted
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at the end of each stage to identify problem areas and initiate corrective actions. All necessary measures should be taken to ensure clear and definitive statements of the design requirements. Methods for evaluating conformance during production should also be specified. Design verification and validation through prototype testing or other techniques is required. Provisions should be made for periodic evaluation of the design in light of field performance data.
3. Quality in purchasing. The standard requires that all purchasing activities be planned and controlled by documented procedures. Successful purchase of supplies begins with clear definition of the requirements. A close working relationship with vendors and subcontractors is required to facilitate and secure continuous quality improvements. Procedures must be established for evaluating the capability of the vendors. In some cases, the vendor is required to establish a demonstrated capability of meeting design requirements. If incoming inspection is to be performed, the costs involved should be considered and the vendor should be notified of the results.
4. Quality of processes. This element stipulates the requirements of operation under controlled conditions. The operation of processes and the operating conditions should be specified by documented work instructions. Process capability studies are required to determine the effectiveness of the process and to identify the need for improvements (see Chapter 6, section G, for more information).
a.1. Basic Elements
The elements of a quality system are the activities used to ensure customer satisfac- tion. Typically, these activities depend on the type of organization, its structure, the market, and the particular type of product or service provided. The basic elements of a quality system (planning, control, and improvement) are relevant from product and process design through quality cost systems and audit programs.
Quality-related activities start with identifying customer needs and extend throughout the life cycle of the product, as depicted in Figure 2.1. The procedures and work instructions followed within each of these functional areas to achieve the stated quality objectives represent elements of the quality system. It is impor- tant to note that the suitability and effectiveness of the system as a whole are deter- mined by the attributes of these individual elements and their relationships. Top management must establish, document, and maintain such systems with the over- all objectives in mind.
System elements closely correspond to the various phases in the traditional product life cycle depicted in Figure 2.1. In other words, a quality system must cover all the activities that affect product or service quality. ISO 9004:2009 (2009) organizes quality management in the following elements: managing for the sustained success of an organization, strategy and policy, resource manage- ment, process management, monitoring, measurement, analysis, and review, and
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improvement, innovation, and learning. These elements are critical for managing quality and for the long- term success of an organization. In the following subsec- tions, we discuss planning, improvement, and control of the quality system.
A.1.a. Planning
Planning a quality system is critical in order to address risks and opportunities in an organization. As described in ISO 9001:2015 (2015), there are three main goals of planning for a quality system: planning actions to address risks and opportuni- ties, planning to achieve quality objectives, and planning changes. Planning for risks and opportunities (1) gives assurance that the quality system can achieve its intended results, (2) enhances desirable effects, (3) prevents or reduces undesired effects, and (4) achieves improvements. Addressing risks includes avoiding risks, taking risks to pursue an opportunity, eliminating risk, reducing risk, sharing risk, or accepting risk. We discuss risk management in more detail in Chapter 7. Plan- ning for opportunities may include adopting new practices, launching new prod- ucts, opening new markets, seeking new customers, building partnerships, and using new technology.
Organizations should plan how to achieve quality objectives. These objec- tives should be documented and be consistent with policy, measurable, relevant, monitored, communicated, and updated as needed; they should also take into account any requirements. Organizations should identify what will be done, what resources will be needed, who will be responsible, when items will be completed, and how the results will be evaluated.
Figure 2.1 Product life cycle and quality system elements. Source: Adapted from ANSI/ISO/ASQC Q9004-1-1994. Used with permission.
Marketing and research
Disposition or recycling at the end of useful life
After sales
Technical assistance and servicing
Installation and commissioning
Sales and distribution
Product design and development
Typical life cycle phases of a product
Process planning and development
Purchasing
Production or provision of service
Verification
Packaging and storage
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Employee training, qualification, and motivation are key factors in develop- ing the human resources of an organization and emphasizing quality awareness among them. Documented procedures for identifying and providing training pro- grams at all levels should be established and maintained. Periodic assessment of personnel skills and capabilities should be considered. Recognition of proper job performance and the use of motivational programs are ways in which manage- ment can support quality improvement efforts.
Organizations should plan for when changes will be required to the quality system. As discussed in ISO 9001:2015, the organization should consider the pur- pose of the changes and their potential consequences, the integrity of the quality system, the availability of resources, and the allocation of responsibilities.
Finally, procedures are needed for identifying the safety aspects of products and processes. These aspects are best identified and considered during the design phase of the product life cycle. Further, the rule of strict liability has created a need to plan for field failures and their legal implications. These procedures may include documenting prototype and product design evaluation testing for safety, providing adequate operational instructions with warnings against known haz- ards, and developing contingency plans for product recall. Failure modes and effects analysis (FMEA), fault tree analysis, and hazard function analysis are all valuable tools for assessing risk (see Chapter 7, section B, for details).
A.1.b. Control and Improvement
Control of processes is a central element in achieving conformance to design requirements. The type and sensitivity of the control technique depend on the quality characteristic involved or generated, the nature and stability of the pro- cess, and its potential capability. Control should extend over the material and parts used, tooling and any shop aids utilized, and environmental conditions. Proper identification of materials from the time of receipt to product delivery and installa- tion is required. Statistical techniques for monitoring process variables are a crucial and extremely important element of the quality system. The analytical techniques used to measure, control, and improve quality throughout the product life cycle include design of experiments, estimation, tests of significance, control charts, and sampling inspection. All of these techniques are discussed in detail in Chapter 6. The control chart is one of the most useful techniques for understanding whether a process is in statistical control and how it is behaving with respect to tolerance limits required by the process.
Control charts are useful for studying a process over time and can be used for both variable critical- to-quality measures (e.g., the diameter of an arterial stent) and attributes (e.g., whether the stent contains a defect). Defects can be labeled as nonconformities. Documented procedures for dealing with nonconforming units should be established and maintained. These procedures include steps for the identification, segregation, and review of the nonconformities. The objective is to avoid the unintended use of such units and the consequent dissatisfaction of internal and external customers. Product verification addresses the allocation of test and inspection points in the process for the purpose of verifying conformance. Verification of incoming materials and products at various stages of the process prevents the unnecessary cost of further processing nonconforming units. Final product verification is performed to prevent shipping nonconforming units to customers.
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In addition to the process and output being in control, all measuring systems used in the development, production, and installation of products should be con- trolled. Documented procedures should be established to maintain the measuring process in a state of statistical control. The procedure includes initial calibration against a reference standard as well as periodic recall for adjustment and recalibra- tion, and may be extended to all vendors.
Post-production activities, including procedures for product storage, deliv- ery, and installation activities, can prevent deterioration of product quality, secure proper identification, and safeguard against improper installation. Also, the qual- ity system should allow for feedback of information regarding field performance, customer satisfaction, and the initiation of corrective actions.
A quality system should define the responsibility and authority for instituting corrective actions. These actions should be planned after identifying the root causes of the problem. Actions to eliminate these causes may involve a variety of functions such as design, purchasing, production, and quality control. The objective should be to prevent the recurrence of these causes and improve quality. Corrective action is required to monitor the effect of actions and ensure a high quality is obtained.
Quality records are records that indicate the results of implementing the system and provide subjective means for evaluating its effectiveness. An organization is required to establish and maintain documented procedures for identification, col- lection, storage, retrieval, and disposition of these records. Analysis of the quality records can help identify trends in quality performance, as well as the need for and effectiveness of corrective actions, and thus is useful for improvement. In addition, records should indicate authorized changes to the quality manual and any modifi- cations made in the procedures or work instructions. Documentation systems are discussed in section B of this chapter.
a.2. design
The design of the quality system should be related to the strategic plan and core processes within an organization. This means that quality initiatives and plans should be part of the management system. Montgomery (2013) argues that the effective management of quality involves successful execution of three activities: quality planning, quality assurance, and quality control/improvement.
Quality planning is a strategic activity that involves identifying the needs of the customer. This activity relies on the combined efforts of management, employees, and customers (internal and external). Determining how the needs of the custom- ers will be achieved should be included in the quality plan.
Quality assurance is the set of activities associated with achieving and main- taining products and/or services that meet the needs of both the business and the customer. Documentation during this activity is important for successful quality efforts. Documenting is discussed in detail in section B.
Quality control/improvement activities are those used to ensure, improve, and maintain a level of quality that is appropriate for the organization. These activi- ties often involve statistical methods such as statistical process control (SPC), design of experiments, and acceptance sampling. All of these statistical methods are discussed in detail in Chapter 6 of this handbook. Continuous improvement methods such as Six Sigma and lean, among others, can also be utilized in a qual- ity system. See Chapter 5 for details on these and other continuous improvement methodologies.
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B. doCuMEnTaTion oF THE QuaLiTy SySTEM It is important to have appropriate and detailed documentation of the quality sys- tem in use. ISO 9000 defines a document as “information and the medium on which it is contained.” Documented information includes any goals, policies, procedures, and other meaningful data required to be controlled and maintained by the organi- zation and the medium on which it is contained. A common document is specifica- tion, which states any requirements. As discussed in ISO 9000:2015, specifications may be related to processes or products. Regular checks to ensure that the docu- mentation is up to date and still pertains to the existing practice are important. A quality plan is “the specification of the procedures and associated resources to be applied when and by whom to a specific object” (ISO 9000:2015). Quality plans often reference sections in the quality manual. We discuss the quality manual in more detail in the following section, along with document components and docu- ment control.
B.1. document Components
Plans for achieving customer satisfaction and ensuring that the quality of products or services is documented in a quality manual are sometimes referred to as the quality program, which represents the first of two major system efforts: documen- tation and implementation. A quality manual is the “specification for the quality management system of an organization” (ISO 9000:2015). Compliance, accuracy, and clarity are critical characteristics of documentation. A generic quality manual may be viewed as a composite document of four tiers—also known as the docu- mentation hierarchy or pyramid—as illustrated in Figure 2.2. Starting from the top, these tiers are policies, procedures, work instructions, and quality records.
The first tier represents a policy statement, which explains what the com- pany stands for and what its commitments are. This is the opening statement of
Figure 2.2 Tiers of the quality documentation hierarchy.
Document contents
Layer I Statements of the quality policy and objectives
Layer II Description of the activities needed to implement the system
Layer III Detailed work documents
Layer IV Results of implementing the quality system
Policies
Procedures
Work instructions
Quality records
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the quality manual, indicating the management policy and objectives for quality. There is usually a policy statement for each of the requirements of the applicable standard.
The second tier is procedures and provides an overview of how a company conducts its business. Direct yet simple statements indicate who is responsible for what in achieving the requirements. In some cases, the procedures will be in the quality manual, but more often they will be distributed, often online.
The third tier represents work instructions, which spell out the how- to in a clear manner. An organization may choose to include detailed work instructions or exclude proprietary information. Work instructions are documented in many ways, depending on the function at hand. Format examples are available in ISO/TR 10013:2001.
Finally, the fourth tier shows the results obtained by implementing the qual- ity system. These results are documented and maintained to form quality records. These records provide subjective evidence that the system has been implemented and is effective. Records must be maintained for a specified time in a protected format retrievable for analysis.
A quality manual need not be partitioned into four separate parts to include the four tiers; this is only a model. Most likely, all the elements will not even be included in one document. The structure of the manual is best selected based on the nature of the organization and the applicable standard. In a small organiza- tion, a separation between the work instructions and the procedures manuals may not be necessary, as it would be for a large organization. However, if the tiers are separated, it is important to provide cross- references or links between the tiers to ensure effective documentation.
B.2. document Control
Upon the completion of the quality manual, a final review to determine its com- petence, accuracy, and clarity is undertaken. Top management should endorse the contents of the reviewed copy and authorize its release. Authorized copies of the manual should be distributed in total or by section to intended users throughout the organization. Proper distribution and control can be aided, for example, by a dedicated document control function. Parsowith (1995) identifies the following four requirements for proper document control:
1. A process is in place for the generation of documents that includes the writing of the policies and procedures, drawings and specifications or other required documentation, approval of the contents of the documents, and the distribution of the documents
2. Documentation fulfilling the needs of contractual or process requirements is available at all locations in which these functions are performed
3. A process is in place for the control of revisions to or redistribution of documents using the same system as the original document distribution
4. A process is in place for the identification and removal of obsolete documents to ensure against unintended use
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C. QuaLiTy STandardS and oTHEr guidELinES The concept of national standards, which spelled out requirements for how things were to be done in the production of goods and services, grew hand- in-hand with the industrialization of national economies. One of the first national standards was a boiler standard, issued late in the nineteenth century by the American Society of Mechanical Engineers. Over time, different nations adopted standards that, while entirely appropriate to their own national needs, were in conflict with those of other nations, which meant that a company would have to either customize its operations to different national standards (which could become very inefficient and expensive) or forfeit the opportunity to do business in some countries.
These conflicts between different national standards naturally gave rise to the concept of international standards. Around the beginning of the twentieth cen- tury, when electricity became a powerful force in many different countries, an international standards organization, the International Electrotechnical Institute, was established. This organization provided precedence for the later establish- ment of the International Organization for Standardization, familiarly known as ISO. In the 1970s and 1980s, as global commerce became a matter of interest to more than a hundred countries and thousands of companies, a major coop- erative effort of international quality professionals led to the ISO 9000 family of standards. ISO is an independent, nongovernmental international organization based in Geneva, Switzerland, with a membership of over 160 national stan- dards bodies (https://www.iso.org/home.html).
There has been some critique of the effectiveness of the ISO 9000 series and of industry- specific standards because of their focus on formal documentation of the quality system. Montgomery (2013) argues that with the primary focus on docu- mentation, not enough attention is always paid to quality improvement and that ISO certification does not guarantee quality products are produced.
C.1. The iSo 9000 Family
ISO 9000:2015 refers to both a family of three related standards and one of the standards in that family. The purpose of this family is “to assist organizations, of all types and sizes, to implement and operate effective quality management systems”; the family consists of three standards: quality management vocabulary, requirements, and guidelines for performance.
• ISO 9000: Quality management systems—fundamentals and vocabulary provides the fundamental concepts, principles, and terminology of quality management systems
• ISO 9001: Quality management systems—requirements specifies the needed requirements for an organization to provide products that aim to enhance customer satisfaction
• ISO 9004: Quality management systems—guidelines for performance improvements suggests ways to improve organizational performance and customer satisfaction beyond the requirements of ISO 9001
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The ISO 9000 standards are based on seven quality management principles:
1. Customer focus. Understand customer needs, meet their requirements, and strive to exceed their expectations
2. Leadership. Establish unity of purpose and direction and create conditions in which people are engaged in achieving the organization’s quality objectives
3. Engagement of people. Recognize, empower, and enhance competence to help people at all levels have a sense of ownership and involvement in achieving quality objectives
4. Process approach. Understand how interrelated processes combine and affect each other in order to optimize the system
5. Continual improvement. Maintain the ideal of continually improving all aspects of the organization to maintain current performance, to react to internal and external conditions, and to create new opportunities
6. Evidence-based decision making. Facts, evidence, and data analysis provide better objectivity and confidence in decision making
7. Relationship management. All relevant, interested parties influence performance, including suppliers, customers, providers, employees, investors, society as a whole, and so on
ISO standards are reviewed every five years and revised if needed. These revisions are made to adapt to changing environments, to reflect increasingly complex orga- nizations in the global market, and to ensure that new standards reflect the chang- ing needs of all interested parties. The latest edition of the ISO 9000 family was published in September 2015 with ISO 9001:2015 replacing ISO 9001:2008, which replaced ISO 9001:2000 (which itself replaced ISO 9001:1994). The major changes from ISO 9001:2005 to ISO 9001:2015 are summarized by Cianfrani and West (2015) in ISO 9001:2015 Explained.
The following sections are overviews of the three standards that make up what is commonly referred to as the ISO 9000 family.
C.1.a. ISO 9000:2015 Fundamentals and Vocabulary
This document is the language foundation for the entire worldwide system of development, implementation, auditing, and registration of ISO 9001:2015. It proposes a well- defined quality management system (QMS) “based on a frame- work that integrates established fundamental concepts, principles, processes, and resources related to quality, in order to help organizations realize their objectives. . . . Its aim is to increase an organization’s awareness of its duties and commitment in fulfilling the needs and expectations of its customers and interested parties, and in achieving satisfaction with its products and services.”
The standard contains definitions of terms based on many person- years of research and consultation. In all, 146 concepts and their associated terms are organized in conceptual order in the following 14 categories: person or people, organization, activity, process, system, requirement, result, data, information and document, customer, characteristic, determination, action, and audits.
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The standard differentiates between a concept and a term as follows: a concept is a unit of knowledge created by a unique combination of characteristics; a term is the verbal designation of the concept as it applies to a specific field of study.
Two criteria apply to all of the ISO 9000 terms and definitions:
1. Avoid technical language in technical descriptions
2. Employ a coherent and harmonized vocabulary that is understood by all actual and potential users
Because the standards are translated into many different languages, it is critical that all users have the same understanding of what the terms mean. This goal has been accomplished by having representatives from more than 100 countries intimately involved in polishing the definitions. A good example of the need for this kind of language- specific polishing is the term “interested party.” In English, a more common term would be “stakeholder.” However, in some languages, a lit- eral translation of “stakeholder” is “someone holding a stick.” Many such conflicts and ambiguities have been resolved over the years.
C.1.b. ISO 9001:2015 Requirements
This document is the set of requirements that organizations must satisfy in order to achieve ISO 9001 registration or certification. Such registration is required in some industries and highly regarded in many others. Over 140 countries have ISO 9001 registration programs. Many people inadvertently refer to this key document as “ISO 9000:2015.” Remember: ISO 9000 refers to a family of three documents (9000, 9001, and 9004); ISO 9001 refers to the requirements document. ISO 9001:2015 is the only standard in the 9000 family that allows an organization to be certified. When a new edition is published, an organization wishing to remain ISO 9001 certified must update its QMS so that it conforms to the new standard. The orga- nization must then seek certification to the new standard; there is a grace period to allow time for organizations to make all necessary changes.
ISO 9001:2015 has several major changes compared with the previous edition. First, the structure of this standard was changed so that it follows the same overall structure as other ISO management system standards (called High- Level Struc- ture). This restructuring was implemented to make it easier for organizations to unify multiple management systems.
In addition to making the standard more user- friendly for service- and knowledge- based organizations, other major changes include emphasis on the fol- lowing (ISO 9001:2015):
• Leadership engagement
• Organizational risks and opportunities in a structured manner
• Addressing supply chain management more effectively
• Understanding the organization’s context: “one size does not fit all”
• Process-oriented approach
• Preventive action and risk identification and mitigation
• Seeking opportunities for improvement
• Managing processes in a planned manner
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ISO 9001:2015 also updates various terminology used in previous editions of this international standard (e.g., ISO 9001:2008). For example, “products” has been updated to “products and services,” and “purchased product” has been updated to “externally provided products and services.” These changes were made to improve alignment with other management systems standards. Note that there is not a requirement for an organization to apply the structure and terminology of the standard to the organization’s QMS. The standard serves as the list of requirements, not a model for documenting the organization’s policies and objectives.
The standard has 10 major parts, or clauses. The first three are general clauses:
1. Scope. This tells what organization(s), location(s), process(es), product(s), and so on, are covered
2. Normative references. These cite other standards that, by being listed, constitute provisions of the ISO 9001 standard
3. Terms and definitions. Here, reference is made to ISO 9000, which contains all definitions applicable to ISO 9001
The remaining seven are technical clauses and are listed here as they appear in the standard. For further details, consult the standards themselves. The seven techni- cal clauses (clauses 4–10) are the following:
4. Context of the organization
4.1. Understanding the organization and its context
4.2. Understanding the needs and expectations of interested parties
4.3. Determining the scope of the quality management system
4.4. Quality management system and its processes
5. Leadership
5.1. Leadership and commitment
5.2. Policy
5.3. Organizational roles, responsibilities and authorities
6. Planning
6.1. Actions to address risks and opportunities
6.2. Quality objectives and planning to achieve them
6.3. Planning of changes
7. Support
7.1. Resources
7.2. Competence
7.3. Awareness
7.4. Communication
7.5. Documented information
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8. Operation
8.1. Operational planning and control
8.2. Requirements for products and services
8.3. Design and development of products and services
8.4. Control of externally provided processes, products and services
8.5. Production and service provision
8.6. Release of products and services
8.7. Control of nonconforming outputs
9. Performance evaluation
9.1. Monitoring, measurement, analysis and evaluation
9.2. Internal audit
9.3. Management review
10. Improvement
10.1. General
10.2. Nonconformity and corrective action
10.3. Continual improvement
C.1.c. ISO 9004:2009 Managing for the Sustained Success of an Organization— A Quality Management Approach
Whereas ISO 9001 is compliance based, ISO 9004 is improvement based. All the great ideas in ISO 9004 are guidelines, not requirements. There is some controversy in the field of quality on the usefulness of ISO 9004 and it has not been widely adopted. The developers of the ISO 9000 family made ISO 9001 and ISO 9004 completely compatible in structure, so it is easy to follow any of the following three paths:
1. Path A. You have no present wish to be certified to ISO 9001 requirements, but you want to install a powerful QMS now, with the option to go for ISO 9001 certification later. So you build your present quality system on the guidelines of ISO 9004. The actions you take to follow the ISO 9004 guidelines will not cause you trouble if you later seek ISO 9001 certification.
2. Path B. You are presently certified to ISO 9001 but want to upgrade to a more powerful system that leads to performance improvement. Without making any changes in your present QMS, you can start to selectively apply the ISO 9004 guidelines.
3. Path C. You are not certified but want to become certified and also want to put quality improvement procedures in place that are not required for registration. You can work with ISO 9001 and ISO 9004 simultaneously, devoting to each the resources you deem appropriate to achieving your goals.
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ISO 9004:2009 replaces the second edition of this standard (ISO 9004:2000). The major change to this edition is focusing on managing for the sustained success of an organization. Like ISO 9001, ISO 9004:2009 relies on the principles of quality management mentioned earlier. Whereas compliance with 9001 raises the issue of corrective action, the 9004 guidelines suggest how and where to go further in improving performance.
ISO 9004 contains two annexes (not present in 9001) in the form of guides for two differing approaches to improvement. The first annex is a set of guidelines for self- assessment. The second lays out a specific process for self- improvement. Either approach can work. Westcott (2003) expands on both approaches, helps management develop plans to implement ISO 9004, and provides some case studies of successful implementation. The best place to start, of course, is with the ISO 9004 standard itself.
C.2. other Quality Standards
There exist a number of industry- specific standards that may have an impact on the QE. The automobile, telecommunications, and biomedical industries have all created industry- specific standards that emulate ISO 9001 but impose additional requirements. The American automobile industry took the lead in creating its own industry standard with QS-9000. This standard was replaced by ISO 16949:2009. In October 2016, the International Automotive Task Force (IATF) revised this stan- dard as IATF 16949:2016. IATF 16949:2016 is not a stand- alone quality manage- ment standard; rather, it is intended to be used in conjunction with ISO 9001:2015 to define the QMS requirements for automotive- related products. The IATF 16949:2016 standard is now mandatory for many original equipment manufactur- ers (OEMs). The telecommunications quality standard is TL 9000 and the biomedi- cal quality standard is ISO 13485:2016.
These standards, as well as other pertinent publications, are available from ASQ Quality Press by phoning 800-248-1946 or visiting the website http://www. asq.org/quality-press. Commentary and offers of assistance on all three of these standards can be found by entering the standard number into a Web browser search engine. For an overview of quality standards, see Boulanger, Johnson, and Luko (2012).
C.3. Malcolm Baldrige national Quality award
The Malcolm Baldrige National Quality Award (MBNQA), another quality man- agement approach, emphasizes results rather than procedures or requirements. Congress established this award in 1987 to recognize US organizations for their achievements in quality and business performance and to raise awareness about the importance of quality and performance excellence as a competitive edge. The award is named in honor of Malcolm Baldrige, who was secretary of commerce at the time of his death in 1987. The award is not given for specific products or ser- vices. Awards may be given annually in each of these categories: manufacturing, service, small business, education, healthcare, nonprofit, and government.
While the Baldrige Award and the Baldrige recipients make up the very visible centerpiece of the US quality movement, a broader national quality program has evolved around the award and its criteria. A report, Building on Baldrige: American
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Quality for the 21st Century, by the private Council on Competitiveness said, “More than any other program, the Baldrige Quality Award is responsible for making qual- ity a national priority and disseminating best practices across the United States.”
The US Commerce Department’s National Institute of Standards and Technol- ogy (NIST) manages the MBNQA program in close cooperation with the private sector. Since its inception, the MBNQA has received over 1600 applications. As of 2015, over 100 organizations from a wide variety of industries have received the award, with seven organizations winning the award twice.
The MBNQA is awarded according to these criteria for performance excellence:
1. Leadership. Examines how senior executives guide the organization and how the organization addresses its responsibilities to the public and practices good citizenship.
2. Strategic planning. Examines how the organization sets strategic directions and how it determines key action plans.
3. Customer and market focus. Examines how the organization determines requirements and expectations of customers and markets, and how it builds and maintains strong, lasting relationships with customers.
4. Measurement, analysis, and knowledge management. Examines the management, effective use, and analysis of data and information to support key organization processes and the organization’s performance management system.
5. Human resource focus. Examines how the organization enables its workforce to develop its full potential and how the workforce is aligned with the organization’s objectives.
6. Process management. Examines aspects of how key production/ delivery and support processes are designed, managed, and improved.
7. Business/organizational performance results. Examines the organization’s performance and improvement in its key business areas: customer satisfaction, financial and marketplace performance, human resources, supplier and partner performance, and operational performance. This category also examines how the organization performs relative to competitors.
Further information about the MBNQA, including procedures for ordering the criteria, is available at http://www.nist.gov/baldrige.
d. QuaLiTy audiTS A quality system audit, as defined by The ASQ Auditing Handbook, is a “systematic, independent, and documented process for obtaining audit evidence and evaluat- ing it objectively to determine the extent to which the audit criteria are fulfilled” (Russell 2013). It is a fact- finding process that compares actual results with speci- fied standards and plans. It provides feedback for improvement. It differs from inspection, which emphasizes acceptance or rejection, and differs from surveil- lance, which is ongoing continuous monitoring.
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d.1. Types of audits
Quality audits may be classified according to the party conducting them, their scope, and the audit method used. In general, three parties are involved in an audit: (1) the organization requesting the audit (the client), (2) the party conduct- ing the audit (the auditor), and (3) the organization to be audited (the auditee).
When the auditor is an employee of the organization being audited (auditee), the audit is classified as an internal quality audit. For the purposes of maintain- ing objectivity and minimizing bias, internal auditors must be independent from the activity being audited. On the other hand, when the auditors are employees of the client or an independent organization or third party hired for the purpose, the audit is classified as an external quality audit. In this case, the auditors are clearly independent of the auditee and are in a position to provide the client with an unbiased, objective assessment. This type of audit is required to permit list- ing in a register or to meet mandatory quality requirements. However, the time required and costs involved in an external audit are much higher compared with internal audits.
Another way to classify quality audits is by scope and extent. An audit may be as comprehensive as needed or requested by the client. The most comprehensive type of audit is the quality system audit, which examines suitability and effec- tiveness of the system as a whole. This audit involves both the documentation and implementation aspects of the quality system. Reasons for initiating a system audit may range from evaluating a potential supplier to verifying an organiza- tion’s own system. Audits of specific elements of a system, processes, products, or services are also possible. These audits are limited in scope and are typically referred to with a modifier preceding the term “quality audit.” Examples include process quality audits and product quality audits.
The method by which the quality audit is conducted provides yet another way to classify audits. Audits may be conducted by location or function. A location- oriented audit provides an in- depth examination of all the quality- related activities within a given location. In a function- oriented audit, an activity is exam- ined in all the locations where the activity is carried out.
It is important to note that these classifications are not mutually exclusive and that, in practice, cross- classifications of a quality audit are possible.
ISO 19011:2011 details guidelines for auditing management systems. ANSI/ASQ QE 19011S is a supplement to 19011:2011 and provides additional guidance for auditing QMSs. The following four purposes of quality audits are listed in ANSI/ ASQ QE 19011S:
1. To meet requirements for certification to a management system standard
2. To verify conformance with contractual requirements
3. To obtain and maintain confidence in the capability of a supplier
4. To contribute to the improvement of the management system
d.2. roles and responsibilities in audits
Each of the three parties involved in an audit—the client, the auditor, and the auditee—plays a role that contributes to its success. The client, the party that
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initiates the audit, selects the auditor and determines the reference standard to be used. The client, typically the end user of the audit results, determines the type of audit needed (system, process, product, etc.) as well as its time and duration.
The selected auditor, whether an individual or a group, needs to adhere to the role of a third party. That is, the auditor must maintain objectivity and avoid bias in conducting the audit. The auditor must comply with any confidentiality requirements mandated by the auditee. An experienced individual is appointed as lead auditor to communicate audit requirements, manage the auditing activities, and report the results. For rules, qualifications, and evaluation criteria for an audi- tor, see ASQ/ANSI/ISO QE19011:2011 or The ASQ Auditing Handbook.
Finally, the auditee has the responsibility of accommodating the audit, which entails providing the auditors access to the facilities involved and copies of all rele- vant documentation. The auditee is also expected to provide the resources needed and select staff members to accompany the auditors.
d.3. audit Planning and implementation
Proper planning is a key factor in achieving an efficient quality audit. Planning should be conducted with consideration of the client expectations. This includes the scope, depth, and time frame. The lead auditor has the responsibility of plan- ning and conducting the audit and should be authorized to perform these activities.
Planning an audit, just like any other activity, should address the questions of what, when, how, and who. That is, what elements of the quality system are to be audited? Against what document or reference standard should the quality system be audited? The answers to both questions are determined by the client and should be communicated clearly to the auditee. A schedule of the audit activi- ties needs to be prepared and communicated to both the client and the auditee. It is the lead auditor’s responsibility to inform the client of any delays, report their reasons, and update the completion date of the audit.
The method of conducting the audit should also be addressed. Working docu- ments need to be prepared, including checklists of the elements to examine, ques- tions to ask, and activities to monitor. A number of references provide generic checklists that can be used as templates. However, it is best to design a checklist to suit the audit at hand and its specific scope and objectives. Forms for collecting auditors’ observations and the supporting evidence should also be included in the working document. Working documents typically are reviewed by an expe- rienced auditor and approved by the lead auditor before implementation. It is recommended that the auditor explain the methods planned to the auditee. This should help the organization better prepare for the audit and ease the fear usually attached to the process.
The question of who will examine specific elements, processes, or products addresses the qualifications and experience of the individual auditors (assessors) needed. With the client expectations in mind, the lead auditor should assign the various tasks among his or her team.
An audit is usually conducted in three steps. The first step is a preexamination or opening meeting with the auditee that marks the beginning of the process. Dur- ing this meeting, the lead auditor introduces team members to the senior manage- ment of the auditee and explains the objectives of the audit and the methods used. The auditee is represented by selected members of the organization who facilitate
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and assist in the process and submit a documented description of the quality sys- tem or element to be examined. Issues regarding proprietary information typically are addressed and resolved before starting the audit.
The second step involves a suitability audit of the documented procedures against the selected reference standard. Observed nonconformities at this stage of the audit should be reported to both the client and the auditee for immediate action. The auditing process should pause to allow for corrective measures.
For the third step, the auditor examines the implementation of the quality system in depth. The auditor maintains records of all nonconformities observed and the supporting data. Provisions should be made in the audit plan to allow additional investigation of clues suggesting nonconformities revealed by the data collected. The auditee management should be made aware of and acknowledge all the nonconformities observed during the audit. This step concludes with a closing meeting with the auditee’s management for a presentation of findings. In some cases, the auditor may be required to recommend corrective measures for improv- ing the system. However, it is up to the auditee to plan and implement these mea- sures in a way that best suits the organization.
d.4. audit reporting and Follow- up
A final report is submitted to the client indicating the facts of the audit and conclu- sions regarding the ability of the subject system, element, process, or product to achieve quality objectives. Proper planning and execution of the audit facilitates the preparation of this report and provides data to support its conclusions. The lead auditor is responsible for the accuracy of the report and the validity of its con- clusions. The report should be submitted to the client, who in turn is responsible for providing a copy to the auditee.
The audit final report should include, at a minimum, the following:
1. Type of audit conducted
2. Objectives of audit
3. Identification of involved parties: auditor, auditee, and third party
4. Audit team members
5. Critical nonconformities and other observations
6. Audit standards and reference documents used
7. Determination of proper corrective action(s)
8. Duration of audit
9. Audit report distribution and date
10. Audit results and recommendations
11. Audit-related records
Should the auditee initiate improvement efforts to correct nonconformities, the three parties should agree on a follow- up audit to verify the results. The plan, audit, report, and improve cycle may be repeated whenever systems and/or requirements change. The results attained provide a measure of the effectiveness
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of the audit. Improvement efforts also should be directed to identifying and elimi- nating the root causes of reported nonconformities and identifying the corrective action(s) to be taken. Root causes represent the main reason behind the occurrence of a nonconformance or an undesirable condition or status. These corrective actions may then be validated by performing tests, inspections, or even more audits.
E. CoST oF QuaLiTy (CoQ) To achieve the most effective improvement efforts, management should ensure that the organization has ingrained in its operating principles the understanding that quality, speed, and cost are complementary, and not conflicting, objectives. Traditionally, recommendations made to management were choices between qual- ity, speed, and cost, where they could pick two of the three. Experience throughout the world has shown, and management is beginning to see, that this is not true. Good quality leads to increased productivity and reduced quality costs, and even- tually to increased sales, market penetration, and profits.
The facts about quality management and quality costs show that the real value of a quality program is determined by its ability to contribute to customer satis- faction and profits. Quality cost techniques provide tools for management in its pursuit of customer satisfaction, quality improvement, and profit contributions.
The purpose of cost of quality (COQ) techniques is to provide a tool to man- agement for facilitating quality program and quality improvement activities. Quality cost reports can be used to point out the strengths and weaknesses of a quality system. Improvement teams can use COQ reports to describe the mon- etary benefits and ramifications of proposed changes. Return- on-investment (ROI) models and other financial analyses can be constructed directly from quality cost data to justify proposals to management. Improvement team members can use this information to rank problems in order of priority. In practice, quality costs can define the activities of quality program and quality improvement efforts in a language that management can understand and act on: dollars. Any reduction in quality costs will have a direct impact on gross profit margins and can be counted immediately as pretax profit.
E.1. The Economics of Quality
The expression “the economics of quality” has contributed to some confusion sur- rounding the true business and economic value of quality management. Some people believe there is no economics of quality; that is, it is never economical to ignore quality. At the other extreme are those managers who believe it is uneco- nomical to have 100% quality.
Whether for manufacturing or service, a quality cost program will lend cre- dence to the business value of the quality management program and provide cost justification for the corrective actions demanded. Quality cost measurements pro- vide guidance to the quality management program much as the cost accounting system does for general management. Quality cost measurements define and quan- tify those costs that are directly affected, both positively and negatively, by the qual- ity management program, thus allowing quality to be managed more effectively.
Simply stated, quality costs are a measure of the costs specifically associated with the achievement or nonachievement of product or service quality, including
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all product or service requirements established by the company and its contracts with customers and society. More specifically, quality costs are the total of the costs incurred by (1) investing in the prevention of nonconformances to requirements (prevention costs), (2) appraising a product or service for conformance to require- ments (appraisal costs), and (3) failure to meet requirements (failure costs). Qual- ity costs represent the difference between the actual cost of a product or service and what the reduced cost would be if there were no possibility of substandard service, failure of products, or defects in their manufacture.
Every company lives with significant costs that fit this description. Unfortu- nately, significant chunks of quality costs are normally overlooked or unrecog- nized simply because most accounting systems are not designed to identify them. As this is generally the case, it is not too difficult to understand why top manage- ment of most companies is more sensitive to overall cost and schedule than to quality. The interrelationship of quality, schedule, and cost is likely to be unbal- anced in favor of schedule and cost, and often unwittingly at the expense of qual- ity. This imbalance will continue to exist as long as the real COQ remains hidden among total costs. In fact, such a condition can easily set the stage for a still greater imbalance whenever the rising, but hidden, true COQ grows to a magnitude that can significantly affect a company’s competitive position.
When the COQ rises without constraint, or is tolerated at too high a level, failure to expose the condition is a sign of ineffective management. Yet, it is entirely possible for this condition to exist without top management’s awareness. A quality cost program can provide specific warnings of oncoming dangerous quality- related financial situations. An argument for needed quality improve- ment is clear when a company suddenly finds itself in serious, expensive quality trouble.
On the premise that any dollar expenditure that could have been avoided will have a direct negative effect on profits, the value of clearly identifying the COQ should be obvious. Achieving this clarity of identification, however, is more eas- ily said than done. A real danger lies in finding and collecting on a small por- tion of the costs involved and assuming it represents the total. There are as many ways of hiding costs in industry as there are people with imagination. This is an all- too-natural phenomenon in organizations that are never fully charged with all inefficiencies (because some inefficiencies are hidden and not measured) and thus are able to maintain an illusion of effective management.
E.2. goal of a Quality Cost System
The goal of any quality cost system is to facilitate quality improvement efforts that will lead to opportunities to reduce operating costs. The strategy for using quality costs is quite simple: (1) directly attack failure costs in an attempt to drive them to zero, (2) invest in the right prevention activities to bring about improvement, (3) reduce appraisal costs according to results achieved, and (4) continually evalu- ate and redirect prevention efforts to gain further improvement.
This strategy is based on the premise that:
• For each failure there is a root cause
• Causes are preventable
• Prevention is generally cheaper
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In a practical sense, real quality costs can be measured and then reduced through the proper analysis of cause and effect. As failures are revealed through appraisal actions or customer complaints, they are examined for root causes and eliminated through corrective action. The further along in the operating process that a fail- ure is discovered (and thus the nearer to product or service use by the customer), the more expensive it is to correct. Usually as failure costs are reduced, appraisal efforts also can be reduced in a statistically sound manner. The knowledge gained from this improvement can then be applied, through prevention activities or dis- ciplines, to all new work. By minimizing quality costs, quality performance levels can be improved.
E.3. Management of Quality Costs
Managing quality costs begins with a general understanding and belief that improving quality performance and improving quality costs are synonymous (the economics of quality). The next step is recognizing that measurable quality improvement also can have a tangible effect on other business measures, such as sales and market share. The caveat, however, is that quality costs must be mea- sured and must reflect cost or lost opportunities to the company.
It should be further understood that COQ is a comprehensive system, not a piecemeal tool. There is a danger in responding to a customer problem only with added internal operations, such as inspections or tests. For service operations, this could mean more operators. While this may solve the immediate customer prob- lem, the added costs may, in fact, destroy profit potential. A comprehensive qual- ity management program will force the analysis of all associated quality costs, making these added internal costs appear clearly as just one step toward the ulti- mate resolution: prevention of the root cause of the problem. Quality costs should, therefore, become an integral part of any quality management program and, in turn, any quality system or quality improvement activity. Overall quality cost data will point out the potential for improvement and provide management with the basis for measuring the improvement accomplished.
E.4. Quality Cost Categories
To manage quality costs, they must be categorized. The three major categories commonly used are prevention costs, appraisal costs, and failure costs.
Prevention costs are the costs of all activities specifically designed to prevent poor quality in products or services. Examples are the costs of quality planning, training programs, and quality improvement projects.
Appraisal costs are the costs associated with measuring, evaluating, or audit- ing products or services to ensure conformance with quality standards and per- formance requirements. These include the costs of inspection, testing, product or service audits, process audits, and calibration of measuring and test equipment.
Failure costs are those costs resulting from products or services not conform- ing to requirements or customer needs. They are usually divided into two types, internal and external. Internal failure costs occur prior to delivery or shipment of the product or furnishing of a service to the customer, such as the costs of scrap, rework, material review, and so on. External failure costs occur after delivery of
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the product and during or after furnishing of a service to the customer. Examples include the costs of processing customer complaints, customer returns, warranty claims, and product recalls. See Table 2.1 for an example list of quality cost ele- ments by category.
Total quality cost is the sum of these costs (prevention, appraisal, and failure) and represents the difference between the actual cost of a product or service and what the reduced cost would be if there were no possibility of substandard service, failure of products, or defects in their manufacture. It is, according to Joseph Juran, “gold in the mine,” waiting to be extracted (Juran and Godfrey 1999). When you zero in on the elimination of failure costs and then challenge the level of appraisal costs, you will not only be managing COQ but mining gold.
E.5. Quality Cost implementation
To implement a quality cost program, the need for the program must first be deter- mined. This need should be presented to members of management in a way that will justify the effort and interest them in participating. One way to do this is by establishing a simple trial program. For this purpose, only major costs need to be gathered and only readily available data need to be included. Much of the required data may already be available. If necessary, some of these costs may even be estimated.
When setting up the trial program, there is no need to do everything immedi- ately. Select a program, facility, or area of particular interest to management. The results should be sufficient to sell management on the need for the program.
With management sold and the accounting department recruited to assist in the quality cost program, the specific quality costs to be collected must be deter- mined. To do this, tasks must be classified as prevention, appraisal, or failure and listed together with the departments responsible for them. Remember that the quality department isn’t the only department to incur quality costs. To determine the prevention costs in the effort to prevent poor quality, such tasks performed in the company should be listed together with the departments responsible for those tasks. In a like manner, appraisal cost elements are determined by listing those tasks associated with the inspection or test of products or services for the detec- tion of poor quality. For failure costs, determine those costs that would not have been expended if quality were perfect. If quality were perfect, there would not be any rework, customer complaints needing response, or need for corrective action. Remember to also divide failure costs into internal and external categories.
Quality cost elements may differ from company to company and particularly from industry to industry. However, the overall categories of prevention, appraisal, and failure are always the same.
E.6. Quality Cost Collection
Now that the specific costs to be collected have been decided, a method to collect them must be developed. Collection of quality costs should be the responsibility of the controller. The finance and accounting department is the cost collection agency of the company. In addition, having the controller collect the costs adds credibility to the data.
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Table 2.1 Quality cost elements by category. (Continued)
1.0 Prevention Costs 1.4.2.1 Design and Development of Quality Measurement and Equipment
1.1 Marketing/Customer/User 1.4.3 Operations Support Quality Planning
1.1.1 Marketing Research 1.4.4 Operator Quality Education
1.1.2 Customer/User Perception Surveys/Clinics 1.4.5 Operator SPC/Process Control
1.1.3 Contract/Document Review 1.5 Quality Administration
1.2 Product/Service/Design Development 1.5.1 Administrative Salaries
1.2.1 Design Quality Progress Reviews 1.5.2 Administrative Expenses
1.2.2 Design Support Activities 1.5.3 Quality Program Planning
1.2.3 Product Design Qualification Test 1.5.4 Quality Performance Reporting
1.2.4 Service Design Qualification 1.5.5 Quality Education
1.2.5 Field Trials 1.5.6 Quality Improvement
1.3 Purchasing Prevention Costs 1.5.7 Quality System Audits
1.3.1 Supplier Reviews 1.6 Other Prevention Costs
1.3.2 Supplier Rating
1.3.3 Purchase Order Tech Data Reviews 2.0 Appraisal Costs
1.3.4 Supplier Quality Planning 2.1 Purchasing Appraisal Costs
1.4 Operations (Manufacturing or Service) Prevention Costs 2.1.1 Receiving or Incoming Inspections and Tests
1.4.l Operations Process Validation 2.1.2 Measurement Equipment
1.4.2 Operations Quality Planning 2.1.3 Qualification of Supplier Product
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Table 2.1 Quality cost elements by category. (Continued)
2.1.4 Source Inspection and Control Programs 3.0 Internal Failure Costs
2.2 Operations (Manufacturing or Service) Appraisal Costs 3.1 Product/Service Design Failure Costs (Internal)
2.2.1 Planned Operations Inspections, Tests, Audits 3.1.1 Design Corrective Action
2.2.1.1 Checking Labor 3.1.2 Rework Due to Design Changes
2.2.1.2 Product or Service Quality Audits 3.1.3 Scrap Due to Design Changes
2.2.1.3 Inspection and Test Materials 3.1.4 Production Liaison Costs
2.2.2 Set-Up Inspections and Tests 3.2 Purchasing Failure Costs
2.2.3 Special Tests (Manufacturing) 3.2.1 Purchased Material Reject Disposition Costs
2.2.4 Process Control Measurements 3.2.2 Purchased Material Replacement Costs
2.2.5 Laboratory Support 3.2.3 Supplier Corrective Action
2.2.6 Measurement (Inspection and Test) Equipment 3.2.4 Rework of Supplier Rejects
2.2.6.1 Depreciation Allowances 3.2.5 Uncontrolled Material Losses
2.2.6.2 Measurement Equipment Expenses 3.3 Operations (Product or Service) Failure Costs
2.2.6.3 Maintenance and Calibration Labor 3.3.1 Material Review and Corrective Action Costs
2.2.7 Outside Endorsements and Certifications Control 3.3.1.1 Disposition Costs
2.3 External Appraisal Costs 3.3.1.2 Troubleshooting or Failure Analysis Costs (Operations)
2.3.1 Field Performance Evaluation 3.3.1.3 Investigation Support Costs
2.3.2 Special Product Evaluations 3.3.1.4 Operations Corrective Action
2.3.3 Evaluation of Field Stock and Spare Parts 3.3.2 Operations Rework and Repair Costs
2.4 Review of Test and Inspection Data 3.3.2.1 Rework
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Table 2.1 Quality cost elements by category. (Continued)
3.3.2.2 Repair 4.0 External Failure Costs
3.3.3 Reinspection/Retest Costs 4.1 Complaint Investigations/Customer or User Service
3.3.4 Extra Operations 4.2 Returned Goods
3.3.5 Scrap Costs (Operations) 4.3 Retrofit Costs
3.3.6 Downgraded End-Product or Service 4.3.1 Recall Costs
3.3.7 Internal Failure Labor Losses 4.4 Warranty Claims
3.4 Other Internal Failure Costs 4.5 Liability Costs
4.6 Penalties
4.7 Customer/User Goodwill
4.8 Lost Sales
4.9 Other External Failure Costs
Source: ASQ Quality Costs Committee, Principles of Quality Costs: Principles, Implementation, and Use, 3rd ed., edited by Jack Campanella (Milwaukee, WI: ASQ Quality Press, 1999).
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If top management is properly sold on the program, the controller will be charged with the task of heading this effort. With the help of the quality man- ager, the controller should review the list of costs to be collected, determine which of these are already available under the existing accounting system, and decide where additions to the existing system are needed. Sometimes, the simple addition of new cost element codes to the present charging system is sufficient. However, if necessary, the present system could be supplemented by separate inputs designed specifically for this purpose.
Ideally, a complete system of cost element codes would be generated. The sys- tem could be coded in such a way that the costs of prevention, appraisal, and inter- nal and external failures could be easily distinguished and sorted (see Table 2.1). Then, these codes could be entered into the labor cost collection system, together with the hours expended against the cost element or task represented by the code. The labor hours could later be easily converted to dollars.
Scrap is an exception to this system of collecting quality costs as they are incurred. All work needs to be inspected, rejected, and dispositioned first. In many companies, the existing scrap reporting documents are forwarded to estimating, where the costs of expended labor and material are estimated to the stage of com- pletion of the scrapped items.
The accounting department should provide all collected quality costs to the quality function in a format suitable for analysis and reporting. Of course, training programs will be necessary to ensure that all personnel are informed as to how to report their quality cost expenditures. The training should be repeated periodi- cally, and the collection system should be audited on a regular basis.
E.7. Quality Cost Summary and analysis
Quality costs can be summarized in many ways, such as by company, division, facility, department, or shop. They may be summarized by program, type of pro- gram, or all programs combined. The best way to summarize these costs is accord- ing to the specific needs of the organization.
Analysis can include comparison of the total quality cost with an appropriate measurement base. Some commonly used bases are sales, cost input, and direct labor. The base selected will depend on what is appropriate for the needs of the organization. Comparing quality costs with a measurement base will relate the COQ to the amount of work performed. An increase in quality costs with a pro- portionate increase in the base is normal. It is the non- proportionate change that should be of interest. The index “total quality cost over the measurement base” is the factor analyzed. The goal is to bring this index to a minimum through quality improvement. The index may be plotted so that trends representing present status in relation to past performance and future goals may be analyzed.
Other methods of analysis include study of the effect that changes in one cat- egory have on the other categories and on the total quality cost. For example, was the increase in prevention costs effective in reducing failure costs? And was this reduction in failure costs sufficient to cause a reduction in total quality costs? This technique can provide insight into where the quality dollar can most wisely be spent. Increases in failure costs must be investigated to determine where costs must be expended to reverse a trend and reduce the total quality cost. Losses
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must be defined, their causes identified, and corrective action taken to preclude recurrence.
Other existing quality systems, such as a defect reporting system, can be used in conjunction with the quality cost program to identify significant problems. The defect reporting system can help define the causes of scrap, rework, and other failure costs. While the losses are distributed among many causes, they are not uniformly distributed. A small percentage of the causes will account for a high percentage of the losses. This concept is the Pareto principle, where these causes are the vital few as opposed to the trivial many. Refer to Chapter 5, section A, for more information on the Pareto principle and Pareto charts. Concentration on pre- vention of the vital few causes will achieve maximum improvement at a minimum of cost. This quality improvement tool will have the effect of improving quality while reducing costs.
E.8. Quality Cost reporting
There are almost as many ways to report quality costs as there are companies reporting them, because how they are reported depends on who they are reported to and what the report is trying to say. The amount of detail included in the quality cost report generally depends on the level of management the report is geared to.
To top management, the report might be a scorecard depicting the status of the quality program through a few carefully selected trend charts—where the quality program has been and the direction it is heading. Savings over the report period and opportunities for future savings might be identified. To middle management, the report might provide quality cost trends by department or shop to enable iden- tification of areas in need of improvement. Reports to line management might provide detailed cost information, perhaps the results of a Pareto analysis identify- ing specific areas where corrective action would afford the greatest improvement. Scrap and rework costs by shop also provide valuable information when included in reports to line management.
E.9. using Quality Costs
Once the quality cost program is implemented, it should be used by management to justify and support improvement in each major area of product or service activ- ity. Quality costs should be reviewed for each major product line, manufacturing area, service area, and cost center. The improvement potential that exists in each individual area can then be looked at and meaningful goals can be established. The quality cost system then becomes an integral part of quality measurement. The proper balance is to establish improvement efforts at the level necessary to effectively reduce the total COQ. As progress is achieved, adjust improvement efforts to where total quality costs are at the lowest attainable level. This approach prevents unheeded growth in quality costs and creates improved overall quality performance, reputation, and profits.
Another benefit to be gained from a quality cost program is its ability to be used as a budgeting tool. As costs are collected against quality cost elements, a his- tory of costs is generated. This history can then be used to determine the average cost per element. In other words, depending on how detailed the elements are that have been established, the organization can identify what it has been spending for various functions or tasks. This information can be used as the basis for future
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quotes and estimates. Budgets can be established for each element. Then, going full circle, the actuals collected against these elements can be bounced against the budget amounts to determine budget variances. Action can then be initiated to bring over- or underrunning elements into line.
The key factor in the reduction of quality costs is quality improvement, and a key factor in quality improvement is corrective action. Quality costs do not reduce themselves. They are merely the scorecard. They can tell you where you are and where your corrective action dollar will afford the greatest return. Quality costs identify targets for corrective action.
Once a target for corrective action is identified, through Pareto or other meth- ods of quality cost analysis, the action necessary must be carefully determined. It must be individually justified on the basis of an equitable cost trade- off. You do not want to resolve a $500 problem with a $5000 solution. At this point, experience in measuring quality costs will be invaluable for estimating the payback on indi- vidual corrective action investments or quality improvement projects. Cost– benefit justification of corrective action and quality improvement projects should be a continuing part of the quality management program.
Some problems have fairly obvious solutions, such as the replacement of a worn bearing or a worn tool, which can be fixed immediately. Others are not so obvious, such as a marginal condition in design or processing, and are almost never discovered and corrected without the benefit of a well- organized and formal approach. Marginal conditions usually result in problems that can easily become lost in the accepted cost of doing business. Having an organized quality improve- ment program and corrective action system, justified by quality costs, will reveal such problems to management. The true value of corrective action is that you only have to pay for it once, whereas failure to take corrective action may be paid for repeatedly.
E.10. Quality Cost Principles and Lessons
Traditional quality cost methods have been around a long time. These principles still apply today and will apply for the foreseeable future. However, through our experiences with quality costs over time, we’ve identified some useful lessons learned that can be applied in the future.
The first lesson is that speaking the language of money is essential. For a suc- cessful quality effort, the single most important element is leadership by upper management. To gain that leadership, some concepts or tools could be proposed, but that is the wrong approach. Instead, management should first be convinced that a problem exists that requires its attention and action, such as excessive costs due to poor quality. A quality cost study, particularly when coupled with a success- ful pilot quality improvement project, is a solid way to gain management support for a broad quality improvement effort. Excessive costs and loss of sales revenue are both quality- related hot buttons for management.
The second lesson learned is that quality cost measurement and publication do not solve quality problems, unless they are used. Improvement projects must be identified, clear responsibilities established, and resources provided to diag- nose and remove the cause of problems, as well as other essential steps. New orga- nizational machinery is needed to attack and reduce the high costs of poor quality.
The third lesson is that the scope of traditional quality costs should be expanded. Traditionally, quality costs have emphasized the cost of nonconformities.
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Important as this cost is, we also need to estimate the cost of inefficient processes. This includes variation of product characteristics (even on conforming products), redundant operations, sorting inspections, and other forms of non- value-added activities. Another area to be considered is the cost of lost opportunities for sales revenue.
The fourth lesson is that the traditional categories of quality costs have had remarkable longevity. About 50 years ago, some pioneers proposed that quality costs be assigned the categories of prevention, appraisal, and failure. Many practi- tioners found the categories useful and even devised ingenious ways to adapt the categories beyond manufacturing, as in engineering design, and also to the service sector, as in financial services and healthcare. The principles still work today. The difference is in their additional applications.
Quality costs have expanded to become a principal management and quality improvement tool. Definitions and standards have been developed and refined along with techniques and methods for implementation. Quality cost principles and concepts have been expanded to include lessons learned over the past half century, with applications now including the software and service sectors. The quality cost program is the bridge between line and executive management. It pro- vides a common language and a measurement and evaluation system that shows how quality pays in increased profits, productivity, and customer acceptance.
F. QuaLiTy Training To keep an organization healthy, its people must be continually trained in new concepts and techniques. Frank Gryna epitomized the role of QE as leader and trainer. In addition to the college students he helped educate and the adult work- ers he trained, he inspired hundreds of professionals to become passionate, skill- ful carriers of the quality torch as trainers. Gryna (1988b) stated, “The need is to extend training in quality- related matters to personnel in all functions.” He stated that while U.S. companies (formerly) trained only quality- related specialists in quality, the Japanese, from the very beginning of their industrial renaissance in 1946, targeted all departments, and that “this difference in training contributed to a quality crisis” for the United States.
Training may be formal or informal, large scale or small. Although it is usually delivered by instructors in classrooms, training can be done through self- directed study with workbooks or as an online course. A powerful training method in the areas of intellectual skills and human relations is mentoring. A mentor pro- vides guidance, inspiration, and motivation. Likewise, well- structured improve- ment projects can include a training component. Project leaders are given special instructions to ensure that team members learn various skills as they complete the project. For details on facilitation, see Chapter 1, section E.
If the training task is small, it may involve only two people—for example, an apprentice working alongside a master. At the other extreme it may require a fully staffed training and development department with a budget in the millions. In some companies, such as McDonald’s and Motorola, this department is called a corporate university, and it manages dozens of instructors delivering courses, seminars, exercises, and workshops year- round. The middle ground would be a QE organizing a small training program or helping a task team build a larger one.
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Regardless of the size of the training program, the same development process is used. Many training programs are organized in five phases, as follows:
1. Assess the need for training
2. Design a curriculum, or training plan
3. Develop the lesson plans and training materials
4. Implement the plan (i.e., deliver the instruction)
5. Evaluate effectiveness of the training
This is known as the ADDIE model, so- called because it uses the first letter of the first word of each phase. Some have proposed a seven- phase model— beginning with “customer identification” and ending with “maintain benefi- cial outcomes”—resulting in the CADDIEM model. Many other variations can also be used successfully.
If large- scale training is contemplated, spend more time than you think nec- essary in assessing the needs. Stay open- minded as to possibilities. A thorough needs analysis can prevent wasting large amounts of time and money on ineffec- tive or inadequate training. For example, you might start out thinking a training program is needed, only to discover through needs assessment that your problem is solved by organizational realignment.
The methods of quality function deployment (QFD) are very suitable to needs assessment. One key QFD tool is the matrix chart, which can be used to organize the results of surveys and focus groups. See Chapter 1, section G, for more details about QFD.
The curriculum flows naturally from the needs analysis. It states learning objectives and how they will be achieved in each training event, whether by lec- turer, demonstration, role- play, or other method. Suppliers, customers, and even competitors may share their curriculum ideas. Training professionals will need to partner with subject matter experts in this phase because of its highly technical and subject- specific nature.
Once the curriculum is set, the lesson plans flow rather naturally. There are numerous issues to resolve, such as detailed content, sequencing, depth, breadth, review points, quizzes, and demonstrations.
Finding suitable materials is not a problem. However, selecting from the pleth- ora of artifacts, paper and electronic media, simulations, workbooks, role- plays, and so on, might be a problem. Remember that training can be structured in many ways. Formal classrooms may be appropriate, but often the best training area is actually the workplace, with carefully designed instructional aids so that training occurs as the work is done. Creativity and imagination are definitely in order.
For developing intellectual skills such as facilitation and conflict resolution, role- playing is highly recommended. It is a good way to increase the learners’ involvement in the process. Designing good role- plays can be time- consuming, but Stolovitch and Keeps (1992) explained how role- plays and simulations can be developed during the learning process in certain situations. Book discussion groups can focus management attention on timely issues with little or no development cost.
A rapidly growing body of training material is now available electronically, through video modules and online material. A number of different organiza- tions offer products and assistance in this exciting new area of training materials.
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Webinars hosted by companies and divisions of ASQ are also popular methods for on- the-job training. In addition, using search engines such as Google and Yahoo will yield many more electronic training materials.
Both technical competence and teaching skills are mandatory! An experienced QE might need special “how to teach” training, since technical competence does not ensure teaching skill. “Train the trainer” programs can be anything from a colleague giving informal mentoring to an eight- week off- site course. Remember, students must respect both teaching ability and competence of the instructor in order for learning to take place.
Most training programs are directed toward adults. Knowles (1996) pointed out significant facts about how adults learn. Three of his conclusions are the following:
1. Adults decide for themselves what is and isn’t important to learn. In order to ensure learning takes place, the trainer should state specifically what should be learned and how it relates to on- the-job performance.
2. Adults buy into training when it is supported on the job by supervisors and management.
3. Adults who are happy in their jobs are more receptive to training, and adults who are well trained for their jobs are happier employees.
According to Gryna (1988b), there are 10 reasons why training programs fail:
1. Cultural resistance by line managers
2. Doubt as to the usefulness of the training
3. Lack of participation by line managers
4. Technique rather than problem orientation
5. Inadequacy of leader/instructor
6. Mixing of participant levels
7. Lack of application during the course
8. Overly complex language
9. Lack of participation by the training function
10. Operational and logistical deficiencies
Consider reason #7 in more detail. To quote Gryna, “The ideal approach is to design the course so that the participants must apply the training during the course. One of the best learning experiences is the application of the material being taught. This was successfully done during World War II in the area of work- simplification pro- grams. More recently, value engineering seminars often have a project included as part of the seminar. Quality circles also use the concept.” In the years since Gryna published those words, the scope and effectiveness of training for quality in the United States have greatly increased.
Referring to the ADDIE model, evaluation (step 5) and needs analysis (step 1) are closely related. Both are learning experiences for the training team. It is very important for credibility to demonstrate that something useful has been accom- plished. And if the outcome was poor, you need to know quickly in order to take immediate corrective action.
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Some informal evaluation can occur while the course is under way. Especially if the instructor is new, a trusted colleague or mentor can help with discussions and role- play exercises and can visit with attendees during breaks. The single most prevalent (and probably most cost- effective) form of evaluation is the pre- and posttest of knowledge.
Attendee evaluation forms (rate the instructor, rate the course) have been widely used in the past, but this technique is not recommended as there is too much room for subjectivity. This is known as level 1 evaluation in Donald Kirk- patrick’s (2006) hierarchy, shown with one additional level in Table 2.2. The fifth level is most informative but also quite difficult and expensive to achieve. Phillips (2003) discusses an ROI methodology for training and improvement programs.
In 1980, a company- wide training program at Tennessee Eastman Company evolved from concern about poor quality attitudes. The project was reported by Hill and McClaskey (1980; available at http://www.asq.org/qic). While this train- ing study is rather dated, we feel that some aspects of the training program are still applicable almost 40 years later and have highlighted them here:
1. Determining the purpose. This phase involved upper and middle management, numerous operational units, and the training department. The stated purpose was not to create a training program but to improve overall quality performance. Only after several weeks of discussions and surveys was a training program decided on.
2. Developing alternatives that will achieve the purpose. Through the use of focus groups, brainstorming, and exchange of memos, several dozen alternatives were proposed. Seven were selected for further study; two of these were the following:
— Require position guides that include quality responsibilities
— Create a central quality organization to coordinate development and dissemination of information and requirements
Table 2.2 Five different levels of evaluation.
Level Name Question Techniques
1 Reaction How did learners feel?
Post-instruction questionnaires or interviews. Learners report impressions of instructor, curriculum, facilities, and content.
2 Learning What did learners retain?
Pretest and posttest, checking for gains in either knowledge or performance.
3 Behavior Did learners change?
Assessors must collect data at the workplace to evaluate changes in skill and performance.
4 Organizational Is the impact beyond the learner?
Large-scale surveys of morale, product quality, turnover, and so on, followed by executive conferences.
5 Return on investment
Is there an effect on the bottom line?
Analyze financial data carefully constructed by professionals. Six Sigma includes this element in program evaluation.
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3. Analyzing the alternatives. A set of five weighted criteria was developed, including such items as “chance of completion” and “flexibility of format.” The alternative with the highest score was “teach about quality responsibilities.”
4. Designing the selected program. Many elements were specified. Some typical design parameters were the following:
— Create awareness of quality responsibilities
— Cover total learning needs in quality awareness
— Be adaptable to specific needs
— Be used only by people requiring the knowledge
— Be portable and usable near work area
— Have a maximum length of two hours
After the design criteria were set, a survey was taken of all employees, with 40 potential courses listed. Using the survey, the task force proceeded to develop 16 new courses.
5. Implementing the solution. The team proceeded to write outlines and scripts, identify needed materials, and publish results, calling on others for help as needed. Teachers were selected and trained, a budget was set up, a cost tracking system was created, and the courses were put in place for delivery. The courses ran for several years with periodic updates.
6. Evaluating the results. Pre- and posttests were regularly used to determine how much was learned.
In conclusion, it is important to keep in mind the distinction between training and education. Training is providing a skill and/or technique for immediate applica- tion in a job or related situation. Education is the development of knowledge and understanding in a topic. Education, rather than training, can enrich one’s life in an unpredictable future.
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The five sections in this chapter cover the different elements that QEs use in qual- ity initiatives involving products and processes. The areas covered in this chap- ter include classification of quality characteristics (as opposed to product defects, which are discussed in Chapter 4), design inputs and review elements, elements of technical drawings and specifications, design verification and validation to ensure fitness for use, and reliability and maintainability.
Basic definitions of reliability for both repairable and nonrepairable systems are presented. In addition, the basic relationships between the failure rate (hazard rate), probability density function, and reliability function are developed. Reliabil- ity estimations of simple systems made of series, parallel, or k-out-of-n compo- nents are obtained using the reliability of individual components. Maintainability of the systems is defined, and three widely used maintenance and repair policies— corrective maintenance, preventive maintenance, and predictive maintenance— are discussed. Conditions for the applicability of these policies are also discussed.
a. CLaSSiFiCaTion oF QuaLiTy CHaraCTEriSTiCS Quality characteristics are features that describe the fit and function of a product or process and aid in differentiating between items of a given sample or population. To differentiate items from each other and/or to compare items with a standard, measurements and/or comparisons are used. Variables data are represented by direct measurement on a continuous scale. Attributes data are most often discrete data usually reported in the form of counts. The counts are classified by category, with the most common categories being pass/fail, go/no-go, and accept/reject. (For more details on variables, attributes, and continuous and discrete data, see Chapter 6, section A).
Measurement is the process of evaluating a property or characteristic of an object and describing it with a numerical or nominal value. A quality characteristic is referred to as a variable if it is measurable over a continuous scale. For example, in healthcare, a patient’s temperature can be measured with an electronic ther- mometer. Other examples include measurements related to weight, length, diam- eter, or cost.
If the quality characteristic of interest can’t be directly measured, then each item under inspection is often classified into one of two or more categories. For example, items inspected may be classified as conforming or nonconforming. Each product unit is assigned one of these two labels according to inspection operation
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results. It is then possible to derive a numerical measure of process quality using a quantitative scale. The numerical measure is achieved by calculating the frac- tion nonconforming as the ratio between the number of units labeled as noncon- forming and the total number of units inspected. When the item inspected can be classified into one of exactly two possible categories, the binomial distribution is often appropriate to model this situation. (See Chapter 6, section A, for more details on attributes data, and Chapter 6, section C, for more details on the bino- mial distribution.)
Another commonly used attribute quality characteristic is the number of non- conformities (or number of defects) observed on an inspected item. In this situa- tion, a single item inspected may have more than one nonconformity or defect. For example, a car door panel may have more than one scratch, dent, or discoloration. These would be considered nonconformities on the single item inspected, in this case, the car door panel. The number of nonconformities is often well modeled by the Poisson distribution. (See Chapter 6, sections A and C, for more details on the number of nonconformities and the Poisson distribution, respectively.)
Classification can be used for prioritization categories of quality character- istics and is also frequently used to describe defect seriousness, which we dis- cuss in detail in Chapter 4, section B. Desirability or consequence categories can also be used to indicate the levels of prioritization. These levels include minor, major, serious, and critical. Another prioritization classification method is to group changes or suggestions in the form of would have, could have, should have, or must have. ISO 9001:2015 gives the following verbal forms of improvement along with definitions:
• “Can” indicates a possibility or capability
• “May” indicates a permission
• “Should” indicates a recommendation
• “Shall” indicates a requirement
B. dESign inPuTS and rEviEW Design is a term that describes the thought processes, procedures, tools, docu- mentation, and specifications associated with products and processes. Designs are developed to document and ensure compliance with customer expectations as they relate to operational capabilities and characteristics of products and processes.
Whether for products or processes, designs progress through phases. Design phases are linked to phases within the product/process development life cycle, wherein the product/process development life cycle typically includes the following:
• Definition phase. A problem or opportunity is clearly defined, documented, and refined, and consensus on the definition is reached among stakeholders.
• Specification phase. Specifications are set that lead to, or provide, a product or process desired by the customer. Specifications are commonly applied to characteristics that affect the form, fit, or function
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of a product or process. Specifications also are commonly applied to characteristics that affect the reliability of a product or process to function in a given environment, within a given range of temperature, or for a given period of time or cycle of operation.
• Concept phase. All possible solutions to solve the problem or exploit the opportunity are explored. Feasible solutions are identified, and nonfeasible solutions are dismissed.
• Detailed design phase. Specifications and the most feasible concept are used as the basis for development of detailed plans for a product or process design.
• Prototype phase. A working model of the detailed design is fabricated and tested in a laboratory or development environment for its ability to perform or operate as intended.
• Production phase. Following successful development, testing, and refinement of a prototype, production units of the design are produced in sufficient volume to satisfy customer demand. Unlike prototype units, which are produced in a laboratory or development environment, production phase units are produced with tools, equipment, methods, and procedures used on the shop floor or service delivery area, by regular production or service delivery personnel.
• Distribution phase. The product or process enters the supply chain for sale and distribution.
• Normal use phase. The product or process is released to the customer for use in its intended role or function. Products and processes require normal maintenance and repair (warranty and non- warranty), and customers frequently require technical assistance and support.
• Obsolescence and disposal phase. Products and processes lose their usefulness due to normal wear, catastrophic failure (planned or unplanned), introduction of enhancements to an existing design, or changes in technology. As products and processes become obsolete, the original designer must consider either how to provide enhancements that extend the life of the product/process or how to safely and ethically address disposal of the product/process.
How a design links with the product/process development life cycle (i.e., which specific design phases are used, and how the work of various design phases is related to the product/process development life cycle) depends completely on the model used as the basis for the design. It should be noted that there are numerous models or approaches for design, far too many to provide an exhaustive list in this book. One model for review of designs, however, is well developed, is sufficiently universal, and applies to products or processes: the Systems Engineering Technical Review Process as described in the United States Navy, Naval Air Systems Com- mand (NAVAIR) Instruction 4355.19E (February 2015). The International Electro- technical Commission (IEC) also has developed a standard on design review in IEC 61160 (September 2005).
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B.1. inputs
Designs for products and processes are influenced by factors known as inputs. Inputs are simply requirements placed on products and processes that relate to the following:
• Customer needs and expectations (e.g., time of delivery, cost, performance characteristics)
• Regulatory agency guidance and law requirements (e.g., safety of end users, safety of production/service delivery personnel, use of hazardous chemicals, distribution to unauthorized personnel/vendors, control of sensitive technology)
• Patents and technology licensing (e.g., protections for existing designs owned by competitors)
• Product/process capabilities
• Product/process reliability
The design inputs identified above translate into basic design concepts relevant to quality engineering, particularly when the nature of a design emphasizes a spe- cific characteristic. When a specific characteristic is emphasized, the nature of the design may be considered “constrained,” wherein the design input must either guide design efforts (such as the case with Design for Six Sigma) or constrain the product or process resulting from the design (such as the case with design for cost). In such cases, the design is referred to as “design for X,” where X may be Six Sigma, cost, manufacturability, or reliability. Awareness of the basic design concepts and how they relate to the design process is included within the CQE BoK and is addressed as follows:
Design for Six Sigma. Design for Six Sigma (DFSS) is not to be confused with the well- known DMAIC (define, measure, analyze, improve, control) approach to Six Sigma. While DMAIC Six Sigma focuses on solving problems in existing products or processes, DFSS focuses on eliminating problems before they occur.
Design for Reliability. Design for Reliability (DFR) is an approach to design that focuses on development of products and processes able to perform under specified conditions, in a specified environment, and for specified periods of time.
Quality Function Deployment. Quality function deployment (QFD) provides a framework for the design process that captures the voice of the customer (VOC) via a series of matrices. Once captured, the VOC is used to guide design efforts to ensure that customer expectations are met. See Chapter 1, section G, for more information on QFD.
Concurrent Engineering. Concurrent engineering is practice of the design function and associated activities by a team of engineers, technicians, management, and administrative personnel such that all aspects of the design phases are considered simultaneously.
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B.2. review
Depending on the type and complexity of the design, any or all of the design reviews may be completed. The depth of detail, analysis, and documenta- tion should increase as the design reviews progress toward completion. Design reviews are not intended for solving design- related problems but rather to verify completion of problem- solving activities by cross- functional teams. Discovery of too many design- related problems during a design review may indicate that a design review is being conducted prematurely.
Although the product development group is responsible for creating a design, no one group can provide all the necessary assurance that the design is adequate. Therefore, design reviews should be conducted periodically by a cross- functional team until the design and process are finalized. Quality and manufacturing should be active participants in the review process. Suppliers also should participate, if possible. Early sourcing commitments enable suppliers to attend the design reviews and contribute their expertise prior to investing in expensive tooling. Drawings should comply with applicable standards for drafting, dimensioning, and tolerances. At each review, the design must be considered from several differ- ent viewpoints:
• Reliability. Will the failure rate be sufficiently low?
• Quality engineering. Can the design be adequately inspected and tested?
• Field engineering. Are proper installation, maintenance, and user- handling features included in the design?
• Procurement. Can the necessary parts be acquired at acceptable costs, delivery schedules, and quality levels?
• Materials engineering. Will the selected materials perform as expected?
• Tooling engineering. Is the equipment capable of meeting the specified tolerances on a consistent basis?
• Packaging engineering. Can the product be shipped without damage?
• Outside consultants. Have appropriate outside consultants been called for when necessary?
• Customer. Should a customer representative participate in the design reviews for military applications and original equipment manufacturers?
• Other design engineers. Are other design engineers needed when there are tight tolerances to mating components or critical system interfaces?
Design reviews are often done at various stages of the life cycle of the product, particularly in the Department of Defense (DoD). Several types of reviews are dis- cussed here, but many more are often done throughout the life cycle of the prod- uct or system. The systems requirements review is a technical review to ensure all system and performance requirements are defined, consistent, and testable. The preliminary design review assesses the maturity of the preliminary design of a system or product. This review establishes baselines to ensure the product is
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ready to move into the next phase of development. The critical design review is a technical review that assesses the maturity of the product or system design and remaining risks. This review determines whether the system is ready to move into prototype development. The production readiness review examines the system to determine whether the design is ready to move into production without incurring unacceptable risks. Finally, the in- service review ensures the product or system is operating with managed risk. The goal of this review is to understand the quality of the deployed product and provide a summary of any risks related to the prod- uct (Defense Acquisition Guidebook 2017).
C. TECHniCaL draWingS and SPECiFiCaTionS It is expected that drawings have dimensions that provide detailed information about sizes, shapes, and the location of different components and parts. It is also expected that part and component dimensions show acceptable variation. To pro- duce any part or component to an exact dimension is nearly impossible, except by remote chance. Variations in materials, machines, manufacturing parameters, and humans make it necessary that dimensions have acceptable variations. Such variation is referred to as tolerance. Higher quality requires tighter tolerances that, in turn, require more expensive and strict production and inspection procedures to obtain. There are two types of tolerances: unilateral tolerance and bilateral toler- ance. Unilateral tolerance specifies allowable variation in a dimension from a basic or nominal size in one direction in relation to that basic size.
For example, 2.000+0.000/-0.005 inches describes an allowable variation only in the lower limit: unilateral tolerance. Specifications on a part with this tolerance will be 2.000 inches and 1.995 inches as desired upper and lower limits, respectively. On the other hand, 2.000+0.005/-0.005 inches describes a bilateral tolerance. It specifies a dimension with allowable variations in both directions of the basic size. Specifica- tions on a part with such bilateral tolerance will be 2.005 inches and 1.995 inches as desired upper and lower limits, respectively.
C.1. geometric dimensioning and Tolerancing (gd&T)
Geometric tolerancing defines tolerances for geometric features or characteristics on a part. Figure 3.1 shows some of the geometric dimensioning symbols as defined in ANSI Y14.5M. Figure 3.2 illustrates the interpretation of a geometric tolerance on a drawing.
The limit dimensions of the simple cylindrical piece at the top of Figure 3.3 define the maximum and minimum limits of a profile for the work. The form or shape of the part may vary as long as no portions of the part exceed the maxi- mum profile limit or are inside the minimum profile limit. If a part measures its maximum material limit of size everywhere, it should be of perfect form. This is referred to as the maximum material condition (MMC) and is at the low limit for a hole or slot, but at the high limit for parts such as shafts, bolts, or pins.
If it is desired to provide greater control on the form than is imposed by the limit dimensions, then certain tolerances of form must be applied. In most cases, these tolerances appear in the form of notations on the drawing, as is illustrated at the bottom of Figure 3.3.
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Figure 3.1 Some geometric tolerancing symbols.
Geometric Symbols
Other Symbols
Straightness
Flatness
Parallelism
Perpendicularity
Angularity
Roundness
Cylindericity
Concentricity
Profile of a line
Profile of a surface
True position
Runout
Total runout
Maximum material condition (MMC)
Least material condition (LMC)
Diameter
Datum is A
M
L
– A –
Figure 3.2 Interpretation of a geometric tolerance on a drawing.
0.005 A B CM
Tertiary datum
Secondary datum
Primary datum
Geometric tolerance
Modifier
Shape of tolerance zone
Geometric feature/characteristic
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C.2. Positional Tolerances
Positional tolerancing is a system of specifying the true position, size, or form of a part feature and the amount it may vary from the ideal. The advantage of the sys- tem is that it allows the one responsible for making the part to divide tolerances between position and size as he or she finds best. The principles for two simple mating parts are illustrated in Figure 3.4. The basic dimensions without tolerances are shown at the bottom and right side of each part. Beneath the size dimension for holes or posts is a box with the notations for positional tolerancing. A number of specifications are possible, but only one set is shown here as an example. The circle and cross in the first cell of the box is the convention that says the feature has a positional tolerance.
Part I in Figure 3.4 introduces the idea of the MMC utilized in most posi- tional tolerancing. This is designated by the letter “M” in a circle and means that the smallest hole (12.70 mm or 0.500 in.) determines the inner boundary for any hole. The “∅ 0.20 mm (0.008 in.)” notation in the box specifies that the axis of any minimum- size hole must not be outside a theoretical cylinder of 0.20 mm (0.008 in.) diameter around the true position. A 12.50 mm (0.492 in.) diameter plug in true position will fit in any 12.70 mm (0.500 in.) diameter hole with its axis on the 0.20 mm (0.008 in.) diameter cylinder. Any hole that passes over such a plug is acceptable, provided that its diameter is within the high and low limits specified.
The letter “A” in the specification box designates that the theoretical cylinder bounding the hole axes must be perpendicular to the datum surface carrying the “A” flag. Features usually are referred to with three coordinate datum surfaces, but for simplicity, in this case, the holes are related only to each other and surface “A” and not to the sides of the part.
Figure 3.3 Part drawing with and without tolerances of form. Source: Reprinted with permission of the Society of Manufacturing Engineers, Manufacturing Processes and Materials, 4th ed., Copyright 2000.
Part X
Flat within 0.002 in.
Flat within 0.05 mm
A A
Straight within 0.05 mm
Straight within 0.002 in.
This face parallel to A within 0.05 mm
FIM*
This face parallel to A within 0.002 in.
FIM*
*FIM = Full indicator movement
(mm)
50.15 49.85
2 5 .0
0 2 4 .9
0
50.15 49.85
2 5 .0
0 2 4 .9
0
Part Y(in.)
2.005 1.995
1 .0
0 0
0 .9
9 0
2.005 1.995
1 .0
0 0
0 .9
9 0
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Part II of Figure 3.4 introduces the idea of zero MMC specified by “∅ 0.000” before the MMC symbol. This means the axis of the largest diameter post (12.50 mm [0.492 in.]) must be exactly in the true position, but smaller sizes of posts may vary in position as long as they do not lie outside the boundary set by the largest. Thus, if the posts are held to a tolerance smaller than the 0.20 mm (0.008 in.) specified, say to a tolerance of 0.05 mm (0.002 in.), the difference (0.15 mm [0.006 in.]) is then available for variations in post positions. The advantage of zero MMC is that only one limit of the feature, in this case the lower limit of the post diameter, needs to be checked along with position.
d. vEriFiCaTion and vaLidaTion Design verification as defined by ISO 9000:2015 is the “confirmation, through the provision of objective evidence, that specified requirements have been fulfilled.” In other words, verification consists of a series of evaluations to ensure that the item is built to design and to specifications. The objective evidence may be the result of an inspection or may come from other forms of determination, such as performing alternative calculations or reviewing documents.
There are four potential verification methods: demonstration, inspection, analysis, and test. Demonstration, one of the simplest methods, is where the prod- uct or process is operated and visual confirmation is made to verify that the design requirements are met. Demonstration is appropriate only when quantitative evi- dence is not required for verification. Demonstration may use models, simula- tions, mockups, or the end product.
Inspection is another simple means of verification, where physical characteris- tics or materials or parts are examined to ensure requirements are met. Inspection requires the least amount of resources and is typically used to verify a physical fea- ture of the design. Inspection to ensure and control product quality, as performed
Figure 3.4 Two parts dimensioned with positional tolerances. Source: Reprinted with permission of the Society of Manufacturing Engineers, Manufacturing Processes and Materials, 4th ed., Copyright 2000.
12.70 mm (0.500 in.)
63.50 mm (2.500 in.)
3 8 .1
0 m
m (
1 .5
0 0 in
.)
3 8 .1
0 m
m (
1 .5
0 0 in
.)
4 × Ø 4 holes 4 posts
Part IM A
A 63.50 mm (2.500 in.)
A
Ø 0.20 mm (0.008 in.)
12.90 mm (0.508 in.) 12.50 mm (0.492 in.)
4 × Ø 12.30 mm (0.484 in.)
Part IIM AØ 0.000
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in industry, is of two kinds: visual inspection and dimensional inspection. Visual inspection, by far the more common of the two, involves visual examination by human operators for conformity to aesthetic requirements. Less common, but equally important and generally more difficult to perform adequately, is dimen- sional inspection. Quality, of course, cannot be inspected into a product. Qual- ity depends on engineering and manufacturing excellence, and inspection simply determines whether it exists. Better inspection is not the solution to large numbers of rejects. The solution must take the form of improvements in design or in the manufacturing process.
Visual inspection takes place, even if inadvertently, each time a part is han- dled during its manufacture. Parts such as bearing elements, which have critical aesthetic requirements, may be given a final visual inspection once manufacture is complete. Visual inspection is concerned primarily with gross appearance, the detection of surface flaws, and the recognition of patterns. These functions have, to date, attracted far less attention from developers of automatic inspection systems than the functions associated with dimensional inspection. As a consequence, the human being currently is the most efficient general- purpose flaw- detection and pattern- recognition “instrument” available to the manufacturer. Human beings have highly developed sensing and data- processing faculties. Human operators are trainable and adaptive, although they are generally less reliable and experience more downtime than their automatic- equipment counterparts. Current research in artificial intelligence will surely cause this situation to change in the future.
Dimensional inspection refers to the measurement of lengths and angles and, in combination, of geometric shapes and may be accomplished automatically by a machine or manually by an operator. Measurements that are taken while the prod- uct is still undergoing manufacture have a greater value than those applied to the finished product, since the former constitute process control whereas the latter are merely process verification. Obviously, it is more expensive to correct or to scrap a bad product than it is to manufacture it properly in the first place.
Analysis uses mathematical models and other analytical techniques such as modeling and simulation to determine whether the design conforms to require- ments. Finally, test is a designed experiment or activity to determine whether the requirements are met under controlled conditions (Defense Acquisition Guide- book 2017). This method of verification requires the most resources; it is described in more detail in Chapter 6. These verification activities should be done through- out the review process of a product or process.
Design validation is the “confirmation, through the provision of objective evi- dence, that the requirements for a specific intended use or application have been fulfilled” (ISO 9000:2015). In other words, validation ensures that the product or system operates as expected in its intended environment and as required by the customer. The objective evidence for validation may be the result of a test or some other form of determination. The test methods, sample sizes, and acceptance cri- teria should be clearly specified during conceptual planning of a product or pro- cess, and the design should be validated at each phase of the design review. When designing the tests, consider incorporating the following factors:
• Dimensional wear, material fatigue, assembly process variation
• Variation of critical characteristics throughout the range of the tolerances
• Contamination
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• Environmental aging and extreme environmental conditions
• Extreme customer usage, such as maximum loads or long- duty cycles
Test results should be analyzed using appropriate statistical methods, including reliability analysis. Reliability testing falls into four major categories (details in Ebeling 2009 and Dodson and Nolan 1999):
• Environmental stress screening to determine the capability of the product under different operating conditions and to help eliminate infant mortality failures
• Reliability growth tests performed periodically between design conception and final production to track gains in reliability
• Reliability qualification tests to provide assurance that production units will meet requirements when they become available
• Production reliability acceptance tests to periodically verify that production units meet specified reliability requirements and are acceptable for use
A more detailed discussion of reliability and maintainability is provided in sec- tion E of this chapter. Test failures should be carefully examined to determine the failure modes. Unanticipated failure modes must be added to the design failure modes and effects analysis (DFMEA). Of course, corrective actions and design improvements must be pursued if the test results do not meet the quality goals.
Results of engineering evaluations, reliability tests, and other methods used to validate the design should be included in the design reviews. In addition, this information should be used to update the classification of quality characteristics. As is the case with many quality disciplines, the process of classifying character- istics should be iterative. Characteristics that are associated with unexpected fail- ures may require reclassification as major or critical characteristics. Characteristics that perform as expected may be candidates for downgrading to minor charac- teristics. In all cases, involve the quality team in the discussions. There can be no substitute for the experience and process knowledge the team members bring to the design review process.
Process and product validation is broken down into several parts: installation qualification (IQ), operational qualification (OQ), and process qualification (PQ). IQ is done to establish that all systems and equipment have been installed correctly. OQ is performed to establish that the equipment control limits meet their respective requirements. PQ is done to establish that the process consistently produces accept- able products under normal operating conditions (Durivage 2016). A final stage often undertaken is the process performance qualification (PPQ), which demon- strates that the validated processes produce a product that meets its specifications. This phase accounts for the entire manufacturing process, not just one subprocess. For more information on product and process validation, see Durivage (2016).
E. rELiaBiLiTy and MainTainaBiLiTy This section focuses on estimating and predicting reliability and defines other reliability measures for repairable systems, such as maintainability and availabil- ity. Other chapters in this handbook focus on quality as a static characteristic of
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a product at the time it is released to the user. However, because reliability is a time- dependent quality characteristic, traditional methods for quality control can- not be used to ensure product reliability and maintainability. Because reliability engineering is a broad field, it is impossible to cover the entire range of reliability topics in one section. Therefore, we focus on reliability and maintainability defini- tions, analysis of failure data, design of systems for reliability, and maintainability. Reliability and maintainability as elements of product and process design will be discussed in four sections: predictive and preventive maintenance tools, reliability systems and measures, reliability models, and reliability, safety, and hazard assess- ment tools. The BoK includes a section on reliability, safety, and hazard assessment tools (III.E.4). While these are an important part of reliability and maintainability, they are also important tools for risk management and thus we discuss them in detail in Chapter 7, section B.
E.1. Predictive and Preventive Maintenance Tools
Reliability is defined as the probability that a product or service will operate prop- erly for a specified period of time (design life) under the design operating condi- tions. The main factors that lead to a system’s failure include the system’s design and configuration, the reliability of its components, the operating environment, and the interactions among environmental factors, manufacturing defects, and preventive and scheduled maintenance. Further, reliability cannot be measured at the release time of the product but can only be predicted (Elsayed 2000).
It is extremely important to consider reliability during the design phase of a product or service because minor, major, and catastrophic failures result in eco- nomic consequences such as repairs or replacements, the loss of production or interruption of service, and potentially severe economic losses and the loss of life. Examples of major failures are failure of a major link of a telecommunica- tions network, failure of a power generating unit, or failure of software for an air traffic control system. A more recent major failure was the Galaxy Note 7 cell phone battery failures (https://www.usatoday.com/story/tech/2016/10/10/ samsung-galaxy-note-7-recall-cost/91876162/).
The consequences of catastrophic failures are much more severe than those of minor or major failures, and may include the loss of human life and significant eco- nomic losses. Examples of catastrophic failures are the explosions at the Chernobyl nuclear reactors site in the former USSR (Elsayed 1996), the explosion of the space shuttle Challenger in 1986, and the failure of the space shuttle Columbia in 2003.
Reliability also has a great effect on consumers’ perception of a manufacturer. For example, consumers’ experiences with automobile recalls, repairs, and war- ranties affect the manufacturer’s future sales. Another example shows the impor- tance of reliability: 6.5 million tires were recalled after 46 deaths were attributed to the separation of the tread from the tire, causing vehicles to skid or roll over (ABC News, “Firestone Recalls Millions of Tires,” http://abcnews.go.com/US/ story?id=96216&page=1).
Three important functions that are the result of traditional calculus deriva- tions help quantify reliability: the reliability function, the failure time distribution function (sometimes referred to as the probability density function), and the haz- ard rate function (or instantaneous failure rate).
Suppose N identical components are tested. During a specified time inter- val t, we observe x failures and (N – x) survivors. Because reliability is defined
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as the cumulative probability function of success, then at time t the reliability R(t) is:
R t
N x
N ( ) = −( )
(3.1)
In other words, the reliability function (or survival function) at time t is the fraction of all components that have survived for a time greater than t. R(t) is also used as the estimate of the probability that a randomly selected component will survive for a time greater than t. In order to describe the distribution of failures, the cumu- lative distribution function (cdf) of failure F(t) can be defined as:
F t
x N
( ) =
(3.2)
(See Chapter 6, section C, for more details on cdfs.) The cdf given in Equation (3.2) can be interpreted as:
• The probability that a randomly selected unit drawn from a population fails by time t, or
• The fraction of all units in the population that fail by time t.
In addition, F(t) is the complement of R(t) so
F t R t( ) + ( ) = 1 (3.3) Equation (3.3) can be rewritten as:
R t F t( ) = − ( )1 (3.4) or
F t R t( ) = − ( )1 (3.5) Suppose N identical units are selected at random from a population described by F(t). Then NF(t) is the average (expected) number of failures through time t, and NR(t) represents the average (expected) number of survivors through time t. That is, we would expect NR(t) of the units to still be operational up to time t.
A probability density function (pdf) that represents the distribution of failure time can be found by taking the derivative of Equation (3.5):
f t
dF t
dt
dR t
dt ( ) = ( ) = − ( )
(3.6)
The hazard rate function is defined as the limit of the failure rate as the time interval approaches zero. In other words, it provides an instantaneous rate of failure at some time t. The hazard rate function (also known as the instantaneous failure rate function) can be expressed as:
h t( ) = Number of failures per unit time
Number of components tested per unit time =
( )f t R t(( )
(3.7)
In some situations, failure times are placed into time intervals and the indi- vidual failure times are no longer preserved. In this case, the failure times become
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grouped data. The reliability, distribution function, failure density, and hazard rate are estimated from the grouped data.
Suppose we wish to estimate the four quantities given in Equations (3.1), (3.2), (3.6), and (3.7) in terms of a reliability test where N identical units are tested. First, record the number of failed units (xi) and the number of survivors (ni = N – xi) at time ti (where i = 1, 2, . . .). Next, R(t), F(t), f (t), and h(t) can be estimated as follows:
R̂ t
n N
i( ) =
(3.8)
ˆ ˆF t R t( ) = − ( )1 (3.9)
ˆ ,f t n n
N t t t t ti i
i i i i( ) =
− −( )
< <+ +
+ 1
1 1for
(3.10)
ˆ ˆ
ˆ ,h t n n
ni t t
f t
R t ti i
i i ( ) = −
−( ) =
( ) ( )
+
+
1
1
for ii it t< < +1
(3.11)
The caret (“hat”) on each term indicates an estimated quantity. There are sev- eral methods for estimating these functions if the data are ungrouped, censored, ungrouped and censored, or grouped and censored. The reader is referred to Ebel- ing (2009) for a complete discussion and derivations of these quantities and more.
ExaMpLE 3.1
Suppose that 300 light bulbs are subjected to a reliability test. The manufacturer will release the bulbs for distribution if the reliability of the bulbs is 0.75 at 2000 hours of usage. The observed failures during 1000-hour intervals are shown in Table 3.1.
Solution: Using the previous equations, we can determine the four functions: reliability function, distribution function, probability density function (failure density), and the hazard rate function. The results are shown in Table 3.2.
To illustrate, consider the values for i = 2, where 1000 < t < 2000:
Estimated reliability: ˆ .R t n N
( ) = = =2 286 300
0 95333
Cumulative distribution function: ˆ ˆ . .F t R t( ) = − ( ) = − =1 1 0 95333 0 04667
Failure density: f̂ t n n t t
( ) = − −( ) =
− −( )
2 3
3 2
286 269 2000 1000
== 0 0000567. N 300
Hazard rate: ĥ t n n t tn
( ) = − −( ) =
− −( )
2 3
3 22
286 269 2000 1000286
0 0000594= .
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Table 3.1 Number of failures in the time intervals.
Upper bound (hours) Number of failures, x
0 0
1000 14
2000 17
3000 21
4000 25
5000 31
6000 37
7000 40
8000 50
9000 65
Table 3.2 Reliability, cdf, failure density, and hazard rate for the light bulb example.
Interval Upper bound
Failures in the
interval Survivors Reliability cdf Failure density
Hazard rate
i ti xi ni R̂ (t) F̂ (t) f̂ (t) ĥ (t)
1 0 0 300 1.00000 0.00000 0.0000467 0.0000467
2 1000 14 286 0.95333 0.04667 0.0000567 0.0000594
3 2000 17 269 0.89667 0.10333 0.0000700 0.0000781
4 3000 21 248 0.82667 0.17333 0.0000833 0.0001008
5 4000 25 223 0.74333 0.25667 0.0001033 0.0001390
6 5000 31 192 0.64000 0.36000 0.0001233 0.0001927
7 6000 37 155 0.51667 0.48333 0.0001333 0.0002581
8 7000 40 115 0.38333 0.61667 0.0001667 0.0004348
9 8000 50 65 0.21667 0.78333 0.0002167 0.0010000
10 9000 65 0 0.00000 1.00000 — —
These values can be easily calculated using a spreadsheet. The reliability function in Figure 3.5 indicates that the bulbs exceed the level of
reliability set by the manufacturers. The distribution function in Figure 3.6 shows how unreliability grows with the passage of time. The failure density function is displayed in Figure 3.7, and the hazard rate is displayed in Figure 3.8.
Continued
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0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0.0
0.2
0.4
0.8
0.6
1.0 R
(t )
Bound (hours)
Figure 3.5 Reliability function versus time.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0.0
0.2
0.4
0.8
0.6
1.0
F (t
)
Bound (hours)
Figure 3.6 Cumulative distribution function versus time.
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0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0.000050
0.000075
0.000100
0.000175
0.000125
0.000225
0.000200
0.000150
f( t)
Bound (hours)
Figure 3.7 Failure density versus time.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0.0000
0.0002
0.0004
0.0008
0.0006
0.0010
h (t
)
Bound (hours)
Figure 3.8 Hazard rate versus time.
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E.2. reliability Systems and Measures
A product is considered a system when it consists of components that are con- nected according to some design rules to produce the desired functions of the product. While the previous section discussed how to determine the reliability of individual components, this section covers reliability estimates for systems with a specific focus on simple systems. Methods for estimating reliability of complex systems are given in Elsayed (1996). In complex systems such as a telecommuni- cations network, the system is composed of units or subsystems connected in a network configuration where the arcs represent the units and the nodes represent connection points along the paths. Reliability estimates of complex systems are often simplified into an aggregation of many simple systems.
This section discusses how to estimate the reliability of two kinds of sim- ple systems (series systems and parallel systems) and addresses k-out-of-n sys- tems and standby systems. Metrics for maintainability and availability are also discussed.
E.2.a. Series Systems
A typical series system is composed of n components (or subsystems) connected end to end. A failure of any component results in the failure of the entire system. A laser printer, for example, has several major components, such as a photoconduc- tor drum, a laser beam, a toner station, and a paper feed system. The printer will
ExaMpLE 3.2
Suppose the time-to-failure information given is no longer a set of data, but now is well modeled by a particular distribution. For example, suppose the failure time is well esti- mated or modeled by an exponential distribution with parameter λ (see Chapter 6, sec- tion C, for details on the exponential distribution). For this distribution the cdf is given by
F t e t( ) = − −1 λ
It can be shown that the pdf (failure density) is given by
f t e t( ) = −λ λ
Using the relationship R(t) = 1 – F(t), we also obtain
R t e t( ) = −λ
Finally, the hazard rate function can be shown to be
h t f t R t e
( ) = ( ) ( )
= = λ e t−λ λ
t− λ
This result represents a constant failure rate. The exponential distribution is the only distribution with a constant failure rate function and whose application in reliability is discussed in more detail in section E.3.
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fail if any of these components fails. We depict the components graphically, with their respective reliabilities, in a block diagram in Figure 3.9.
Under the assumption that each of the component failures is independent, the reliability of the system is the product of the reliabilities of its components. The system reliability is expressed as:
R t R t R t R ts ( ) = ( ) ( ) ( )1 2 ... n (3.12)
where Ri(t) is the reliability of the ith component (for i = 1, 2, . . . n). Equation (3.12) assumes that the components are independent; that is, the degradation of one com- ponent does not affect the failure rate of the other components.
ExaMpLE 3.3
For Figure 3.9, the series reliability Rs(t) is computed as follows:
Rs(t) = (0.96)(0.92)(0.94)(0.90) = 0.7472
The reliability of a series system is lower than its “weakest” component.
E.2.b. Parallel Systems
In a parallel system, components or units are connected in parallel so that the failure of one or more paths still allows the remaining path(s) to perform properly. The system fails when all units fail. Under the assumption of independence, the reliability of a parallel system Rs(t) with n units can be estimated by:
R t F t F t F ts ( ) = − ( ) ( ) ( ) 1 1 2 ... n (3.13)
or equivalently:
( )R t R t R t R ts ( ) = − − ( )( ) − ( )( ) − ( )1 1 11 2 ... 1 n7 7 (3.14)
where Fi(t) is the probability of failure of the ith component and Ri(t) is the reli- ability of the ith component (for i = 1, 2, . . . , n). Equation (3.14) results from the relationship given in Equation (3.5); that is, Fi(t) = 1– Ri(t). If the components are
Figure 3.9 A typical series system.
Photoconductor drum
0.96
Laser beam
0.92
Toner station
0.94
Paper feed
0.90
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identical and p is the probability that a component is operational (that is, Ri(t) = p for all units), then the system reliability becomes
R t F t F t F t
R t
s ( ) = − ( ) ( ) ( ) = − − ( )
1
1 1
1 2
1
...
(( ) ( )( ) − ( )( ) = − −( ) −
1 1
1 1 1
2R t R t
p p
n...
( ) −( ) = − −( )
... 1
1 1
p
p n
n
−
(3.15)
The reliability block diagram of a parallel system is shown in Figure 3.10. The reader is referred to Ebeling (2009) or Tobias and Trindade (2011) for complete discussion of parallel systems.
Figures 3.9 and 3.10 show what we refer to as pure series and pure paral- lel systems, respectively. There are many situations where the design of the system is composed of combinations of series and parallel subsystems, such as parallel- series, series- parallel, and mixed parallel.
ExaMpLE 3.4
For Figure 3.10, the parallel system reliability is computed using Equation (3.14) (since the units are not identical):
1 1 ( ) (
.( )5 ( )R t− ( )3
( )3 1 1 1R t R t R ts( ) = − − ( )( ) − ( )1 1 1 2
= − − − −
=
0 9 0 9 0 91
0 99
.
. 9969
1 .
6 6 66
E.2.c. k-out-of-n Systems
Sometimes the system design requires, at a minimum, k-out-of-n functioning units for the system to operate properly. This is a direct application of the binomial
Figure 3.10 A typical parallel system.
Unit 1
0.95
Unit 2
0.93
Unit 3
0.91
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distribution, where p represents the probability of success of a component (see Chapter 6, section C, for complete details of the binomial distribution). Assum- ing the units are identical and independent, the system reliability in this case is given by:
R t C p ps n i
i n i
i k
n
( ) = −( ) − = ∑ 1
(3.16)
where
(3.17) n iC n
i n i =
−( ) !
! ! and represents the number of ways i units can be chosen from a group of n units.
n! = n (n – 1) (n – 2) . . . (2)(1)
0! = 1
ExaMpLE 3.5
Suppose a system has four identical and independent components. System design requirements indicate that a minimum of two components must function for the suc- cessful operation of the system. What is the reliability of this system if each component has a reliability of 0.90?
Solution: In this situation, n = 4, k = 2, and p = 0.90. Direct substitution into Equations (3.16) and (3.17) yields the following result:
R t C
C
s i i i
i
( ) = ( ) −( )
= ( )
−
= ∑ 4 4
2
4
4 2 2
0 90 1 0 90
0 90
. .
. 11 0 90 0 90 1 0 90 0 90 1 2
4 3 3 1
4 4 4−( ) + ( ) −( ) + ( ) −. . . .C C 00 90
6 0 90 1 0 90 4 0 90 1 0 90
0
2 2 3
.
. . . .
( ) = ( ) −( ) + ( ) −( )11 4 01 0 90 1 0 90
0 9963
+ ( ) −( ) =
. .
.
E.2.d. Standby Systems
Parallel systems are treated as redundant systems. Only one operational path is needed for the system to operate properly. Redundancy can take other forms, such as hot standby redundancy, where all units are operating in parallel at all times. Under this design, all units share the load equally.
In standby systems, the standby components function only upon the failure of the main component. The simplest form of a standby system is one where the com- ponents are assumed to be identical, the switch is assumed never to fail, and the standby component is also assumed never to fail while in standby status. Devia- tions from these assumptions present a variety of systems configurations whose analyses go beyond the scope of this section (see Tobias and Trindade [2011] or Ebeling [2009]). Figure 3.11 shows a standby system with perfect switching
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(i.e., the switch will turn on the standby component instantaneously upon the failure of the main component).
Standby system reliability with n standby components is given by
R t e
t i
t
i
i
n
( ) = ( )− = ∑
λ !0
λ
(3.18)
where λ is the component failure rate and t is the time. If the system has only one standby component, the system reliability is given by
R(t) = e–λt[1 + λt] (3.19)
If the system has two standby components, the system reliability is given by
R t e t
tt( ) = + + ( )− λλ λ1 2
2
! > >
(3.20)
Cold standby is another form of redundancy where the minimum number of units needed to properly operate the system share the load equally, and other units are available on a standby basis but can only share the load when one or more of the operating units fail. The third type of redundancy is called warm standby. This is similar to hot standby, but not all units share the load equally. Those carrying more than 50% of the load are the primary units, while the others are considered to be in a warm standby state. When a primary unit fails, the warm standby unit shares the load equally with the remaining primary units.
The following paragraphs present some important measures of reliability. The mean time to failure (MTTF) should not be confused with the mean time between failures (MTBF). The expected time between two successive failures is the MTTF when the system is nonrepairable. The expected time between failures, the MTBF, can be calculated when the system is repairable.
Figure 3.11 A standby system with n components in standby mode.
Main component failure rate
Input Output
Standby 1 failure rate
Standby n failure rate
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First consider n identical nonrepairable systems and observe the times to fail- ure for them. Assume that the observed times to failure are t1, t2, . . . , tn. The esti- mated MTTF is:
MTTF =
= ∑1
1n ti
i
n
(3.21)
For constant failure rate the MTTF is:
MTTF =
1 λ
(3.22)
which can be interpreted as the reciprocal of the failure rate. It should be noted that this is only true for the constant failure rate model. The accurate method for estimating MTTF for discrete time intervals is given in Equation (3.21). MTTF can be estimated by using integration for continuous time functions.
E.2.e. Maintainability and Availability
We have presented several measures of reliability for nonrepairable systems that include reliability function and MTTF. Other measures of reliability are defined for repairable systems, such as system availability (instantaneous, average up- time, inherent, operational, and achieved availabilities), mean time to repair (MTTR), and maintainability.
MTTR is defined as the average time to repair a failure, not including wait- ing time for parts or tools to start the repair. Maintainability is defined as the probability that a failed system is restored to its operational condition within a specified time.
ExaMpLE 3.6
Suppose we have a standby system with three components in standby mode. All com- ponents are identical with a constant rate of failure of λ = 0.02. What is the system reli- ability at 75 hours of continuous operation?
Solution: Substitute n = 3, λ = 0.02, and t = 75 in Equation (3.18):
R e i
e
i
75
1
0 02 75
0
3
1 5
( ) = [(0.02)(75)] i
=
−( )( )
=
−
∑.
.
!
.55
1 5 0
1 5 1
1 5 2
0
3
1 5 0 1 2
( )
= ( )
+ ( )
+ ( )
=
−
∑ i
i i
e
!
. !
. !
.. !!
. !
. . ..
+ ( )
= + + +[−
1 5 3
1 1 5 1 125 0 5625
3
1 5e ] = [ ] =
0 2231 4 1875
0 9344
. .
.
< <
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Common to all these definitions is that the system is subject to repair or replace- ment upon failure. Availability at time t is defined as the probability that the sys- tem is properly operating at that time. The steady state availability is the long- term availability of the system (t → ∞). The steady state availability A is defined as:
A =
+ MTBF
MTBF MTTR (3.23)
Maintenance actions or policies can be classified as corrective maintenance, preventive maintenance, and predictive maintenance (which is also called on- condition maintenance). Maintenance actions are dependent on many factors, such as the failure rate of the machine, the cost associated with downtime, the cost of repair, and the expected life of the machine.
A corrective maintenance policy requires no repairs, replacements, or preventive maintenance until failures occur, which allows for maximum run time between repairs. Although a corrective maintenance policy does allow for maximum run time between repairs, it is neither economical nor efficient, as it may result in a catastrophic failure that requires extensive repair time and cost.
A preventive maintenance policy requires maintaining a machine according to a predetermined schedule, whether a problem is apparent or not. On a scheduled basis, machines are removed from operation, disassembled, inspected for defec- tive parts, and repaired accordingly. Actual repair costs can be reduced in this manner, but production loss may increase if the machine is complex and requires days or even weeks to maintain. Preventive maintenance also may create machine problems where none existed before. It is important to note that preventive main- tenance is applicable only when the following conditions are satisfied:
1. The cost to repair the system after its failure is greater than the cost of maintaining the system before its failure.
2. The failure rate function of the system is monotonically increasing with time. Clearly, if the system’s failure rate is decreasing with time, then the system is likely to improve with time and any preventive action or replacement is considered a waste of resources. Likewise, performing preventive maintenance when the failure rate is constant is improper, as replacing or maintaining the system before failures does not affect the probability that the system will fail in the next instant, given that it is now good (Jardine and Buzacott 1983).
The third repair policy is the predictive maintenance policy. Obviously, tremendous savings can result if a machine failure can be predicted and the machine can be taken off- line to make only the necessary repairs. Predictive maintenance can also be done when failure modes for the machine can be identified and monitored for increased intensity, and when the machine can be shut down at a fixed control limit before critical fault levels are reached.
Predictive maintenance results in two benefits. The first benefit is the result of taking a machine off- line at a predetermined time, which allows production loss to be minimized by scheduling production around the downtime. Since defective components can be predetermined, repair parts can be ordered and manpower scheduled for the maintenance accordingly. Moreover, sensors for monitoring the machines eliminate time spent on diagnostics, thus reducing the time to perform
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the actual repair. The second benefit is that only defective parts need to be repaired or replaced and the components in good working order are left as is, thus minimiz- ing repair costs and downtime.
Three main tasks must be fulfilled for predictive maintenance. The first task is to find the condition parameter that can describe the condition of the machine. A condition parameter can be any characteristic, such as vibration, sound, tempera- ture, corrosion, crack growth, wear, or lubricant condition. The second task is to monitor the condition parameter and to assess the current machine condition from the measured data. The final task is to determine the limit value of the condition parameter and its two components, the alarm value and the breakdown value. A running machine reaching the alarm value is an indication that the machine is experiencing intensive wear. At this point, the type and advancement of the fault must be identified in order to prepare the maintenance procedure. If a machine reaches the breakdown value, the machine must be shut down for maintenance. See Ebeling (2009) for a detailed discussion of availability and maintainability.
E.3. reliability Models
One of the earliest models of failure rate, the bathtub curve (see Figure 3.12), is so named because of its shape. The failure rate versus time can be divided into three regions. The first region is characterized by a decreasing failure rate with time and is conventionally referred to as the infant mortality phase or the early life region of the product, component, or system during its early period of use. Experience shows that the length (0 to T1) of this region is about 10,000 hours (approximately one year) for most electronic components. The failures in this region are usually attributed to defects in the manufacturing processes, assemblies, and shipping of the product.
The second region of the bathtub curve is the constant failure rate region, which is characterized by the inherent failure rate of the product’s composite components. In this region, the failures occur randomly over time, as shown in Example 3.2. The third region is referred to as the wear- out region. It is character- ized by an increasing failure rate over time. Most electronic components do not exhibit such a region, with the exception of electromechanical devices, such as
Figure 3.12 The general failure rate model (the bathtub curve).
T 1
Early life region
Constant failure rate region
Wear-out region
H a za
rd r
a te
T 2 Time
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relays. On the other hand, most, if not all, mechanical components that are sub- jected to rotating and alternating motions wear out with time. This is exemplified by the behavior of cutting tools, fatigue loading on structures, and wear- out due to friction between mating surfaces.
In Example 3.2, we showed a case of constant failure rate (we use “hazard rate” and “failure rate” interchangeably). This is the simplest failure model, as its pdf and reliability function can easily be shown in the following section, whereas other failure rate models (decreasing or increasing) are sometimes difficult to obtain from their corresponding functions.
The second region in the general failure rate model (bathtub curve) shows constant failure rate. Let λ be the constant failure rate. Thus h(t) = λ.
The reliability function and the pdf, originally derived in Example 3.2, are given in Equations (3.24) and (3.25), respectively:
R t e t( ) = −λ (3.24)
f t h t R t e t( ) = ( ) ( ) = −λ λ (3.25)
This is the standard exponential failure time distribution. The graphs of Equations (3.24) and (3.25), shown in Figures 3.13 and 3.14, are similar to those in Figures 3.6 and 3.7, which are obtained from actual failure data.
The first and third regions (the decreasing and increasing failure rate regions) of the general failure rate models can be described by time- dependent failure rate functions. The Weibull failure rate is the most widely used failure rate model that describes these regions. It is expressed as:
h t t( ) = −γ γ
θ 1
(3.26)
Figure 3.13 Probability density function for constant failure rate.
0.02
0 100 20 4030 50 60
P ro
b a b
il it
y d
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s it
y f
u n
c ti
o n
Time
0.06
0.04
0.08
0.1
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where γ and θ are the shape and scale parameters of the two- parameter Weibull distribution. For discussion of the Weibull distribution, see Chapter 6, section C. The appeal of the Weibull hazard rate function comes from the fact that it can represent several other known functions. For example, when γ = 1, the Weibull hazard rate function becomes constant. When γ = 2, the resultant hazard function is linear with time and its pdf becomes the Rayleigh distribution. Indeed, Makino (1984) shows that the normal distribution can be approximated to Weibull when γ = 3.43927.
The reliability function and the pdf of the Weibull distribution are expressed respectively as:
R t e t( ) ( )= > t− γ
θ 0 (3.27)
and
f t t( ) = −γθ θ
γ 1) a a
( e ( )t− γθ
(3.28)
Figures 3.15 and 3.16 demonstrate the use of the Weibull failure model to describe decreasing and increasing failure rates. Of course, the constant failure rate is also included.
Other probability distributions can be used to appropriately describe the fail- ure times, including gamma, beta, log- logistic, lognormal, extreme value, and nor- mal distribution (Elsayed 1996).
In order to be effective, a comprehensive reliability program must be based on data that are collected, verified and/or validated, analyzed, and used as the basis of decision making for design improvements and corrective action. At a minimum,
Figure 3.14 Reliability function for constant failure rate.
0.2
0 100 20 4030 50 60
R e li a b
il it
y
Time
0.6
0.4
0.8
1
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Figure 3.15 Probability density functions for the Weibull model with different shape and scale parameters.
0.3
0 20 4 8
Shape, scale
2,1
3,1
0.7, 1
6 10 12
P ro
b a b
il it
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s it
y f
u n
c ti
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Time
0.9
0.6
1.2
1.5
Figure 3.16 Hazard rate functions for the Weibull model with different shape and scale parameters.
1
0 20 4 8
Shape, scale
2,1
3,1
0.7, 1
6 10 12
H a za
rd r
a te
f u
n c ti
o n
Time
3
2
5
4
6
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reliability data must be thoroughly evaluated at key milestones such as the design phase and program reviews.
In the context of failure analysis and reporting, reliability data are most com- monly evaluated in a closed- loop failure reporting and corrective action system. For purposes of this chapter, a closed- loop failure reporting and corrective action system provides the means to ensure that failures are not only documented and tracked over time but also analyzed to a sufficient depth to determine whether corrective action is required, and if so, what corrective action is necessary as deter- mined by appropriate design engineers or a reliability review board.
E.4. reliability, Safety, and Hazard assessment Tools
During the design phase of the system, and when the system fails during opera- tion, it is important to identify potential failures and their causes in order to elimi- nate critical failures (those that cause total interruption of the function or potential injuries to users) or develop appropriate methods to reduce their effects. Several approaches have proven to be effective in identifying potential failures: environ- mental stress screening (ESS), failure modes and effects analysis (FMEA), failure modes effects and criticality analysis (FMECA), and fault tree analysis (FTA).
In the risk management chapter of this handbook (Chapter 7, section B), we discuss ESS, FTA, hazard and operability analysis (HAZOP), FMEA, and FMECA with applications and examples in manufacturing and with a focus on risk identi- fication and assessment. The BoK requires CQEs to define, construct, and interpret the results of FMEAs, FMECAs, and FTAs. Please refer to Chapter 7, section B, for this material.
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137
This chapter includes details on the following elements: product and process control methods, material control (including material identification, status, and traceability), material segregation, classification of defects, acceptance sampling (including sampling concepts, sampling standards and plans, and sample integ- rity), measurement and test (including measurement tools and destructive and nondestructive tests), metrology, and measurement system analysis (MSA).
a. METHodS In this section, product and process control methods, including control plans, critical control point identification, and work instructions, are presented. Such methods depend on the classification of quality characteristics and the results of validation tests (see Chapter 3 for more details). Characteristics that are critical to the operation of the process or the function of the product are subject to more intense monitoring and control.
Control plans are used to document and communicate the plan for monitoring and controlling the process. The control plan summarizes information from vari- ous sources into a single, handy document for quick reference on the production line. The format of the control plan is not important; standard spreadsheets are acceptable. However, the control plan should include the following elements:
• Station or operation number and process description
• Machinery, equipment, or fixtures
• Reference drawing numbers
• Product or process characteristic to be controlled (including tolerances)
• Evaluation method (gages, sensors, visual checks, etc.)
• Sample size and sample frequency
• Control method ( x– and R chart, check sheet, go/no-go, poka- yoke, etc.)
• Reaction plan to be followed when the control method detects a problem
The control plan is the final link in a seamless chain that begins with the design failure modes and effects analysis (FMEA) (refer to Chapter 7 for details on FMEAs). Potential failure modes that cannot be prevented through design are
Chapter 4 Product and Process Control
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carried over to the process FMEA. Some failure modes can be prevented in the process through the use of poka- yokes or reduced to very low frequency through the use of designed experiments to optimize the process. Other failure modes can be detected with high confidence. Despite best efforts, some potential failures may still have unacceptable risk priority numbers (RPNs), and process controls must be added to monitor the process. The control plan should be checked to verify that all critical and significant characteristics identified during the design and process FMEAs are included.
At this point in the process, all nondestructive measurement systems listed in the control plan should have successfully passed the gage repeatability and reproducibility (R&R) requirement. Gage R&R studies are discussed in section F of this chapter. Sample sizes and sample frequencies should be based on statistically sound principles. Keep in mind that the sample frequency should be often enough to enable containment of suspect product prior to shipment to the customer. The QE plays a critical role in selecting the control method that is best suited for the characteristic being monitored.
Perhaps the most important aspect of the control plan methodology is the reaction plan. The reaction plan lists the steps to be taken by the operator when the control method indicates a problem. For example, what should happen when the X
– chart goes out of control? Unfortunately, many references and training semi- nars do not adequately develop this concept. The examples simply state “adjust and recheck” or “recalibrate and recheck.” Simplistic directions may lead to pro- cess tampering (overadjustment). In addition, opportunities for permanent correc- tive actions will be missed. (See Chapter 6, section F, for a complete discussion of control charts.)
Good reaction plans include four critical elements: containment, diagnosis, verification, and disposition.
1. Containment. As soon as the problem is identified, quarantine and segregate all suspect product. This may include everything produced since the previous acceptable sample. A good inventory management system that uses the principle of “first in, first out” will simplify the task of containment should it ever be needed. Provide specific direction to the operator on how to accomplish containment. It also may be wise to intensify inspection until the problem is resolved.
2. Diagnosis. Determine the root cause of the failure. It may be necessary to repeatedly ask, “Why?” For example, if the failure occurred because the operator was not adequately trained, then ask, “Why was the operator not properly trained?” Repeat this process until an appropriate root cause is identified that will lead to a permanent corrective action. Incorporate lessons learned from previous failures to facilitate the diagnostic process. Remember that in the heat of battle, common sense is not very common. Therefore, it is helpful to provide written guidance to the operator as to likely causes of the failure. In other words, specify the diagnostic steps and tests the operator should conduct during preliminary efforts to identify the root cause. If the root cause is still not identified, specify who should be called in to help, such as the product engineer or QE.
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3. Verification. Do not assume that the corrective action resolved the problem—prove it! Collect additional samples after the corrective change is implemented to verify that the problem is fixed. If possible, the reaction plan should specify how many additional samples are necessary before resuming normal operations.
4. Disposition. The obvious but nonetheless mandatory final step of the reaction plan is to determine an appropriate disposition for the material that was contained in the first step of the reaction plan. Typical dispositions include scrap, rework, sort, use as- is, and return to vendor. Written instructions are recommended for performing sorts or rework.
Figures 4.1 and 4.2 show an example of a control plan that was developed by a valve manufacturing company and incorporates many of the suggestions outlined above. The author uses code letters in the reaction plan section of the control plan. Detailed reaction plan instructions are provided on the second page.
Once the initial version of the control plan is released to production, the opera- tors should take ownership of the document and treat it as a living document, constantly reviewing and updating it with new information. There also should be a feedback mechanism in the process; as new or unexpected failure modes are discovered on the line, update the control plan and feed the information back to update the FMEAs. Keeping the documentation current will facilitate the advanced quality planning (AQP) process during future programs.
Hazard analysis and critical control points (HACCP) is traditionally a food safety management system, but is commonly used in many other industries. A critical control point is a step or point in the process in which a major or serious failure of the product can be introduced. The goal of HACCP is to prevent known hazards and reduce the risk of them occurring at points in the production cycle. Rodriguez- Perez (2012) describes a 12-step process for successfully developing an HACCP plan:
1. Create an HACCP team
2. Fully describe the product (including specifications)
3. Determine the product’s intended use
4. Create a process flow diagram (discussed in Chapter 5, section A)
5. Confirm the flow diagram on- site
6. Identify and analyze hazards
7. Determine the critical control points
8. Establish critical limits for each critical control point
9. Establish a monitoring procedure
10. Establish corrective action
11. Verify the HACCP plan
12. Document the HACCP process
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Figure 4.1 Control plan example: page 1.
Control plan number: CP714
Sta #
14 1 1 per hour
Drill press ACheck sheet
714648 0.060” min diameter
Machine needle bleed port on cover
0.60 (minus) gage pin S/N 15-50-2118
18 5 1 per shift
Torque driver
Report issue to floor manager
Report issue to floor manager
Report issue to floor manager
x– chart714647 714648
20 ± 5 IN LBPressure gage torque
Torque gage S/N 15-50-2019
23 3 per screw
2 per shift
Torque driver
Separate x– charts
714647 714648
60 ± 15 IN LBBody-cover screw torque
Torque gage S/N 15-50-2120
27 5 1 per shift
Torque driver
x– chart209647 209648
14 ± 7 IN LBSolenoid assembly torque
Torque gage S/N 15-50-2019
29 1 100%Test tank A, B, C, DGo/no-go209647 209648
Functional test and leak check
Final air test Visual: ref. QA spec 203795 Functional: ref. assy instruction
All 1 100%All See note 2
Go/no-go209647 209648
WorkmanshipAll Visual
Process description
Machine tools/
equipment
Reaction plan code
Methods
Print no.
Characteristic specification
Control method
Evaluation measurement equipment
Sample
Size Freq.
Part/assembly number/rev: 714647-H & 714648-J
Control plan revision level: C
Product line: Soft start air dump valve
Revision date: 12/01/2016
Originator: J. Hausner
Note 1: At all times, quarantine one hour’s worth of product before releasing to shipping. In the event of a final test failure, the last hour of production should be set aside for possible retest. This should be done on all final test failures with the exception of porosity.
Note 2: Compare suspect unit with visual accept/reject standards. If unit is unacceptable, stop the line and follow standard four-step reaction plan: (A) contain suspect units, (B) diagnose the root cause and implement corrective action, (C) verify that the corrective action is effective, (D) disposition suspect material (sort, scrap, rework, use as-is).
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Figure 4.2 Control plan example: page 2.
Control plan number: CP714
Failure mode Valve fails to open
Valve fails to close
Containment: Segregate nonconforming unit and previous hour of production for MRB. Disposition: Verify that wire leads and power supply are hooked up correctly. Verify needle port diameter > 0.060”. If port diameter is under spec, switch to 100% inspection for the next 50 units and notify the product engineer (PE) if another failure is found. Replace drill bit if hole is not drilled through or burrs are present. Verify that piston ring is installed and free of nicks. Verify that needle valve is open at least one complete turn. Verify that the solenoid port resistor is installed. Try another solenoid. If other tests fail, check diameter of diaphragm. Contact the PE if additional diagnosis is required. Verification: Verify that corrective action eliminates problem. Disposition: Scrap nonconforming components. Rework assemblies as necessary and retest 100% of the previous hour of production.
Containment: Segregate nonconforming product for MRB. Diagnosis: Verify that wire leads and power supply are hooked up correctly. Verify that flow control is open. Verify that diaphragm is installed correctly and check for voids in the seal bead. Verify that the dump hole is drilled completely through bonnet. Check that the fluid resistor is in place. Try another solenoid. If solenoid sticks open, quarantine current batch and switch to a new batch of solenoids. Contact PE if further diagnosis is required to determine cause. Verification: Verify that corrective action eliminates problem. Notify PE if another failure is found on the next 50 units. Disposition: Scrap nonconforming components. Rework assembly and retest.
Body–bonnet leak
Containment: Segregate nonconforming product for MRB. Diagnosis: Verify torque. For torque adjustments, see Reaction Code “E” below. Ensure that diaphragm is installed correctly and that there are no voids present on the bead. Verify that the bead grooves on the bonnet and body are free of nicks or porosity and the diameters are within tolerance. Verify that the milled slot on the body is within tolerance. Contact PE if further diagnosis is required. Verification: Verify that corrective action eliminates problem. Disposition: Scrap nonconforming components. Rework assembly and retest. Contact line lead or PE if there are two or more consecutive failures or three failures within one hour.
Leak at fittings
Containment: Segregate nonconforming product for MRB. Diagnosis: Verify that fittings are installed correctly and have the correct torque applied. Verify that the threads on the fitting and assembly are free of nicks or porosity. Contact PE if further diagnosis is required. Verification: Verify that corrective action eliminates problem. Notify PE if another failure is found on the next 50 units. Disposition: Scrap nonconforming components. Rework assembly and retest.
Torque out of spec
Containment: Segregate nonconforming product for MRB. Diagnosis: Verify torque using another torque gage. For torque adjustments, take at least 10 samples and adjust torque gun if average is more than one standard deviation away from the nominal. Notify maintenance if average is close to nominal and there are any observations out of spec. Contact PE for further diagnosis. Verification: Measure a minimum of three subgroups and verify that the process is near nominal and in control. Disposition: If undertorqued, retorque assembly. If overtorqued, replace screw(s) and retorque.
SPC out of control, but parts in spec
Refer to QA/SPC procedure 231573. Comply with SPC procedure requirements. Document the root cause and corrective action in a note on the control chart.
Reaction plan Code
Control plan revision level: C
Revision date: 12/01/2016
Part/assembly number/rev: 714647-H & 714648-J
Product line: Airlogic control valve series
Key contact: J. Hausner
Part name/description: Soft start air dump valve HG & HJ series
Originator: J. Hausner
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To determine critical control points, each step in the process is examined to see if any of the identified hazards could occur and if any control measures exist. The HACCP team assesses which steps in the process are critical control points. As Rodriguez- Perez writes, “If the hazard can be controlled adequately, and is not best controlled at another step, and is essential for process safety, then this step is a critical control point for the specified hazard.” If there exists a step in the pro- cess where a hazard could occur and there are no control measures in place, this high- risk condition must be addressed.
Work instructions provide details for personnel who have direct responsibility for the operation of the process. The instructions must be documented and posted or readily accessible at the work site. Assembly instructions list each task to be per- formed in sequential order. Setup instructions list appropriate machine settings, such as feed rates, temperatures, and pressures. Setup instructions should also list any tasks or inspections that must be performed during production start- up to verify that the process is properly adjusted. Work instructions must be clear and understandable. Liberal use of sketches, charts, photographs, and other visual aids is strongly encouraged. The effort to eliminate opportunities for error in the process should include the work instructions. Therefore, organizations that pro- duce a variety of similar products should consider creating unique instructions for each model, rather than using generic examples, look- up tables for bills of mate- rial, and cross- referenced setup instructions.
B. MaTEriaL ConTroL Material control addresses the raw materials, work in process (WIP), and final products and how they are physically controlled, identified, and tracked. The first step in control is classification; the last step is disposition.
Material control is based on identification and classification. Systems, compo- nents, nonconformities, and features are all subject to classification schemes, and only after a classification has been done can the appropriate control be applied. There are many different classification factors that should be considered, including:
• Volume of production
• Complexity
• Cost
• Expected lifetime
• Amount of maintenance required
• Risk to safety and/or the environment
If the product tends to be complex, expensive, and long- lived, then a great deal of effort must be expended in developing the material control scheme. Commodity- type products may require very little in the way of material control, but even the simplest products must be controlled in simple and inexpensive ways. Such issues as process selection, inspection method, amount of sampling, strictness of inspection, and control of deviating material must be decided with respect to the importance of each characteristic and each component. Every good
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QE should spend time thinking about issues of relative importance and criticality. Think about the Pareto chart (see Chapter 5, section A, for discussion of Pareto charts). Collect data and opinions so that even before production begins, a scheme of relative importance is clearly established.
This process requires careful study by a number of individuals. It is an exer- cise in clarification, in making distinctions, and in clearing up confusion. The task requires input from several sources of expertise to ensure that a balanced result is obtained. The people involved should include representatives of product design, safety, marketing, and field service.
B.1. Material identification, Status, and Traceability
Configuration management (CM) is defined in ANSI/EIA-649 (National Consensus Standard for Configuration Management) as a process for establishing and main- taining consistency of a product’s performance, functional, and physical attributes with its requirements, design, and operational information throughout its life. The Department of Defense (DoD) created the handbook MIL- HDBK-61A(SE) (2001) to provide guidance for acquisition managers, logistics managers, and others responsible for CM. Material identification, status, and traceability can be achieved through CM. We discuss these three important items in this section.
Without product traceability, many manufacturers would be exposed to unac- ceptable risk, especially in the event of a recall. Modern technology has produced a wide array of identification methods. The physical application of markings and subsequent tracking by means of scanners and sensors provide many options. It is necessary to maintain records not only of items produced and their identification, but also of how the record- keeping system itself is operated and modified. The storage and retrieval of information is still a rapidly changing field.
One of the most effective identification methods is radio frequency identifica- tion (RFID). By using radio frequency tags, information can be provided about, for example, identification, tracking, and security. RFID technology has been used in supply chain management, inventory tracking, and the healthcare industry. In healthcare, RFID technology has been implemented for asset management (e.g., determining where mobile medical devices are at all times), patient care (e.g., determining where a patient is at all times while hospitalized), and inventory management (e.g., reducing the chance of inventory being out of stock at criti- cal times).
To illustrate the mechanics of product identification, consider the case of the Sauer Danfoss Company in Ames, Iowa. This company makes moderately com- plex mechanical products that require 100% testing and periodic design modifi- cations. The company improved its materials management system by creating a multifunctional task team of four people. The team collected data for two and a half years and finally decided to scrap the existing system for tracking material, which was dependent on manual entry on paper “move tags” and then manual keying into a computer database. Determination of current status required fre- quent physical count of all items.
The team switched to a system of using bar code and RFID technologies. Now, whenever an item of hardware moves, it is automatically accounted for, either by a bar code scanner or by an RFID receiver. A sophisticated database system
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automatically processes each scan and maintains a variety of characteristics about each unit, including:
• Model number
• Unit number
• Date produced
• Result of test
• Date of test
• Rework record
Product identification is vital when producing complex products, but is often unnecessary for mundane commodities. The StarLink corn seed recall in 2000 provides an example of the consequences of failure to identify and distinguish product (A. Pollack, “Kraft Recalls Taco Shells with Bioengineered Corn,” New York Times, September 23, 2000, http://www.nytimes.com/2000/09/23/business/ kraft-recalls-taco-shells-with-bioengineered-corn.html).
StarLink was a form of seed corn that was approved for growing animal feed but not for human consumption. Inadequate controls were put in place when the seed corn was sold to farmers, and as a result the animal feed corn was inextrica- bly intermixed with human- consumption corn at grain elevators throughout the Midwest. At the time they were delivering the corn, neither the farmers nor the grain elevator operators realized there was a problem. But later, consumer groups that were testing products made out of this corn detected the use of the unaccept- able corn. A great outcry resulted, and many losses were incurred as both types of the intermixed corn had to be converted to animal feed.
Some companies use alternative product identification schemes. For example, a 10-digit alphanumeric product identification code can be used to allow traceabil- ity to a diverse set of factors, including the date of fabrication, the supplier of each subsystem, the product model, and the date of final assembly. Several things must be considered when setting up such a code, such as the amount of liability expo- sure, the number of levels of components and subcomponents, and the process design, which must incorporate the ability to trace products back to their point of creation and installation.
Traceability is an explicit part of the ISO 9000 and ISO/TS 16949 standards. See paragraph 8.5.2 in ANSI/ISO/ASQ 9001:2015, for example. Traceability is like a pedigree for a dog breed: it allows one to learn the history of any item. Com- modity products such as nuts and bolts have limited needs for traceability, but even here, wise manufacturers will keep different lots segregated and identified as long as it is economically possible. Complex products such as automobiles must have multiple paths to trace back through many levels and many different sources. Sensitive material such as pharmaceuticals and food products must be traceable at all times. As an example of a traceability issue, consider the Heparin contami- nation in 2008 (US Food and Drug Administration, “Information on Heparin,” https://www.fda.gov/Drugs/DrugSafety/PostmarketDrugSafetyInformation forPatientsandProviders/ucm112597.htm). Heparin is a blood- thinning drug uti- lized during surgery to help prevent clotting. After reports of patient reactions, an investigation by the Food and Drug Administration led to the discovery of
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an ingredient in the drug that was the likely cause of adverse patient outcomes, including death. For further reading, see the article “3 Ways Quality Control Supports Traceability for Recalls” by Doug Fair (https://www.pharmpro.com/ article/2016/10/3-ways-quality-control-supports-traceability-recalls).
The ISO 9001 standard requires product identification and traceability, where appropriate, for recall of nonconforming product, hazardous product, or product in conflict with laws, regulations, or statutes. Product identification must be pro- vided when required by a customer. Properly identified items must have a unique number and are tracked by their location in the process. Differences between items and between lots must be distinguishable.
The place to start with traceability, that is, the ability to preserve the identity of the product and its origins, is when the process is first designed. Today, appro- priate software and database designs are available. Training of workers may be required in order to create the proper climate and means to accomplish this.
Gryna (1988a) listed four reasons why traceability is needed:
1. Assure that only materials and components of adequate quality enter the final product, for example, sterility of drug materials, adequate metallurgical composition, and heat treatment of structural components
2. Assure positive identification to avoid mix- up of products that otherwise look alike
3. Permit recall of suspected product on a precise basis. Lacking traceability programs, huge recalls of automobiles and other products have been required in the past while the number of defectives in the recalled set was often quite small
4. Localize causes of failure and take remedial action at minimal cost
There are other uses of traceability, such as in inventory control and scheduling. Some of these uses also affect quality. For example, use of materials on a first- in, first- out basis reduces the risk of quality deterioration of perishable materials.
It is important to consider factors related to the product. For example, the fol- lowing questions can help guide traceability:
• What is the cost of the product? A more expensive product requires more accountability over time, and thus better traceability.
• How long will the product last? If it is going to be around a long time, there is more concern about its origin, as new discoveries often are made of chemical characteristics and environmental effects. The discovery that asbestos was a carcinogen after its routine use for decades is a good example.
• Will the product be built into another product?
• Does the product have items or materials in it that have not been thoroughly evaluated over a long period of time?
• Is there a significant possible health hazard associated with the product?
• Are field modifications often required, with different replacement items required on different models? (Automobiles are a prime example.)
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Ten items to consider in a traceability program are the following:
1. Product category
2. Product life
3. Unit cost
4. Recall or modification in the field
5. Product complexity
6. Level of downstream traceability
7. Documents providing traceability
8. Type of identification
9. Coded versus uncoded identification
10. Method of identification (e.g., tags, name plates, ink stamps)
The use of a tracing code is required for efficient operation (Feigenbaum 2004). This code is established at the beginning of material flow, and a traceability flow- chart is established. The major activities on the flowchart include the following:
1. Critical component selection and listing by part number.
2. Vendor part coding (recording vendor name and date of receipt).
3. Coding internally manufactured parts, subassembly, assembly, and storage in a daily tally. At the end of the assembly line, each shipping container is date coded. This sequential coding procedure provides sufficient data to tie critical components to specific dates of receiving inspection, manufacturing, and final assembly.
4. Computerized shipping records, including date codes, customer name, and destination. Correlation of these data with tracing code numbers results in very effective traceability of critical components.
B.2. Material Segregation
There are two major situations that demand disposition of nonconforming prod- ucts. The first is when a product fails to pass inspection or a test and a decision regarding it must be made. This is the function of the material review board, dis- cussed in section B.4 of this chapter. The second situation, considerably more seri- ous, is when a problem develops after the product is out of the plant, on store shelves, in dealer showrooms, and in use by customers. Now a product recall may be required. In view of the very negative aspects of product recall, all the prior work concerning product traceability and product integrity will pay off quite handsomely in organizing the recall.
B.3. Material Classification
In certain types of products, more than one defect could be present and a relatively small number of these minor defects could be acceptable to the customer. Product quality in these cases may be judged by the total number of defects or the number of defects per unit. Control charts for attributes are a tool that may be used for this
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purpose (see Chapter 6, section F, for details). In such cases, the objective of inspec- tion is to determine the number of defects or nonconformities present rather than to classify units as conforming or nonconforming.
“Defect” and “nonconformity” are two terms that may be used synonymously in many situations. However, in other situations, they have slightly different defi- nitions. A nonconformity is defined as a failure of a quality characteristic to meet its intended level or state, occurring with severity sufficient to cause the product not to meet a specification. A defect is a nonconformity severe enough to cause the product not to satisfy normal usage requirements. Thus, the difference between the term “nonconformity” and the term “defect” is based mainly on perspective. The former is defined based on specifications, while the latter is defined based on fitness for use. The numerical result generated by inspection consists of the count of defects or nonconformities for each product unit. Often it is possible to classify the different types of defects according to their severity, and then assign a weight to each class based on the importance of the affected quality characteristic that relates to the product specifications. The selection of the weights should reflect the relative importance of the various defect categories and their likelihood of caus- ing product failure or customer dissatisfaction. A typical seriousness classification includes four levels of defect seriousness:
1. Critical defects may lead directly to severe injury or catastrophic economic loss.
2. Serious defects may lead to injury or significant economic loss.
3. Major defects may cause major problems during normal use. A major defect will likely result in reducing the usability of the product.
4. Minor defects may cause minor problems during normal use.
See Montgomery (2013) for discussion of defect levels.
B.4. Material review Board
The material review board (MRB) is an appointed group of individuals with differ- ent backgrounds and expertise. Their assignment is to determine what corrective actions must be taken after nonconforming parts or components are discovered. In a larger sense, the purposes of the MRB are to determine the disposition of nonconforming parts, components, and subassemblies; determine the causes of the nonconformance of these items; and take the necessary corrective actions to prevent such nonconformance from taking place in future production.
The basic function of an MRB is to (1) review material that does not conform to standard, (2) determine what its disposition should be, and (3) drive the devel- opment of effective corrective action to prevent recurrence.
The MRB is a broad- based reviewing agency whose membership usually con- sists minimally of representatives from the following:
• Engineering. The cognizant designer is often the representative.
• Quality assurance. The representative is often from quality control engineering.
• Customers. The representative may be from the customer’s organization (e.g., the government inspector) or from marketing. Note that customers can also represent an impacted group if “bad” material was used.
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In some companies, the role of the MRB is solely one of judging fitness for use of nonconforming products. Bond (1983) discusses board composition, philosophy, and problem documentation.
In general, the MRB procedural steps can be summarized as follows: After a defect is discovered, verification by inspection may be needed. A complete description of any nonconformance is then initiated. A QE picked by the MRB will review the facts and include the case in an appropriate tracking system. The MRB committee may then follow up with investigation and analysis. After the investi- gation and analysis, the case goes back to the QE, who recommends the appropri- ate corrective action(s) and steps for implementation.
The term “standard repair” is common within the MRB framework. It signi- fies a procedure where a certain type of defect occurs time and again. A standard repair procedure is then initiated, documented, and implemented for such situ- ations. Minor defects are most likely to be treated with a standard repair proce- dure. Within the context of defect classification, defects may further be classified as major or minor. Minor defects, unlike major ones, may not adversely affect the integrity of the part, component, or assembly.
In many cases, the MRB concludes that the lot containing nonconforming products should not be shipped as is. Then, along with inspection personnel, the MRB makes the decision to sort (100% inspection), downgrade, repair, rework, or scrap the nonconforming products. A decision to ship also may be authorized by the MRB. In such cases, a unanimous decision should be reached by all members. The decision should also create factual data and thus is an important source of information. A successful MRB program requires that the board not only make decisions about immediate disposition of rejected material, but also direct ongo- ing programs of root cause analysis to eliminate future rejections of the same type.
C. aCCEPTanCE SaMPLing Acceptance sampling is a method for inspecting the product. Inspection can be done with screening (also called sorting or 100% inspection), in which all units are inspected, or with sampling. Acceptance sampling is the process of inspecting a portion of the product in a lot for the purpose of making a decision regarding classification of the entire lot as either conforming or nonconforming to quality specifications.
Whether inspection is done with screening or with sampling, the results of inspection can be used for different purposes as follows:
1. To distinguish between good lots and bad lots using acceptance sampling plans (as in incoming material inspection and final product inspection).
2. To distinguish between good products and bad products.
3. To determine the status of process control and if the process is changing. This is usually done in conjunction with control charts.
4. To evaluate process capability. In this case, inspection is used to determine if the process exhibits excessive variation and if it is approaching or exceeding the specification limits.
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5. To determine process adjustment. Based on inspection results of process output, as depicted by a histogram, for example, the process mean may require adjustment and/or process variation may need to be reduced. A process might require adjustment even though all the units produced to date conform to the quality standards agreed on with the customer.
6. To rate the accuracy of inspectors or of inspection equipment by comparing the inspection results with corresponding standards. An inspection operation can result in two types of errors: (1) classification of a conforming unit as nonconforming and (2) classification of a nonconforming unit as conforming. The probabilities of both types of errors can be easily estimated using probability theory and other statistical methods.
7. To serve as a mechanism for evaluating vendors in terms of their products’ quality. Vendors that consistently deliver high- quality products can receive preferred status involving reduced inspection and priority in bidding for new contracts, while vendors that do not stand up to quality requirements could be warned or discontinued altogether. This type of procedure is known as vendor qualification or vendor certification.
The last three uses of inspection might be seen as feedback about the production processes, the measurement processes, and the supplier.
Sampling provides the economic advantage of lower inspection costs due to fewer units being inspected. In addition, the time required to inspect a sample is substantially less than that required for the entire lot, and there is less damage to the product due to reduced handling. Most inspectors find that selection and inspection of a random sample is less tedious and monotonous than inspection of the complete lot. Another advantage of sampling inspection is related to the supplier/customer relationship. By inspecting a small fraction of the lot and forc- ing the supplier to screen 100% in case of lot rejection (which is the case for recti- fying inspection), the customer emphasizes that the supplier must be concerned about quality. On the other hand, the variability inherent in sampling results in sampling errors: rejection of lots of conforming quality and acceptance of lots of nonconforming quality.
Acceptance sampling is most appropriate when inspection costs are high and when 100% inspection is monotonous and can cause inspector fatigue and bore- dom, resulting in degraded performance and increased error rates. Obviously, sampling is the only choice available for destructive inspection. Rectifying sam- pling, where sample units detected as nonconforming are discarded from the lot and either replaced with conforming units or repaired, is a form of acceptance sampling. Rejected lots are subject to 100% screening, which can involve discard- ing, replacing, or repairing units detected as nonconforming.
In certain situations, it is preferable to inspect 100% of the product. This would be the case for critical or complex products, where the cost of making the wrong decision would be too high. Screening is appropriate when the fraction noncon- forming is extremely high. In this case, most of the lots would be rejected under acceptance sampling and those accepted would be so as a result of statistical varia- tions rather than better quality. Screening is also appropriate when the fraction nonconforming is not known and an estimate based on a large sample is needed.
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C.1. Sampling Concepts
Sampling may be performed according to the type of quality characteristics to be inspected. There are three major categories of sampling plans: sampling plans for attributes, sampling plans for variables, and special sampling plans. It should be noted that acceptance sampling is not advised for processes in continuous produc- tion and in a state of statistical control. For these processes, Deming (1986) pro- vides decision rules for selecting either 100% inspection or no inspection.
There are risks involved in using acceptance sampling plans. These risks are producer’s risk and consumer’s risk, and correspond with type I and type II errors, respectively, in hypothesis testing (type I and type II errors are discussed in Chapter 6). Producer’s risk and consumer’s risks are defined as follows:
• Producer’s risk (α). The producer’s risk for any given sampling plan is the probability of rejecting a lot that is within the acceptable quality level. This means that the producer faces the possibility (at level of significance α) of having a lot rejected even though the lot has met the requirements stipulated by the acceptable quality limit (AQL).
• Consumer’s risk (β). The consumer’s risk for any given sampling plan is the probability of acceptance (often 10%) for a designated numerical value of relatively poor submitted quality. The consumer’s risk, therefore, is the probability of accepting a lot that has not met the requirements stipulated by the lot tolerance percent defective (LTPD) level, the poorest quality in an individual lot that should be accepted. The LTPD has a low probability of acceptance. In many sampling plans, the LTPD is the percent defective having a 10% probability of acceptance.
The average outgoing quality (AOQ) is the expected average quality of outgoing products, including all accepted lots and all rejected lots that have been sorted 100% and have had all of the nonconforming units replaced by conforming units. There is a given AOQ for specific fractions nonconforming of submitted lots sam- pled under a given sampling plan. When the fraction nonconforming is very low, a large majority of the lots will be accepted as submitted.
The few lots that are rejected will be sorted 100% and have all nonconforming units replaced with conforming units. Thus, the AOQ will always be less than the submitted quality. As the quality of submitted lots declines in relation to the AQL, the percentage of lots rejected increases in proportion to accepted lots. As these rejected lots are sorted and combined with accepted lots, an AOQ lower than the average fraction of nonconformances of submitted lots emerges. Therefore, when the level of quality of incoming lots is good, the AOQ is good; when the incoming quality is bad and most lots are rejected and sorted, the result is also good.
To calculate the AOQ for a specific fraction nonconforming (p) and a sampling plan, the first step is to calculate the probability of accepting the lot at that level of fraction nonconforming. Then, multiply the probability of acceptance (Pa) by the fraction nonconforming for the AOQ. Thus,
AOQ = −P p
n Na
1< <
(4.1)
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where N is the lot size and n is the sample size. If the desired result is a percentage, multiply by 100. If the lot size is assumed infinite (theoretically), then AOQ ≅ Pap.
The average outgoing quality limit (AOQL) is the maximum AOQ for all pos- sible levels of incoming quality. The AOQ is a variable dependent on the quality level of incoming lots. When the AOQ is plotted for all possible levels of incoming quality, a curve as shown in Figure 4.3 results. AOQL is the highest value on the AOQ curve. The probability of acceptance can be calculated using the binomial or Poisson distribution and will be discussed in section C.1.a.
C.1.a. Single Sampling Plans and the OC Curve
A single sampling plan is one where the decision to either accept or reject the lot is based on the results of the inspection of a single sample of n items randomly selected from a submitted lot. A single sampling plan is defined by a sample num- ber n and acceptance number c. If more than c items in the sample are determined to be defective, the lot is rejected. If c or fewer items in the sample are determined to be defective, the lot is accepted. Single sampling plans have the advantage of ease of administration, but due to the unchanging sample size, they do not take advan- tage of the potential cost savings of reduced or tightened inspection when incoming quality is either excellent or poor.
For continuing processes, sampling plans based on average quality protection have characteristics calculated from the binomial and/or Poisson distributions. For processes not considered to be continuing, sampling plans based on lot- by-lot protection have characteristics calculated from the hypergeometric distribution, which takes the lot size into consideration.
Figure 4.3 AOQ curve for a single sampling plan.
0.00
0.00 0.05 0.10 0.15 0.20
0.01
0.02
A O
Q
p (fraction nonconforming)
0.03
0.04
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Sampling plans based on the Poisson and binomial distributions are more common than those based on the hypergeometric distribution. No matter which type of attribute sampling plan is being considered, an important evaluation tool is the operating characteristic (OC) curve.
The OC curve allows a sampling plan to be evaluated at a glance by graphi- cally displaying the probabilities of accepting lots submitted at varying levels of percent nonconforming. The OC curve illustrates the risks involved in acceptance sampling. Figure 4.4 shows an OC curve for a single sampling plan with sample size n of 50 drawn from an infinite lot size, with an acceptance number c of 3.
As seen in the OC curve, if the lot were 100% to specifications, the probability of acceptance Pa also would be 100%. But if the lot were 13.4% defective, there would be approximately a 10% probability of acceptance.
There are two types of OC curves to consider: (1) type A OC curves and (2) type B OC curves. Type A OC curves are used to calculate the probability of acceptance on a lot- by-lot basis when the lot is not a product of a continuous pro- cess. These OC curves are calculated using the hypergeometric distribution.
Type B OC curves are used to evaluate sampling plans for a continuous process or for a process where a lot of size N is large. These curves are based on the bino- mial and/or Poisson distributions when the requirements for usage are met. In general, the ANSI/ASQ Z1.4-2003 (R 2013) standard OC curves are based on the binomial distribution for sample sizes through 80 and the Poisson approximation to the binomial for sample sizes greater than 80.
The Poisson approximation to the binomial was often employed for calculat- ing the probability of acceptance (Pa) when the sample sizes of interest were quite large. This approximation was used because the computations needed to calculate the binomial probabilities were often impractical. With modern software, the bino- mial computations are no longer a problem. However, Poisson approximations are still sometimes used for the binomial distribution. This approximation is accept- able when p < 0.1.
In the examples that follow, it is assumed that the process of interest is con- tinuous (in theory). Since the process is considered continuous, the lot size is not
Figure 4.4 An operating characteristic (OC) curve for n = 50 and c = 3.
0.0
0.00 0.05 0.10 0.15 0.20
0.2
0.4
0.6
P a (
p ro
b a b
il it
y o
f a c c e p
ta n
c e )
p (fraction nonconforming)
0.8
1.0
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taken into consideration in the calculations of Pa. Suppose a sample of n = 50 is randomly selected from the lot. Furthermore, the lot is accepted if three or fewer nonconformities are found in the sample (c = 3). To create the OC curve, plot the probability of acceptance versus various values of p, the fraction nonconforming. Statistical software can readily create these curves.
The probability of acceptance Pa can be calculated using the binomial prob- ability mass function (pmf) (see Chapter 6, section C, for complete details on the binomial distribution and the cumulative distribution function [cdf]). The prob- ability of acceptance is given by
P P d c p pa
d n d
d
c
= ≤( ) = ( ) − =
∑ 1 0
− n
d n d−( ) !
! ! d d
(4.2)
where
d = the number of nonconforming items
c = the acceptance number
p = fraction nonconforming
The probability of acceptance of the lot for a single sampling plan with n = 50 and c = 3 can be found for values of p using Equation (4.2); the probabilities are shown in Table 4.1.
Table 4.1 Probability of acceptance for various levels of fraction nonconforming.
p Pa
0.01 0.9984
0.02 0.9822
0.03 0.9372
0.04 0.8609
0.05 0.7604
0.06 0.6473
0.07 0.5327
0.08 0.4253
0.09 0.3303
0.10 0.2503
0.15 0.0460
0.20 0.0057
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The OC curve can be constructed similarly to the one in Figure 4.4 for various values of p.
The probabilities of acceptance can also be calculated using the Poisson distri- bution (discussed in Chapter 6, section C). The probability of acceptance is
P P d c
np e
da
d np
d
c
= ≤( ) = ( ) −
= ∑
!0 (4.3)
where np is the mean of the binomial distribution and therefore the necessary parameter for the Poisson distribution. Tables providing probabilities for the bino- mial distribution can be found in Appendix I and in Appendix L for the Poisson distribution.
The OC curve is useful for a number of quantities of interest such as the AQL and LTPD. As part of the revision of ANSI/ASQC Z1.4-1993, “acceptable quality level” was changed to “acceptable quality limit” in ANSI/ASQ Z1.4-2003 and is defined as the quality level that is the worst tolerable process average when a continuing series of lots is submitted for acceptance sampling. This means that a lot that has a fraction defective equal to the AQL has a high probability (gen- erally in the area of 0.95, although it may vary) of being accepted. As a result, plans that are based on the AQL, such as ANSI/ASQ Z1.4-2003 (R2013), favor the producer in getting lots accepted that are in the general neighborhood of the AQL for fraction defective in a lot.
C.1.b. Lot Size, Sample Size, and Acceptance Number
For any single sampling plan, the plan is completely described by the lot size, sample size, and acceptance number. In this section, the effect of changing the sample size, acceptance number, and lot size on the behavior of the sampling plan will be explored along with the risks of constant percentage plans.
Consider the previous example of a single sampling plan with n = 50, c = 3, and p = 0.01 to 0.10 by 0.01. The AOQ values are given in Table 4.2. The resulting AOQ curve is displayed in Figure 4.4. The AOQL is approximately 0.03884.
The effect on the OC curve for a single sampling plan caused by changing the sample size while holding all other parameters constant is shown in Figure 4.5. The probability of acceptance changes considerably as sample size changes. The probability of acceptance for the given sample sizes for a 10% nonconforming lot and an acceptance number of zero is shown in Table 4.3.
The effect of changing the acceptance number on a single sampling plan while holding all other parameters constant is shown in Figure 4.6. Another point of interest is that for c = 0, the OC curve is concave in shape, while plans with larger acceptance numbers have a “reverse s” shape. Figure 4.6 and Table 4.4 show the effect of changing the acceptance number of a sampling plan on the indifference quality level (50–50 chance of accepting a given percent defective).
The parameter having the least effect on the OC curve for the single sampling plan is the lot size N. Figure 4.7 shows the changes in the OC curve for a sample size of 10, acceptance number of 0, and lot sizes of 100, 200, and 1000. For this reason, using the binomial and Poisson approximations, even when lot sizes are
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Table 4.2 AOQ levels for various levels of fraction nonconforming.
p Pa AOQ
0.01 0.9984 0.00998
0.02 0.9822 0.01964
0.03 0.9372 0.02812
0.04 0.8609 0.03444
0.05 0.7604 0.03802
0.06 0.6473 0.03884
0.07 0.5327 0.03729
0.08 0.4253 0.03402
0.09 0.3303 0.02973
0.10 0.2503 0.02503
0.15 0.0460 0.00690
0.20 0.0057 0.00114
Figure 4.5 Effect on an OC curve of changing sample size (n) when accept number (c) is held constant.
0
0.10
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.0
n = 1
n = 2
n = 4
n = 10
0.20
0.30
0.40
0.50
0.60
0.70
P a
p (fraction nonconforming)
0.80
0.90
1.00
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Table 4.3 Probability of acceptance for various n.
Sample size (n) Probability of acceptance (Pa)
10 0.35
4 0.66
2 0.81
1 0.90
Figure 4.6 Effect of changing accept number (c) when sample size (n) is held constant.
0
0.10
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.0
c = 0
c = 1
c = 2
n = 10
Indifference quality level
0.20
0.30
0.40
0.50
0.60
0.70
P a
p (fraction nonconforming)
0.80
0.90
1.00
Table 4.4 Fraction defective at indifference quality level.
Sample size (n) Acceptance number (c)
Percent defective at indifference quality level
10 2 0.26
10 1 0.17
10 0 0.07
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known (and are large compared with sample size), results in little error in accu- racy. Some key probabilities of acceptance points for the three lot sizes are dis- played in Table 4.5. The differences due to lot size are minimal.
Computing the sample size as a percentage of the lot size has a large effect on risks and protection, as shown in Figure 4.8. In this case, plans having a sample
Figure 4.7 Effect of changing lot size (N) when acceptance number (c) and sample size (n) are held constant.
0
0.10
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.0
N = 1000
N = 200
N = 100
n = 10
c = 0
0.20
0.30
0.40
0.50
0.60
0.70
P a
p (fraction nonconforming)
0.80
0.90
1.00
Table 4.5 Probability of acceptance for various lot sizes.
Fraction defective (p)
Probability of acceptance (Pa) Lot size (N)
0.10 0.330 100
0.30 0.023 100
0.50 0.001 100
0.10 0.340 200
0.30 0.026 200
0.50 0.001 200
0.10 0.347 1000
0.30 0.028 1000
0.50 0.001 1000
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size totaling 10% of the lot size are shown. The degree of protection changes dra- matically with changes in lot size, which results in low protection for small lot sizes and gives excessively large sample requirements for large lot sizes.
C.2. Sampling Standards and Plans
Acceptance sampling by attributes generally is used for two purposes: (1) pro- tection against accepting lots from a continuing process whose average qual- ity deteriorates beyond an acceptable quality level, and (2) protection against isolated lots that may have levels of nonconformances greater than can be con- sidered acceptable. The most commonly used form of acceptance sampling is sampling by attributes. The most widely used standard of all attribute plans is ANSI/ASQ Z1.4. The following sections provide more details on the character- istics of acceptance sampling and discussion of military standards in acceptance sampling.
There are several types of attribute sampling plans available, with the most common being single, double, multiple, and sequential sampling plans. The type of sampling plan used is determined by ease of use and administration, general quality level of incoming lots, and average sample number.
C.2.a. Double and Multiple Sampling Plans
When using double sampling plans, a smaller first sample is taken from the submitted lot, and one of three decisions is made: (1) accept the lot, (2) reject the
Figure 4.8 OC curves for sampling plans having the sample size equal to 10% of the lot size.
0
0.10
0.10 0.20 0.30 0.40 0.50
N = 250, n = 25, c = 0
N = 100, n = 10, c = 0
N = 50, n = 5, c = 0
0.20
0.30
0.40
0.50
0.60
0.70
P a
p (fraction nonconforming)
0.80
0.90
1.00
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lot, or (3) draw another sample. If a second sample is to be drawn, the lot will either be accepted or be rejected after the second sample. Double sampling plans have the advantage of a lower total sample size when the incoming quality is either excellent or poor because the lot is either accepted or rejected on the first sample.
To calculate the OC curve for a double sampling plan, Equations (4.2) and (4.3) can again be utilized. To calculate probabilities of acceptance, some arbitrary points for p are chosen to cover the range of the OC curve. The fraction defective p is then multiplied by n1 (the first sample) or n2 (the second sample) to determine the expected value np.
The generalized formula for calculating the probability of acceptance (Pa) is:
Pa = p0 + (p1p2 + p1p1 + p1p0) + (p2p1 + p2p0) (4.4)
where:
p0 = probability of zero nonconformities in the first sample
pi pj = probability of i nonconformities in the first sample times the probability of j nonconformities in the second sample for all i and j
ExaMpLE 4.1
A double sampling plan is to be executed as follows: take a first sample (n1) of 75 units and set c1 (the acceptance number for the first sample) at 0. The lot will be accepted based on the first sample results if no nonconformances are found in the first sample.
If three nonconformances are found in the first sample, the lot will be rejected based on the first sample results. If after analyzing the results of the first sample one or two nonconformances are found, take a second sample (n2 = 75). The acceptance num- ber for the second sample (c2) is set to 3. If the combined number of nonconformances in the first and second samples is three or fewer, the lot will be accepted and if the com- bined number of nonconformances is four or more, the lot will be rejected. The plan is represented as shown in Table 4.6.
Table 4.6 Double sampling plan.
Sample size Acceptance number (c)
Rejection number (r)
n1 = 75 c1 = 0 r1 = 3
n2 = 75 c2 = 3 r2 = 4
Continued
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To determine the technique of plotting the OC curve, 3 points for p may be used (0.01, 0.0 4, and 0.08), although in practice 6 –10 should be used. The points for the OC curve are calculated using the generalized equation for each fraction nonconform- ing (Equation (4.2)), selected as shown in Table 4.7.
Table 4.7 OC curve calculations for double sampling plan.
Generalized equation values p = 0.01 p = 0.04 p = 0.08
p0 0.470587 0.04681 0.001923
p1p0 0.167767 0.006848 0.000024
p1p1 0.127096 0.021399 0.000157
p1p2 0.0475 0.032989 0.000506
p2 p0 0.062701 0.010557 0.000078
p2 p1 0.0475 0.032989 0.000506
Totals for Pa 0.923151 0.151592 0.003195
These points are used to construct the OC curve for the double sampling plan, as shown in Figure 4.9.
0
0.10
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.20
0.30
0.40
0.50
0.60
0.70
P a
p (fraction nonconforming)
0.80
0.90
1.00
n1 = 75, c1 = 0, r1 = 3
n2 = 75, c2 = 3, r2 = 4
Figure 4.9 OC curve for double sampling plan where n1 = 75, c1 = 0, r1 = 3, n2 = 75, c2 = 3, r2 = 4.
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Multiple sampling plans work in the same way as double sampling, but with an increase in the number of samples to be taken up to seven, according to ANSI/ ASQ Z1.4-2003 (R2013). In the same manner that double sampling is performed, acceptance or rejection of submitted lots may be reached before the seventh sam- ple, depending on the acceptance/rejection criteria established for the plan.
C.2.b. AOQ, AOQL, and Average Sample Number
The AOQ curve and AOQL for double and multiple sampling plans are plotted and determined in the same manner as for single sampling plans. An AOQ curve for the double sampling plan described in Example 4.1 is shown in Figure 4.10; the AOQL is approximately 1.3%.
The average sample number (ASN) is a determination of the expected average amount of inspection per lot for a given sampling plan. The ASN for single sam- pling plans is a constant value that is equal to the single sample size for the plan. The ASN for double sampling plans is the sum of the first sample size plus the sec- ond sample size times the probability that a second sample will be required. The ASN is also a function of fraction nonconforming when working with a double sampling plan. The double sampling plan ASN formula is:
ASN = n1 + n2P2 (4.5)
where:
n1 = size of first sample
n2 = size of second sample
P2 = probability of requiring a second sample
Figure 4.10 Average outgoing quality curve for double sampling plan.
0
1.0
1 2 3 4 5 6 7 8 9 10
n1 = 75, c1 = 0, r1 = 3
n2 = 75, c2 = 3, r2 = 42.0
p (percent nonconforming)
3.0
A O
Q (
% )
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ExaMpLE 4.2
The double sampling plan in the earlier section was
n1 = 75 c1 = 0 r1 = 3 n2 = 75 c2 = 3 r2 = 4
• A second sample is required if on the first sample one or two nonconformances are noted.
• If zero nonconformances are found in the first sample, the lot is accepted.
• If three or more nonconformances are found in the first sample, the lot is rejected.
Create the average sample number curve for this double sampling plan.
Solution: Denote the probability of making a decision, accept or reject, on the first sample as P(D1). Then,
P(D1) = P(0) + P (3 or more)
P(0) = the probability of zero nonconformances on the first sample
P (3 or more) = the probability of three or more nonconformances on the first sample.
P2 = 1 – P(D1), then, ASN = n1 + n2P2
When using Equation (4.2) to calculate the probability of three or more nonconfor- mances, remember that the probability of three or more nonconformances is given by:
(1 – probability of two or less nonconformances) in the sample
The average sample number will be plotted for several values of fraction nonconform- ing p and an ASN curve will be plotted. An example of the ASN calculation for the frac- tion nonconforming p = 0.01 is shown below. Several other points need to be plotted for other values of p. Figure 4.11 shows the ASN curve for this example.
When p = 0.01:
P(0) = Probability of zero nonconformances in sample = 0.4706
P(3 or more) = Probability of three or more nonconformances in sample = 0.0397
P(D1) = Probability of a decision on the first sample (using the above equation) = 0.4706 + 0.0397 = 0.5103
Then P2 = probability of requiring a second sample = 1 – 0.5103 = 0.4897.
Thus the ASN is
ASN(0.01) = Average sample number for a lot quality p = 0.01
= n1 + n2(P2)
= 75 + 75(0.4866) = 111.73
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lC.2.c. ANSI/ASQ Z1.4-2003 (R2013)
ANSI/ASQ Z1.4-2003 (R2013) is probably the most commonly used standard for attribute sampling plans. It is a revision of ANSI/ASQC Z1.4-1993, incorporating eight changes that include (ANSI/ASQ 2003):
1. Acceptable quality level (AQL) has been changed to acceptable quality limit (AQL).
2. The definition and explanation of AQL have been changed. See the new definition of AQL above.
0
10
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
n1 = 75, c1 = 0, r1 = 3
n2 = 75, c2 = 3, r2 = 4
20
30
40
50
60
70
Fraction nonconforming
80
90
100
110
120
130
A S
N
Figure 4.11 Average sample number curve for double sampling plan.
Values of ASN at different p values ASN(p) may be calculated in a similar way; the results are given below.
ASN(0.01) = 111.73 ASN(0.06) = 86.65
ASN(0.02) = 119.28 ASN(0.07) = 81.93
ASN(0.03) = 112.99 ASN(0.08) = 78.97
ASN(0.04) = 102.89 ASN(0.09) = 77.20
ASN(0.05) = 93.62 ASN(0.10) = 76.18
When comparing sampling plans with equal protection, double sampling plans will gen- erally result in smaller average sample sizes when quality is excellent or poor. When qual- ity is near the indifference level, double sampling plans will rarely result in greater ASN.
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3. The discontinuation of inspection rule has been changed. See later sections in this chapter.
4. ANSI/ASQC A2-1987 has been changed to ANSI/ASQ A3534-2-1993.
5. ANSI/ASQC Q3 has been changed to ASQC Q3-1988.
ANSI/ASQ Z1.4-2003 (R2013) received a minor revision in 2013 to correct some notes in the tables. Other than the above changes and some changes to the footnotes of some tables, all tables, table numbers, and procedures used in MIL- STD-105E (which was canceled in 1995) and ANSI/ASQC Z1.4-1993 have been retained. While MIL- STD-105E was canceled, a new standard on acceptance sampling was instated in the military called MIL- STD-1916 “DOD Preferred Methods for Accep- tance of Product.”
The wide recognition and acceptance of ANSI/ASQ Z1.4-2003 (R2013) could be due to government contracts stipulating the standard rather than its statistical importance. Producers submitting products at a nonconformance level within the AQL have a high probability of having the lot accepted by the customer.
When using ANSI/ASQ Z1.4-2003 (R2013), the characteristics under consider- ation should be classified. The general classifications are critical, major, and minor defects:
• Critical defect. A defect that judgment and experience indicate is likely to result in hazardous or unsafe conditions for the individuals using, maintaining, or depending on the product, or a defect that judgment and experience indicate is likely to prevent performance of the unit. In practice, critical characteristics are commonly inspected to an AQL level of 0.40 to 0.65%, if not 100% inspected. One hundred percent inspection is recommended for critical characteristics if possible. Acceptance numbers are always zero for critical defects.
• Major defect. A defect other than critical that is likely to result in failure or to reduce materially the usability of the unit of product for its intended purpose. In practice, AQL levels for major defects are generally about 1%.
• Minor defect. A defect that is not likely to reduce materially the usability of the unit of product for its intended purpose. In practice, AQL levels for minor defects generally range from 1.5% to 2.5%.
C.2.c.i. Levels of Inspection
There are seven levels of inspection used in ANSI/ASQ Z1.4-2003 (R2013): reduced inspection, normal inspection, tightened inspection, and four levels of special inspection. The special inspection levels should be used only when small sample sizes are necessary and large risks can be tolerated. When using ANSI/ASQ Z1.4- 2003 (R2013), a set of switching rules must be followed as to the use of reduced, normal, and tightened inspection.
The following guidelines are taken from ANSI/ASQ Z1.4-2003 (R2013):
• Initiation of inspection. Normal inspection level II will be used at the start of inspection unless otherwise directed by the responsible authority.
• Continuation of inspection. Normal, tightened, or reduced inspection shall continue unchanged for each class of defect or defectives on
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successive lots or batches except where the following switching procedures require change. The switching procedures shall be applied to each class of defects or defectives independently.
• Switching procedures. Switching rules are shown in Figure 4.12.
• Normal to tightened. When normal inspection is in effect, tightened inspection shall be instituted when two out of five consecutive lots or batches have been rejected on original inspection (i.e., ignoring resubmitted lots or batches for this procedure).
• Tightened to normal. When tightened inspection is in effect, normal inspection shall be instituted when five consecutive lots or batches have been considered acceptable on original inspection.
• Normal to reduced. When normal inspection is in effect, reduced inspection shall be instituted providing that all of the following conditions are satisfied:
a. The preceding 10 lots or batches (or more), as indicated by the note on ANSI/ASQ Z1.4-2003 (R2013) Table VIII, have been on normal inspection and none has been rejected on original inspection.
b. The total number of defectives (or defects) in the sample from the preceding 10 lots or batches (or such other number as was used for condition (a) above) is equal to or less than the applicable number given in Table VIII of ANSI/ASQ Z1.4-2003 (R2013). If double or multiple sampling is in use, all samples inspected should be included, not “first” samples only.
Figure 4.12 Switching rules for normal, tightened, and reduced inspection.
• Preceding 10 lots accepted, with • Total nonconforming less than limit number (optional), and • Production steady, and • Approved by responsible authority
• Two out of five consecutive lots not accepted
• Five consecutive lots accepted
• Five consecutive lots remain on tightened
• Discontinue inspection under Z1.4
Reduced Normal
Start
Tightened
• Lot not accepted, or • Lot accepted but nonconformities found lie between Ac and Re of plan, or • Production irregular, or • Other conditions warrant
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c. Production is at a steady rate.
d. Reduced inspection is considered desirable by the responsible authority.
• Reduced to normal. When reduced inspection is in effect, normal inspection shall be instituted if any of the following occur on original inspection:
a. A lot or batch is rejected.
b. A lot or batch is considered acceptable under reduced inspection but the sampling procedures terminated without either acceptance or rejection criteria having been met. In these circumstances, the lot or batch will be considered acceptable, but normal inspection will be reinstated starting with the new lot or batch.
c. Production becomes irregular or delayed.
d. Other conditions warrant that normal inspection shall be instituted.
• Discontinuation of inspection. If the cumulative number of lots not accepted in a sequence of consecutive lots on tightened inspection reaches five, the acceptance procedures of this standard shall be discontinued. Inspection under the provisions of this standard shall not be resumed until corrective action has been taken. Tightened inspection shall then be used as “normal to tightened” above.
C.2.c.ii. Types of Sampling
ANSI/ASQ Z1.4-2003 (R2013) allows for single sampling, double sampling, or multiple sampling. The choice of the type of plan depends on many variables. Single sampling is the easiest to administer and perform but usually results in the largest average total inspection (ATI). Double sampling in ANSI/ASQ Z1.4-2003 (R2013) results in a lower ATI than single sampling, but requires more decisions to be made, such as the following:
• Accept the lot after first sample
• Reject the lot after first sample
• Take a second sample
• Accept the lot after second sample
• Reject the lot after second sample
Multiple sampling plans further reduce the ATI but also increase the number of deci- sions to be made. As many as seven samples may be required before a decision to accept or reject the lot can be made. This type of plan requires the most administration.
A general procedure for selecting plans from ANSI/ASQ Z1.4-2003 (R2013) is as follows:
1. Decide on an AQL.
2. Decide on the inspection level.
3. Determine the lot size.
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4. Find the appropriate sample size code letter. See Table 1 from ANSI/ASQ Z1.4-2003 (R2013).
5. Determine the type of sampling plan to be used: single, double, or multiple.
6. Using the selected AQL and sample size code letter, enter the appropriate table to find the desired plan to be used.
7. Determine the normal, tightened, and reduced plans as required from the corresponding tables.
ExaMpLE 4.3
A lot of 1750 parts has been received and is to be checked to an AQL level of 1.5%. Determine the appropriate single, double, and multiple sampling plans for general inspection level II.
Steps to define the plans are as follows:
1. Table I on page 10 of ANSI/ASQ Z1.4-2003 (R2013) stipulates code letter K. 2. Normal inspection is applied. For code letter K, using Table II-A of ANSI/ASQ
Z1.4-2003 on page 11 of the standard, a sample of 125 is specified. 3. For double sampling, two samples of 80 may be required. Refer to Table III-A on
page 14 of the standard. 4. For multiple sampling, at least two samples of 32 are required and it may take up
to seven samples of 32 before an acceptance or rejection decision is made. Refer to Table IV-A on page 17 of the standard.
A breakdown of all three plans is provided in Table 4.8.
Table 4.8 Acceptance and rejection number for single, double, and multiple sampling plans.
Sampling plan Sample(s) size Ac Re
Single sampling 125 5 6
Double sampling First 80 2 5
Second 80 6 7
Multiple sampling First 32 * 4
Second 32 1 5
Third 32 2 6
Fourth 32 3 7
Fifth 32 5 8
Sixth 32 7 9
Seventh 32 9 10
Ac = Acceptance number Re = Rejection number *Acceptance not permitted at this sample size.
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C.2.d. Dodge- Romig Tables
Dodge-Romig tables were designed as sampling plans to minimize ATI. These plans require an accurate estimate of the process average nonconforming in selec- tion of the sampling plan to be used. The Dodge- Romig tables use the AOQL and LTPD values for plan selection, rather than the AQL as in ANSI/ASQ Z1.4-2003 (R2013). When the process average nonconforming is controlled to requirements, Dodge- Romig tables result in lower ATI, but rejection of lots and sorting tend to minimize the gains if process quality deteriorates.
Note that if the process average nonconforming shows statistical control, accep- tance sampling should not be used. The most economical course of action in this situation is either no inspection or 100% inspection (Deming 1982). See Stephens (2016), Duncan (1986), or Montgomery (2013) for more details on Dodge- Romig tables.
C.2.e. Variables Sampling Plans
Variables sampling plans use the actual measurements of sample products for decision making rather than classifying products as conforming or nonconform- ing, as in attribute sampling plans. Variables sampling plans are more complex in administration than attribute plans; thus, they require more skill. They pro- vide some benefits, however, over attribute plans. Two of these benefits are the following:
1. Equal protection to an attribute sampling plan with a much smaller sample size. There are several types of variables sampling plans in use, three of these being (1) σ known, (2) σ unknown but can be estimated using sample standard deviation s, and (3) σ unknown and the range R is used as an estimator. If an attribute sampling plan sample size is determined, the variables plans previously listed can be compared as a percentage to the attribute plan. The sample size percentages based on these scenarios are shown in Table 4.9, which can be compared to the 100% inspection of items in an attribute sampling plan.
2. Variables sampling plans allow the determination of how close to nominal or a specification limit the process is performing. Attribute plans either accept or reject a lot; variables plans give information on how well or poorly the process is performing.
Table 4.9 Percentage of acceptance sampling for previously discussed plans.
Plan Sample size (percent)
σ unknown, range method 60
σ unknown, s estimated from sample 40
σ known 15
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Variables sampling plans, such as ANSI/ASQ Z1.9-2003 (R2013), have some dis- advantages and limitations:
1. Separate characteristics on the same parts will have different averages and dispersions, resulting in a separate sampling plan for each characteristic
2. Variables plans are more complex in administration than attribute plans
3. Variables gauging is generally more expensive than attribute gauging
In addition, for variables sampling plans, it is assumed that the quality character- istic under study is normally distributed.
ANSI/ASQ Z1.9-2003 (R2013) is a revision of ANSI/ASQC Z1.9-1993 that includes changing the term “acceptable quality level” to “acceptable quality limit” (AQL), changing the definition and explanation of AQL, and changing the dis- continuation of inspection rule, as explained previously in terms of ANSI/ASQ Z1.4-2003 (R2013).
The most common standard for variables sampling plans is ANSI/ASQ Z1.9- 2003 (R2013), which has plans for (1) variability known, (2) variability unknown (standard deviation method), and (3) variability unknown (range method). Using these methods, this sampling plan can be used to test for a single specification limit, a double (or bilateral) specification limit, estimation of the process average, and estimation of the dispersion of the parent population.
Like ANSI/ASQ Z1.4-2003 (R2013), ANSI/ASQ Z1.9-2003 (R2013) uses sev- eral AQL levels and follows specific switching procedures for normal, reduced, or tightened inspection. ANSI/ASQ Z1.9-2003 (R2013) allows for the same AQL value for each specification limit of double specification limit plans or the use of different AQL values for each specification limit. The AQL values are designated ML for the lower specification limit and MU for the upper specification limit.
There are two forms used for every specification limit ANSI/ASQ Z1.9-2003 (R2013) plan. Form 1 provides only acceptance or rejection criteria, whereas form 2 estimates the percentage below the lower specification limit and the percentage above the upper specification limit. These percentages are compared with the AQL for acceptance/rejection criteria. Figure 4.13 summarizes the structure and organi- zation of ANSI/ASQ Z1.9-2003 (R2013).
An example of the sampling method where the variability is unknown and thus uses the standard deviation is shown in Example 4.4.
There are 14 AQL levels used in ANSI/ASQ Z1.9-2003 (R2013) that are con- sistent with the AQL levels used in ANSI/ASQ Z1.4-2003 (R2013). Section A of ANSI/ASQ Z1.9-2003 (R2013) contains both an AQL conversion table and a table for selecting the desired inspection level. Inspection level II should be used unless otherwise specified. See section A7.1 of the standard for further information about levels.
Table A-3 on page 7 of ANSI/ASQ Z1.9-2003 (R2013) contains the OC curves for the sampling plans in sections B, C, and D.
Section B contains sampling plans used when the variability is unknown and the standard deviation method is used. Part I is used for a single specification limit, Part II is used for a double specification limit, and Part III is used for estima- tion of process average and criteria for reduced and tightened inspection.
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Section C contains sampling plans used when the variability is unknown and the range method is used. Parts I, II, and III are the same as Parts I, II, and III in Section B.
Section D contains sampling plans used when variability is known. Parts I, II, and III are the same as Parts I, II, and III in Section B.
We now describe in detail the standard deviation method when the variability is unknown. This continuous sampling plan is for the situation where the vari- ability is not known and the standard deviation is estimated from the sample data. The sampling plan will be that for a double specification limit, and it is found in Section B of the standard with one AQL value for both upper and lower specifica- tion limits combined.
The acceptability criterion is based on comparing an estimated percent non- conforming with a maximum allowable percent nonconforming for the given AQL level. The estimated percent nonconforming is found in ANSI/ASQ Z1.9-2003 (R2013) Table B-5.
The quality indices for this sampling plan are:
Q
x s
Q sU L
= −
= −USL
and LSLx
(4.6)
where
USL = upper specification limit
LSL = lower specification limit
x– = sample mean
s = estimate of lot standard deviation
Figure 4.13 Structure and organization of ANSI/ASQ Z1.9-2003 (R2013).
Section A
AQL conversion and inspection
levels
Variability unknown Standard deviation
method Variability known
Single specification limits
k-method Procedure 1
M-method Procedure 2
M-method Procedure 2
Double specification limits
Process average estimation and criteria
for reduced and tightened inspection
Variability unknown Range method
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It should be noted that QL and QU follow a standard normal distribution since the quality characteristic being measured is assumed to be normally distributed.
The quality level of the lot is in terms of the lot percent defective. Three val- ues are calculated: PU, PL, and p. PU is an estimate of conformance with the upper specification limit, PL is an estimate of conformance with the lower specification limit, and p is the sum of PU and PL.
The value of p is then compared with the maximum allowable percent defec- tive. If p is less than or equal to M (ANSI/ASQ Z1.9-2003 [R2013] Table B-5), or if either QU or QL is negative, the lot is rejected, since this would be the result of x lying beyond the specification limits. Example 4.4 illustrates the above procedure.
ExaMpLE 4.4
The minimum temperature of operation for a certain device is specified as 180°F. The maximum temperature is 209°F. A lot of 40 items is submitted for inspection. Inspection level IV, normal inspection with AQL = 1%, is to be used. ANSI/ASQ Z1.9-2003 Table A-2 gives code letter D, which results in a sample size of five from ANSI/ASQ Z1.9-2003 Table B-3. The results of the five measurements in degrees Fahrenheit are as follows: 197, 188, 184, 205, 201. Determine if the lot meets acceptance criteria.
Given:
• Sample size, n = 5
• Upper specification limit, USL = 209
• Lower specification limit, LSL = 180
• From Table B of the ANSI/ASQ Z1.9-2003 standard, we find the maximum allowable percent nonconforming (M) to be M = 3.32%
Let the random variable X represent the temperature of operation. The steps for calculating the percent nonconforming are as follows:
1. Calculate the sample mean (see Chapter 6, section A, for calculation of the sample mean):
x x
n
xi i
n
i i= = = == =
∑ ∑ 1 1
5
5 975
5 195
2. Calculate the sample standard deviation (see Chapter 6, section A, for calculation of the sample standard deviation):
s x x
n
i i
n
= −( )
− ==
∑ 2 1
1 8 803.
3. Calculate QU:
Q x
sU USL
= −
= −
= 209 195
8 803 1 59
. .
4. Calculate QL:
Q x
sL LSL
= −
= −
= 195 180
8 803 1 70
. .
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From Table B-5 of the ANSI/ASQ Z1.9-2003 standard, determine the percent non -conforming:
• The percent above the upper specification limit with n = 5 and QU = 1.59 is 2.19%
• The percent below the lower specification limit with n = 5 and QL = 1.70 is 0.66%
The total percent nonconforming is then 2.19% + 0.66% = 2.85%. Therefore, since our percent nonconforming (2.85%) is less than the maximum
allowable (3.32%), we conclude that the lot is acceptable.
C.2.f. Sequential Sampling Plans
When tests are either destructive in nature or costly, it may be advantageous to use sequential sampling plans popularized by Wald (1973). These plans have the advantage of greatly reduced sample sizes while giving good protection.
To determine a sequential sampling plan, the following parameters must be defined:
α = producer’s risk
AQL = p1 β = consumer’s risk
RQL = rejectable (or unacceptable) quality level = p2; this is also referred to as limited quality level
The following example uses α = 0.05, AQL = 0.05, β = 0.1, and RQL = 0.2. This results in a plan that will have a 5% chance of rejecting a lot that is 5% nonconform- ing, and a 10% chance of accepting a lot that is 20% nonconforming.
Figure 4.14 shows the accept, reject, and continue testing areas for a sequential sampling plan. The y-axis represents the number of nonconforming items in the sample, and the x-axis represents the number of units inspected.
Figure 4.14 Decision areas for a sequential sampling plan.
Number of units inspected
Reject zone
Continue testing
Accept zone
sn + h2
sn – h1
N u
m b
e r
o f
n o
n c o
n fo
rm in
g u
n it
s
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The equations for the acceptance and rejection zone lines are:
Reject zone line = sn + h2 (4.7)
Accept zone line = sn – h1 (4.8)
where:
n = sample size
h b
p p
p p
= −( ) −( )
1 2 1
1 2
1
1 log >>
(4.9)
= −( ) −( )
h a
p p
p p
2 2 1
1 2
1
1 log >>
(4.10)
= −(
s p11log ) −( )
− −( )
/
log
1
1
1
2
2 1
1 2
p
p p
p p >>
7 7 ( )
(4.11)
a =
−( ) log
1 β α >>
(4.12)
b =
−( ) log
1 α β >>
(4.13)
C.2.g. Continuous Sampling Plans
Many production processes do not produce lots, and thus lot- by-lot acceptance sampling plans discussed earlier cannot be applied. In such cases, continuous sampling plans are developed. In continuous sampling plans, 100% inspection and sampling inspection are alternately applied. A common standard for devel- oping continuous sampling plans is ASTM- E2819-11, established in 2011. This standard is a conversion of MIL- STD 1235B, which was canceled by the DoD in 1997.
Continuous sampling plans are characterized by two parameters: i, the clear- ance number or the number of conforming units under 100% inspection, and f, the ratio of the units inspected to the total number of units produced or passing through the inspection station.
Dodge’s continuous sampling plan and ASTM- E2819-11, discussed in the fol- lowing subsections, are two different standards for continuous sampling plans.
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ExaMpLE 4.5
Assume that the following values are desired for a sequential sampling plan:
α = 0.05, p1 (AQL) = 0.05
β = 0.1, p2 (RQL) = 0.2 Then:
a
b
= −9 =
= −
log .
. .
log .
.
1 0 10 0 05
1 2553
1 0 05 0 10
9 C=
= −( ) −( )[ ]
0 9777
1 0 05 1 0 20
0
.
log . / .
log .
s 220 1 0 05
0 05 1 0 20
0 1103
0 97 1
−( ) −( )9
=
=
. . .
.
. h
777
0 20 1 0 05 0 05 1 0 20
1 4448
log . . . .
. −( ) −( )9 C
=
hh2 1 2553
0 20 1 0 05 0 05 1 0 20
= −( ) −( )9
.
log . . . .
==
= + = +
1 855
1 8552
.
Reject line
Accept line
sn h
= − = −sn h1 0.1103n
0.1103n
1 4448.
.
C
9
9 Table 4.10 Points for accept and reject lines.*
n Acceptance
number Rejection number n
Acceptance number
Rejection number
1 A B 14 0 4
2 A B 20 0 5
3 A 3 24 1 5
4 A 3 40 2 7
5 A 3 50 4 8
6 A 3
*Acceptance values are rounded down to the nearest integer. Note A: Acceptance not possible when acceptance number is negative. Note B: Rejection not possible when rejection number is greater than sample number.
As can be seen by the preceding plan, rejecting the lot is not possible until the third sample unit and acceptance of the lot is withheld until the 14th sample unit.
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C.2.g.i. Dodge’s Continuous Sampling Plan
Dodge’s continuous sampling plan includes CSP-1 and CSP-2 sampling plans. These plans take AOQL as a quality index. That is, for every AOQL value, there are dif- ferent combinations of i and f.
Dodge’s CSP-1 operates as follows for a selected AOQL value:
1. Start with 100% inspection
2. When i (clearance number) consecutive number of units are found free from nonconformities, 100% inspection is then substituted with sampling inspection
2.1. A fraction of f units is randomly selected and then inspected
2.1.1. If one nonconformity is found, the 100% inspection procedure is restarted and the cycle is repeated
Dodge’s CSP-2 operates as follows for a selected AOQL value:
1. Start with 100% inspection
2. When i (clearance number) consecutive number of units are found free from nonconformities, 100% inspection is then substituted with sampling inspection
2.1. A fraction of f units is randomly selected and then inspected
2.1.1. If one nonconformity is found, the sampling inspection continues and the following procedure (2.1.2) is initiated
2.1.2. The number of conforming units (after finding the nonconformity) is counted
2.1.2.1. If i consecutive number of units are found free of noncon- formities, sampling inspection continues
2.1.2.2. If one nonconformity is found, 100% inspection is reinstated
C.2.g.ii. ASTM- E2819-11
The American Society for Testing and Materials (ASTM) preserved MIL- STD-1235B in 2011 because although the DoD no longer maintains its continuous sampling standard, it continues to be widely used in industry throughout the world. This standard uses the same parameters, i and f, as previously defined. The stan- dard includes CSP-1, CSP-2, CSP- F, CSP- T, and CSP- V plans.
CSP-1 and CSP-2 plans operate in the same way as Dodge’s CSP-1 and CSP-2 plans, but they are selected based on a sample size code letter and an AQL value as a quality index. The sample size code letter is selected based on the number of units in the production interval.
CSP-F plans work the same way as CSP-1 plans, providing alternate sequences of 100% and sampling inspection procedures, but the difference is that the AOQL and the number of units in the production interval are used to characterize the plans in this case. Once AOQL and f values are selected, go to the corresponding table to read i, the clearance number. CSP- F is a single- level continuous sampling scheme.
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CSP-T plans provide for reduced sampling frequency once the product shows superior quality. The CSP- T plan works as follows:
1. Start with 100% inspection
2. When i (clearance number) consecutive number of units are found free from nonconformities, 100% inspection is then substituted with sampling inspection
3. A fraction of f units is randomly selected and then inspected
3.1. If one nonconformity is found, the inspector reinstates 100% inspection
3.2. If the inspector finds i consecutive units free from nonconformities, the frequency f is reduced to f/2
3.2.1. If one nonconformity is found, the inspector switches back to 100% inspection
3.2.2. If the inspector finds i consecutive units free from nonconformities, the frequency f is reduced to f/4
3.2.2.1. If one nonconformity is found, 100% inspection is reinstated
CSP-V plans work the same way as CSP- T plans but with reduced i instead of reduced f. The procedure is as follows:
1. Start with 100% inspection
2. When i (clearance number) consecutive number of units are found free from nonconformities, 100% inspection is then substituted with sampling inspection
3. A fraction of f units is randomly selected and then inspected
3.1. If one nonconformity is found, the inspector reinstates 100% inspection
3.2. If the inspector finds i consecutive units free from nonconformities, the inspection continues with inspecting the same fraction f
3.2.1. If one nonconformity is found, the inspector switches back to 100% inspection
3.2.2. If the inspector finds i/3, the sampling inspection continues with the same fraction f
For more information on standards for acceptance sampling, see Neubauer and Luko (2013a, 2013b).
C.3. Sample integrity
Products are always at risk of contamination and misuse. Sample integrity is vital whenever sampling is done for any purpose, whether to go through a fitness pro- gram, for customer evaluation, or for destructive/nondestructive testing. In order to maintain sample integrity, carefully thought- out controls are necessary. Many people recall the murder trial of O. J. Simpson, where extremely complex and expensive DNA testing was challenged by the defense because the prosecution could not prove that the DNA sample was completely safe from any contamina- tion at all times. While this is an extreme example, it highlights the importance of
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maintaining sample integrity. In this section, we discuss sample integrity in terms of batch control, change control, and configuration control.
When products are created in batches (as opposed to discrete item production or continuous processes), it is necessary to keep records on all aspects of the batch. The concept of a batch includes mixing, heating, distilling, and other comparable operations. A recipe is used; documentation that the recipe was followed is vital in all but the most trivial cases. A qualified operator must maintain a log or journal indicating the quantities and products (or identification) of each material that is inserted into the batch. The time that each insertion is made is usually important, as well as the time that different inputs (heat, pressure, and so on) are applied to the batch.
Tests may be required to verify that the batch has developed the needed prop- erties over time. The results of such tests must be tightly linked to the physical batch and to all the other records. In some cases, these details can be automated, but often they must be recorded manually. When the batch is finished, it must be labeled with an identification code that is separate from other batches. The batch (lot) number must be printed or engraved on appropriate cartons, drums, jugs, pallets, and so on. A linkage between batch number and customer name is often necessary when the product is sold, so that it can be tracked through the entire distribution chain.
Change control is a technique for dealing with relatively simple to moderately complex products to which minor changes are made that must be tracked. For example, such products as refrigerators and desktop computers may be changed slightly and new version numbers issued on the same model name/number. For warranty purposes, product repair, and replacement, it is necessary to record each time the product is changed.
Engineers must decide when a change is required and how rapidly it is to be implemented. One priority scheme is to categorize the changes as emergency, pri- ority, or routine. An emergency change is appropriate when a hazardous condition is discovered in the present version. In such cases, no time must be lost in correct- ing the deficiency. A priority change is called for if there is sound economic reason to make the change promptly, but life and property are not at risk. For example, a product upgrade that reduces power consumption or maintenance could be implemented as a priority change. The final category, routine, is for changes that must be made, but need not be rushed. These are often to accommodate newly designed parts or to allow the product to have slightly more functionality, but not enough to justify an entirely new model.
Configuration control is an extension of change control. The term “configura- tion” refers to how a complex product is composed of various units and subassem- blies. In an evolving product with high research and development content, such as aerospace vehicles, defense weapons, and so on, the field version of the same unit of product gradually changes over time as new engines, new avionics, and new hydraulic systems are installed into existing units of product.
In order to manage such ongoing field product modifications, a lot of effort must be put into configuration control systems. This is really an adaptation of materials resource planning techniques. Extensive documentation is mandatory for proper control. Usually both computerized database records and hard- copy backup records (often at multiple locations) are required.
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Three now- canceled DoD standards addressed the subject of configuration control, DOD- STD-480A, DOD- STD-481, and DOD- STD-973. Standard EIA-649 “National Consensus Standard for Configuration Management,” however, can be used as a replacement for these canceled military standards since it is almost a complete duplication of the military standard. Russell (2013) discusses com- mon components in configuration management plans. These elements include a configuration plan, procedures and guidelines, an identification process, the change- control process, a records status process, and an audit process.
A key principle of configuration control is to avoid changes in a given product model unless a clear and compelling benefit can be shown. Management must consider the downsides of change to a product model. Change can result in more complexity in the product line and more chance for confusion. However, some potential benefits are reduced cost, increased performance, better safety, and lower maintenance.
d. MEaSurEMEnT and TEST A measurement process is a repeated application of a test method using a mea- suring system. A test method includes requirements for a test apparatus and a well- defined procedure for using it to measure a physical property.
Measurement is the process of evaluating a property or characteristic of an object and describing it with a numerical or nominal value. If the value is numeri- cal, reflecting the extent of the characteristic, the measurement is said to be on a quantitative scale and the actual property is referred to as a variable. Examples of variables inspection are measurements related to weight, length, temperature, and so on.
If the value assigned to each unit is not numerical, the measurement is on a qualitative or classification scale and is referred to as an attribute. In most inspec- tion situations involving nominal or attribute data, there are two possible nominal values: conforming (good) and nonconforming (defective). Each product unit is assigned one of these two labels according to inspection operation results. It is also possible to derive a numerical measure from a qualitative scale. This is achieved by calculating the fraction nonconforming (fraction defective) as the ratio between the number of units labeled as nonconforming and the total number of units inspected.
A measuring system should be able to provide accuracy capabilities that will ensure the attainment of a reliable measurement. In general, the elements of a measuring system include the instrumentation, calibration standards, environ- mental influences, human operator limitations, and features of the workpiece or object being measured. Each of these elements may involve detailed studies of extended scope and thus fall beyond the purpose of this book. The design of mea- suring systems also involves proper analysis of cost- to-accuracy considerations (Darmody 1967).
The functional design of measuring systems can include consideration of many approaches and employment of a variety of physical phenomena useful in establishing parametric variables from the measured quantity. In linear measuring
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systems, the basic function may be mechanical, optical, pneumatic, electronic, radiological, or combinations of these (Darmody 1967).
In contrast to the rather imprecise measurements made in everyday life, measurements and standards applied to manufactured parts must necessarily be extremely precise because they must conform to definite geometric and aesthetic design specifications. The production of quality products in any manufacturing operation requires an efficient and continuous testing program, and such pro- grams have become increasingly important in recent years.
Society has changed its attitude, not only with respect to product safety and cost, but also with respect to product reliability. Variations in product quality that were once accepted as the natural result of industrial systems are no longer tolerated. What is required today is the consistent extraction of the best tech- nological quality available on a routine production basis. In this circumstance, testing serves two functions: (1) to check on the performance of materials or components to obtain design data and (2) to check on the conformity of a prod- uct to its design specifications. Testing of the latter type is commonly called inspection.
Because dimensional measurement is very important to every manufactur- ing operation, much effort has historically been expended toward both improv- ing the techniques and the instrumentation involved and refining the standards employed.
The term “standard” has a dual meaning in the manufacturing environment. It is used to denote universally accepted specifications for devices, components, or processes that ensure conformity and therefore interchangeability throughout a particular industry. Thus, one manufacturer’s screw will fit another’s nut, all mak- ers of bricks will produce them in the same sizes, and all microscope objectives will fit all microscopes.
As used in metrology, on the other hand, a standard provides a reference for assigning a numerical value to a measured quantity. The term “measurement” implies the comparison of an unknown with a known to determine the qualitative relationship between the two. Each basic, measurable quantity has associated with it an ultimate standard that embodies the definition of a particular unit. Work- ing standards, those used in conjunction with the various measurement- making instruments, are calibrated in terms of the particular unit definitions involved. Obviously, if measurements made at different locations are to be comparable, they must ultimately be traceable to the same standard.
d.1. Measurement Tools
Selection of a measuring tool or measuring instrument is based on several factors. With advanced technology, we can now classify the measurement tools into two general categories: contact (e.g., touch probes) and noncontact (e.g., laser scan- ners). In general, the Rule of Ten serves as a baseline for the selection process of a measurement tool. The Rule of Ten of the Automotive Industry Action Group (AIAG) states that inspection measurements should be better than the tolerance of a dimension by a factor of 10 and calibration standards should be better than the inspection instrument by a factor of 10. Once this rule is implemented, candidate instruments need to be evaluated based on the following criteria:
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• Accuracy and precision
• Repeatability
• Sensitivity
• Resolution
• Stability and consistency
• Part or workpiece material
• Shape and dimensions of the part being measured
• Capabilities of the metrology laboratory
The following is a review of the different measurement instruments and technolo- gies employed in common measurement practices such as length and angle mea- surement, surface texture measurement, and measurement of out- of-roundness.
Current technological advances, including, but not limited to, supercomput- ers, hybrid manufacturing, semiconductors, and fiber optics, require more sophis- ticated measurements and/or measurement tools. However, as Whitehouse (2002, 2010) discusses, even with more sophisticated tools, we still need to address some of the same, or similar, problems and issues regarding measurements.
D.1.a. Length and Angle Measurements
The standard environmental conditions for length measurements include a tem- perature of 68°F (20°C) and a barometric pressure of 760 mm Hg (Doiron 2007). Because these conditions are assumed for all precision dimensional measure- ments, dimensional metrology laboratories are temperature controlled as nearly as is practical to 68°F, and thermal expansion corrections are made for any devia- tions that may occur. It is seldom necessary to correct for thermal expansion to achieve the accuracy required in industrial movement. Since the majority of preci- sion parts, like the masters against which they are measured, are made of steel, it is generally safe to assume that their thermal expansion coefficients are identical and that no temperature correction need be made. Temperature corrections are also unnecessary when angles alone are measured, since a uniform temperature change cannot change the size of an angle. This will definitely change with the introduction of new materials.
Dimensional (or linear) measuring instruments are used to measure length. They are of two types: absolute instrument and comparative instruments, or comparators.
Absolute instruments have their working standards built in and thus require no mastering; they are generally used for long- range measurements. Comparators are short- range devices that measure deviations between a working master and a given part. The yardstick is a crude example of the first type, and the dial indicator is an example of the second.
Table 4.11 details typical units, standards, and instruments for length and angle measurements. Measuring instruments range from very basic tools to more sophisticated measuring machines, such as coordinate measuring machines and laser scanners. In the following paragraphs, we review the basic measuring tools that are commonly used for many applications:
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• Surface plates
• Micrometers
• Verniers
• Comparators
• Dial indicators
• Gage blocks
• Ring, plug, and snap gages
Most of the basic or general- purpose linear measuring instruments are typified by the use of steel rulers, vernier calipers, and micrometer calipers.
Steel rulers are commonly used for linear measurements in which the ends of a dimension being measured are aligned with graduations of the scale from which the length is read directly. A specialized type of steel ruler is the depth ruler, which is used for measuring holes, slots, and so on.
Vernier calipers are used for inside or outside linear measurement. Other types of verniers include digital reading calipers that provide LCD readouts in micrometers (μm) or microinches (μ in) and vernier height gages that can measure external, internal, and distance dimensions, as well as perpendicularity, flatness, straightness, centers, and diameters.
Micrometers come in various types. The measuring element of a micrometer consists of a fixed anvil and a spindle that moves lengthwise as it turns. Vernier micrometer calipers use a vernier scale on the sleeve. Digital micrometers use digital readouts to make readings faster and easier. Indicating micrometers have a built- in dial indicator to provide a positive indication of measuring pressure applied.
Angular measurements use the degree as the standard unit. Angular measur- ing devices range from simple tools such as protractors, bevel protractors, and squares to sine bars and dividing heads. The protractor reads directly in degrees. A bevel protractor utilizes a vernier scale that shows angles as small as five or less minutes.
The sine bar is a more precise device for precision measuring and checking of angles. It consists of an accurately ground flat steel straight edge with precisely affixed round buttons that are a known distance apart and of identical diameters.
Table 4.11 Typical standards and instrumentation for industrial length and angle measurements.
Length measurements Angle measurements
Unit of measurement Meter Radian
Ultimate standard Speed of light Circle
Single-valued working standards
Length gage blocks Angle gage blocks
Many-valued working standards Line scales, step bars Optical polygons, serrated- type index tables
Displacement-measuring Interferometers Autocollimator instruments
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Dividing heads are either optical or mechanical devices that often are used for the circular measurement of angular spacing, common in machine tool operations.
D.1.b. Layout and Locating Devices
Surface plates provide a relatively accurate surface plane from which measure- ments can be made. Surface plates may employ a cast iron or granite surface. Granite surface plates provide better hardness, resistance to corrosion, and non- magnetic characteristics compared with cast iron plates. Granite surface plates also provide less response to temperature changes than cast iron surface plates.
D.1.c. Gages
Gages are used to determine the conformance or nonconformance of a dimension to required specifications without attempting actual measurements. Typical com- mon functional gages are classified according to their use for checking outside dimensions, inside dimensions, or special features. Ring and snap gages are used for checking outside dimensions, plug gages are used for checking inside dimen- sions, and other gages are used for checking special features like tapers, threads, and splines. They normally provide a decision on part specifications.
A common method of inspection by attributes involves the use of limit gages, also known as fixed limit gages or go/no-go gages. Limit gages are made to sizes essentially identical to the design specification limits of the dimension to be inspected. If a specific gage can properly mate with a part, then the part can be assembled with another part whose physical boundaries do not exceed those of the gage. Consequently, the part is acceptable for assembly. Limit gages designed to identify this condition are called go gages. (See Figure 4.15.)
The “go” end of a go/no-go gage is designed to check the characteristic at the maximum material condition (minimum size for interior features, maximum size for exterior features). The maximum material condition produces the minimum clearance required for assembly. The “no-go” end is designed to detect conditions of excessive clearance. It checks the characteristic at its minimum material condi- tion. A part will not mate with a no- go gage unless the actual condition of the part feature is below the specified minimum. Thus, if the no- go gage mates with the part, then the part dimension is out of specification and the part should be rejected.
In practice, go/no-go gages are used together and often appear at opposite ends of an inspection instrument. An acceptable part should mate with the go end but should not mate with the no- go end. Parts that mate with both ends or that do not mate with either end do not meet design specifications and should be rejected.
Figure 4.15 Go/no-go gage to check the diameter of a shaft.
Go No-go
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Other than gauging, most methods of inspection by attributes are largely sub- jective and depend on the ability of human inspectors to make the right decisions. In many cases, inspection by attributes involves visual characteristics, such as color, shape, or smoothness, and other visual defects.
D.1.d. Dial Indicators, Comparators, and Gage Blocks
Dial indicators magnify the dimension deviation from a standard to which the gage is set. Dial indicators are used for many kinds of checking and gauging operations, checking machines and tools, verifying alignments, and cutter runout. Some indicators employ mechanical mechanisms for their operation and others come with a digital readout.
Comparators normally employ dial indicators for their operation and come in different varieties: mechanical, optical, electronic, and pneumatic. Optical projec- tors, also known as optical comparators, employ a system in which light rays are directed against the object and then reflected back through a projection lens onto a screen. The projections are large enough to accurately measure small configura- tions of objects.
Gage blocks are a system for producing precision lengths. They may be used as a reference for the calibration of dial indicators. They are rectangular, square, or round blocks of steel, carbide, or ceramic materials. Each has two faces that are flat, level, and parallel with an accuracy and length grade, depending on the application.
D.1.e. Surface Texture Measurement
Surface metrology is defined as the measurement and characterization of a surface topology (Whitehouse 2002). It is treated separately from length measurement, which is concerned with the relationship of two surfaces on a workpiece. Surface measurement, however, is involved with the relationship between a surface on the workpiece and a reference that is not actually on the workpiece. One aspect of surface metrology is the measurement of surface roughness as an average devia- tion from a mean center line (Bosch 1984). Townsend et al. (2016) provide a review paper on surface metrology for metal additive manufacturing.
Of all the methods used for the numerical assessment of the surface, the fol- lowing are the most widely used (Reason 1960):
1. Peak-to-valley measure
2. Mean-line measures (center line average [CLA] and root mean square [RMS])
3. Crest-line measures
4. Envelope method, in which the crest line should be defined as the locus of the center of a circle or defined radius rolling across the surface, the locus being displaced toward the surface until it contacts the crests
The international standard for the assessment of surface texture, ISO/R 468, defines three parameters: Ra (CLA), Rz, and Rmax, all measured relative to a straight mean line. Note that ISO/R 468 was replaced by ISO 4287:1997. These parameters are shown in Figure 4.16 and can be defined as follows (Spragg 1976):
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l 1. Ra (CLA) value is the arithmetic mean of the departures of a profile from the mean line. It is normally determined as the mean result of several consecutive sample lengths L.
2. Rz (10-point height) is the average distance between the five height peaks and the five deepest valleys within the sampling length and measured perpendicular to it.
3. Rmax is the maximum peak- to-valley height within the sampling length.
Other parameters of surface roughness are shown in Figure 4.17. They are defined as follows (Machinability Data Center 1980):
1. Rtm is the average value of Rmax’s for five consecutive sampling lengths.
2. Rp is the maximum profile height from the mean line within the sampling length. Rpm is the mean value of Rp’s determined over five sampling lengths.
3. Peak count (PC) is the number of peak/valley pairs per inch projecting through a band of width b centered about the mean line.
The most common method of surface measurement is to move a stylus over the sur- face and measure an average electrical signal produced by a transducer attached to the stylus. Other means used less frequently include stylus profiling (where a
Figure 4.16 ISO/R 468 surface roughness parameters.
+y
–y
L
+y
–y
L
Ra =
R1
(R1 + R3 +... R9) – (R2 + R4 +... R10)
R3 R2R5 R4 R7R6 R8 R9 R10
I L
L | y | dL_
RZ =
Rmax.
5
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lchart record is produced instead of an average number), reflectance meters, pneu- matics, and optical interference. The stylus averaging unit is fast, repeatable, quite easy to interpret, and relatively inexpensive (Bosch 1984).
D.1.f. Measurement of Roundness
Geometrically, a part can be said to be round in a given cross section if there exists within the section a point from which all points on the periphery are equidistant. In practice, however, the radius of nominally round parts tends to vary from point to point. Thus, the problem found by the metrologist is one of displaying and assessing these variations, and correctly interpreting the results (Bosch 1984).
Although many methods have been used for roundness measurement, only those that provide valid radial- deviation data lend themselves to standardiza- tion and consistent, accurate measurement of all out- of-roundness conditions.
Figure 4.17 Other parameters of surface roughness.
Rmax2 Rmax3 Rmax4
Rmax5Rmax1 Rt
L L L Assessment length
L L
Rp2 Rp3 Rp4 Rp5Rp1
L L L L L
1 count 1 count
Mean line
1 count
R R
R R R R R
tm
i i=
= + + + +
= ∑ max
max max max max ma
1
5
1 2 3 4
5
xx 5
5
R R
R R R R R
pm
pi i
p p p p p
=
= + + + +
= ∑
1
5
1 2 3 4 5
5
5
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For this reason, current industry, national, and international standards primarily cover measurements taken with precision spindle- type instruments with the data recorded on a polar chart.
Precision spindle instruments include both those in which the spindle sup- ports and rotates the part with the gage tip remaining stationary, and those in which the spindle rotates the gage tip about the part, which remains stationary. Figure 4.18 illustrates these two types of out- of-roundness measurement (Drews 1978).
The center of rotation of the precision spindle and the indicator gage tip pro- vides a master radius to which all the radii of a cross section profile of the part are compared. It is necessary that the center of the part cross section and the spindle axis be adjusted to be concentric within narrow limits. The variations of the cross section radii from the master radius are usually recorded in a highly magnified form on a polar chart. Because the out- of-roundness value is defined as the differ- ence between the largest and smallest radius that will just contain the measured profile, these radii must be measured from a specified center. The choice of these reference circles is arbitrary, but is chosen to fulfill some functional requirements. As shown in Figure 4.19, there are four ways in which a center can be chosen (Drews 1978):
1. Minimum radial separation (MRS) (also known as minimum zone circle [MZC])
2. Least squares circle (LSC)
3. Maximum inscribed circle (MIC)
4. Minimum circumscribed circle (MCC)
The magnified profile produced on the polar chart is evaluated by two concentric circles that just contain the profile when centered in accordance with the MRS center criteria. Other center criteria can be specified. For example, the concentric
Figure 4.18 Two types of roundness-measuring instruments: (a) rotating table, (b) rotating workpiece.
a b
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circles could be engraved on a transparent overlay (a more common method). The out- of-roundness value is the separation of the two concentric circles divided by the magnification setting of the instrument. The polar chart clearly shows the number and magnitude of the roundness deviations.
There are many advantages to the precision spindle methods. Accurate mea- surements of all types of out- of-roundness are possible and a permanent polar chart, which is easily interpreted, is provided. It is also the most accurate method of measurement available. With proper equipment, accuracies of one microinch are attainable. In addition to roundness, the equipment also permits ultraprecise measurement of centricity, squareness, flatness, and other related geometric part feature characteristics.
Figure 4.19 Four ways by which a center may be chosen.
Profile graph
Reference circle
Minimum radial separation (MRS or MZC)
Two concentric circles are chosen so as to have the least radial separation and yet contain between them all of the polar trace. This radial separation is the measure of the out-of-roundness value. The radial difference between concentric circles determined by this method is numerically unique, in that by definition a smaller value cannot exist.
Least squares circle (LSC)
A theoretical circle is located with the polar profile such that the sum of the squares of the radial ordinated between the circle and the profile is a minimum. The out-of-roundness value would be determined by the sum of the maximum inward and maximum outward ordinates divided by the proper chart amplification factor.
Maximum inscribed circle (MIC)
This procedure determines the center of the polar profile by the center of the largest circle that can be fitted inside the profile. From this circle the maximum outward departure of the profile denotes the out-of-roundness.
Minimum circumscribed circle (MCC)
The profile center is determined by the smallest circle that will just contain the measured profile. From the circle, the maximum inward departure of the profile can be measured; this maximum departure is the out-of-roundness.
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D.1.g. Coordinate Measuring Machines
Coordinate measuring machines (CMMs) have become a primary means of dimensional quality control for manufactured parts of complex form, where the volume of production does not warrant the development of functional gauging. The advent of increasingly inexpensive computing power and more fully inte- grated manufacturing systems will continue to expand the use of these machines into an even larger role in the overall quality assurance of manufactured parts.
CMMs can most easily be defined as physical representations of a three- dimensional rectilinear coordinate system. CMMs now represent a significant fraction of the measuring equipment used for defining the geometry of different- shaped workpieces. Most dimensional characteristics of many parts can be mea- sured within minutes with these machines. Similar measurements would take hours using older measuring equipment and procedures. Besides flexibility and speed, CMMs have several additional advantages:
1. Different features of a part can be measured in one setup. This eliminates errors introduced due to setup changes.
2. All CMM measurements are taken from one geometrically fixed measuring system, eliminating the accumulation of errors resulting from using functional gauging and transfer techniques.
3. The use of digital readouts eliminates the necessity for the interpretation of readings, such as with the dial or vernier- type measuring scales.
4. Most CMMs have automatic data recording, which minimizes operator influence.
5. Part alignment and setup procedures are greatly simplified by using software supplied with computer- assisted CMMs. This minimizes the setup time for measurement.
6. Data can be automatically saved for further analysis.
Although CMMs can be thought of as representations of a simple rectilinear coor- dinate system for measuring the dimensions of different- shaped workpieces, they naturally are constructed in many different configurations, all of which offer dif- ferent advantages. CMMs provide means for locating and recording the coordinate location of points in their measuring volumes. Traditional CMMs are classified according to their configurations, as follows (ANSI/ASME 1985):
1. Cantilever configuration, in which the probe is attached to a vertical machine ram (z-axis) moving on a mutually perpendicular overhang beam (y-axis) that moves along a mutually perpendicular rail (x-axis). Cantilever configuration is limited to small and medium- sized machines. It provides for easy operator access and the possibility of measuring parts longer than the machine table.
2. Bridge-type configuration, in which a horizontal beam moves along the x-axis, carrying the carriage that provides the y-motion. In other configurations, the horizontal beam (bridge structure) is rigidly attached to the machine base and the machine table moves along the x-axis. This
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is called fixed bridge configuration. A bridge- type CMM provides more rigid construction, which in turn provides better accuracy. The presence of the bridge on the machine table makes it a little more difficult to load large parts.
3. Column-type configuration, in which a moving table and saddle arrangement provide the x and y motions and the machine ram (z-axis) moves vertically relative to the machine table.
4. Horizontal-arm configuration features a horizontal probe arm (z-axis) moving horizontally relative to a column (y-axis) that moves in a mutually perpendicular motion (x-axis) along the machine base. This configuration provides the possibility of measuring large parts. Other arrangements of horizontal- arm configuration feature a fixed horizontal- arm configuration in which the probe is attached and moving vertically (y-axis) relative to a column that slides along the machine base in the x-direction. The machine table moves in a mutually perpendicular motion (z-axis) relative to the column.
5. Gantry-type configuration comprises a vertical ram (z-axis) moving vertically relative to a horizontal beam (x-axis) that in turn moves along two rails (y-axis) mounted on the floor. This configuration provides easy access and allows the measurement of large components.
6. L-shaped bridge configuration comprises a ram (z-axis) moving vertically relative to a carriage (x-axis) that moves horizontally relative to an L-shaped bridge moving in the y-direction.
Figure 4.20 shows CMM types according to this classification. The most advanced configuration, that of the ring- bridge, is not illustrated.
In addition to classifying CMMs according to their physical configuration, they can also be classified according to their mode of operation: manually ori- ented, computer- assisted, or direct computer- controlled. With manual machines, the operator moves the probe along the machine’s axes to establish and manually record the measurement values that are provided by digital readouts. In some machines, digital printout devices are used.
Computer-assisted CMMs can be either manually positioned (free-floating mode) by moving the probe to measurement locations or manually driven by pro- viding power- operated motions under the control of the operator. In either case, data processing is accomplished by a computer. Some computer- assisted CMMs can perform some or all of the following functions: inch to metric conversion, auto- matic compensation for misalignment, storing of premeasured parameters and measurement sequences, data recording, means for disengagement of the power drive to allow manual adjustments and manipulations of the machine motions, and geometric and analytical evaluations.
Direct computer- controlled CMMs use a computer to control all machine motions and measuring routines and to perform most of the routinely required data processing. These machines are operated in much the same way as computer numeric control machine tools. Both control and measuring cycles are under pro- gram control. Off- line programming capability is also available.
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Figure 4.20 Coordinate measuring machine classifications.
y
x
Fixed bridge
y
x
Cantilever
x
z
Moving bridge
y
z
x
y
Column
z y
x
Fixed horizontal arm
z y
x
Moving horizontal arm
z
y x
Gantry
x y
z
L-shaped bridge
z
z
The effective use of computers for CMM applications is a principal feature dif- ferentiating available CMM systems. The value of a measurement system depends a great deal on the sophistication and ease of use of the associated software and its functional capabilities. The functional capabilities of a CMM software package depend on the number and types of application programs available.
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d.2. destructive and nondestructive Tests
Testing involves evaluation of product conformance to certain design or produc- tion requirements. In addition, the output of testing can be used to evaluate new designs during product development and to define a product’s potential failure causes in product reliability evaluation.
Inspection is the evaluation of product quality by comparing the results of mea- suring one or several product characteristics with applicable standards. From this definition, it is evident that the inspection function involves a number of tasks:
1. Measurement, which could be on a qualitative or quantitative scale. The objective is to make a judgment about the product’s conformance to specifications.
2. Comparison of the measurement results with specific standards that reflect the intended use of the product by the customer and the various production costs. If the product is found to be nonconforming, a decision as to whether nonconforming products are fit for use may be reached.
3. Decision making regarding the disposition of the unit inspected and, under sampling inspection, the lot from which the sample was drawn.
4. Corrective action(s) in order to improve the quality of the product and/or process based on the aggregate results of inspection over a number of units.
Testing is also carried out to determine the conformity of a product by compar- ing the results of measuring one or several product characteristics with applicable standards. It involves tasks similar to those of inspection. The difference is that testing can be performed on a part, a product, a subassembly, or an assembly, while inspection is typically performed on a component or a part of a product.
Two terms normally associated with inspection are “gauging” and “testing.” Gauging determines product conformance with specifications with the aid of mea- suring instruments such as calipers, micrometers, templates, and other mechanical, optical, and electronic devices. Testing refers to the determination of the capabil- ity of an item to meet specified requirements by subjecting it to a set of physical, chemical, environmental, or other operating conditions and actions similar to or more severe than those expected under normal use.
Testing may be destructive or nondestructive. In testing, the product is sub- jected to measuring procedures that render its usefulness to the customer. Gaug- ing, however, is the more common form of inspection and is less costly. This operation has no effect on the product’s service capability. Of course, certain prod- uct characteristics, mainly those related to failure modes, may only be observed and measured by exposing the product to conditions beyond its designed limits, such as determining the maximum current that an electronic component can carry or the maximum tensile force that a mechanical part can withstand. Most of these procedures normally are destructive testing procedures and may be performed in cases where mandatory requirements are to be met. Nondestructive testing (NDT) of products usually is performed by subjecting the product to tests such as eddy current, ultrasonic resonance, or x-ray testing.
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Nondestructive Testing Techniques
Screening or 100% inspection cannot be used when the product is subjected to a destructive testing procedure or the time involved in performing inspection is too long. Another constraint is that the cost of inspection may be too high to justify the economics of inspection. NDT techniques are more common for automated inspection or 100% inspection. The most common NDT techniques include the following:
• Eddy current testing involves the application of an alternating current passing through a coil that is placed near the surface of the part to be inspected. Thus, its application is limited to conducting materials, and the test results are made by comparison.
• Ultrasonic testing is used to check for surface defects that cause deflection of an ultrasonic wave directed on the part surface, thus indicating the presence of a surface defect. For ultrasonic testing, reference standards are required.
• Radiographic or x-ray techniques cause the internal characteristics of the part to be displayed and thus provide information about the presence of defects, cracks, or other impurities.
• Liquid penetration is commonly used for detecting defects on the part surface. It is used for different part configurations, and, unlike magnetic particle testing, it can be used for nonmagnetic materials. However, liquid penetration cannot be used to locate subsurface discontinuities.
• Magnetic particle testing is used when the part material can be magnetized. Part defects, like cracks or discontinuities, can then be detected by the presence of paring magnetic fields. Magnetic particle testing is limited to parts made of iron, steel, or allied materials.
• Other types of NDT include (Hellier 2012):
– Visual inspection
– Penetrant testing
– Thermal infrared testing
– Acoustic emission testing
– Digital radiography
– Phased Array Ultrasonic Testing (PAUT)
– Ultrasonic phased array testing
– Ultrasonic guided wave inspection
Other common NDT techniques include the application of some phenomenon, such as thermal, chemical, holographic interferometry (employing interfer- ence patterns for checking surface displacements), or optical phenomena. These are used for special testing procedures and often are too expensive to be widely applied.
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E. METroLogy The science of precision measurement, usually referred to as metrology, encom- passes all scientific disciplines. The word “metrology” is derived from two Greek words: metro, meaning “measurement,” and logy, meaning “science.” The term is used in a more restricted sense to mean that portion of measurement science that is often used to provide, maintain, and disseminate a consistent set of units, to provide support for the enforcement of equity in trade by weights and measure- ment laws, or to provide data for quality control in manufacturing (Simpson 1981).
A measurement is a series of manipulations of physical objects or systems according to a defined protocol that results in a number. The number is purported to uniquely represent the magnitude (or intensity) of a certain property, which depends on the properties of the test object. This number is acquired to form the basis of a decision affecting some goal or fulfilling some need, the satisfaction of which depends on the properties of the test subject.
These needs or goals can be viewed as requiring three general classes of mea- surements (Simpson 1981):
1. Technical. This class includes those measurements made to ensure dimensional compatibility or conformation to design specifications necessary for proper function, or, in general, all measurements made to ensure fitness for intended use of some object.
2. Legal. This class includes those measurements made to ensure compliance with a law or regulation. This class is the concern of weights and measures bodies, regulators, and those who must comply with regulations. The measurements are identical in kind with those of technical metrology but usually are embedded in a much more formal structure. Legal metrology is more prevalent in Europe than in the United States, although this is changing.
3. Scientific. This class includes those measurements made to validate theories of the nature of the universe or to suggest new theories. These measurements, which can be called scientific metrology (properly the domain of experimental physics), present special problems.
E.1. Standards of Measurement
The National Institute of Standards and Technology (NIST) is the American custo- dian of the standards of measurement. It was established by an act of Congress in 1901, although the need for such a body had been noted by the founders of the Con- stitution. NIST’s two main campuses are in Gaithersburg, Maryland, and Boulder, Colorado, where research into the phenomenon of measurement, the properties of materials, and calibration of the reference standards submitted by laboratories from throughout the United States is carried out. The following is a generalization of the echelons of standards in the national measurement system (Rice 1986):
• National standards. Include prototype and natural phenomena of SI (Systems International, the worldwide system of weight and measures standards) base units and reference and working standards for derived and other units
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• Metrology standards. Reference standards of industrial or governmental laboratories
• Calibration standards. Working standards of industrial or governmental laboratories
In order to maintain accuracy, standards in a vast industrial complex must be traceable to a single source, usually the country’s national standards. Since the national laboratories of well- developed countries maintain close connections with the International Bureau of Weights and Measures, there is assurance that items manufactured to identical dimensions in different countries will be compatible (McNish 1967).
Application of precise measurement has increased so much during the past few years that it is no longer practical for a single national laboratory to perform all the calibrations and standardization required by a large country with a high technical development. Malshe et al. (2013) discuss manufacturing and metrology at the micro- scale in their review paper of engineering surface architecture with biological references and inspiration.
Increased precision, tolerance, and technology in measurements have led to the establishment of a considerable number of standardizing laboratories in indus- try and in various branches of the state and national governments (see Figure 4.21). In order for results of calibrations to be uniform, the standardizing laboratories must maintain close rapport with the national laboratory. This is facilitated by the use of uniform terminology in discussing standards (McNish 1967).
Figure 4.21 Classification of standards.
National standards
National reference standards
Working standards
Interlaboratory standards
Laboratory reference standards
Working standards
Reference or working standards of lower order
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The term “standard” includes three distinct areas, all of which are of impor- tance in metrology (NIST 1981):
1. Definitions of base units
2. Physical artifacts
3. Paper standards
E.1.a. Definitions of Base Units
The definitions of the base units of measurement form a reference from which all other units can be derived. These base units, together with two supplemen- tary units related to angle measurement that are necessary to specify a complete system of units, are listed in Table 4.12. Table 4.13 gives the definitions of all the SI units listed in Table 4.12. The definitions can also be found at the NIST website (https://www.nist.gov/pml/nist-guide-si-appendix-definitions-si-base-units). A current chart showing the relationships of all the SI units to which names have been assigned can be found at the NIST website in NIST Special Publication 304A (https://www.nist.gov/sites/default/files/documents/pml/wmd/metric/ NIST-SP-304A-Brief-History-Measurement-Systems-w-Color-Chart-1997.pdf).
All of the SI units listed in Tables 4.12 and 4.13 are defined in terms of experi- ments that can be performed in any suitably equipped laboratory, except for the definition of the unit mass, the kilogram. The kilogram is the only base unit defined in terms of a physical artifact.1 It must therefore be carefully preserved and protected, and the unit can only be disseminated by direct comparisons with the defining artifact. The kilogram is the mass of the International Prototype of the Kilogram, which is kept at the International Bureau of Weights and Measures near Paris, France.
Table 4.12 Base units of the international system.
Quantity Name
Length Meter
Mass Kilogram
Time Second
Electric current Ampere
Thermodynamic temperature Kelvin
Amount of substance Mole
Luminous intensity Candela
Plane angle* Radian
Solid angle* Steradian
*Supplementary units
1 The kilogram will be redefined by NIST in 2018 in terms of the Planck constant (https://www.nist. gov/pml/productsservices/redefining-kilogram).
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The standard for angle measurements is present in the form of the circle, and units of angle are defined in terms of this standard. Thus, one degree is the angle that subtends 1/360 of the circumference of a circle, and one radian is the angle that subtends 1/(2π) times the circumference.
Measurements of length are defined and determined by people. Until 1960, the meter was defined as the distance, under certain specified environmental con- ditions, between two lines engraved on the neutral axis of the International Pro- totype Meter, a bar of 90% platinum/10% iridium alloy, which is preserved at the International Bureau of Weights and Measures; Prototype Meter No. 27, whose length was known in terms of the international prototype, served as a standard for the United States. The origins of the meter can be read about at http://physics.nist. gov/cuu/Units/meter.html.
This method of defining the meter length was not entirely satisfactory since it required periodic recalibration of the various national standards in terms of the
Table 4.13 Definitions of the SI base units.
Unit Definition
meter–m The distance traveled by light in a vacuum during a time interval of 1/299,792,458 of a second.
kilogram–kg A cylinder of platinum-iridium alloy kept by the International Bureau of Weights and Measures at Paris. A duplicate in the custody of the National Institute of Standards and Technology serves as the mass standard for the United States.
second–s The duration of 9,192,631,770 cycles of the radiation associated with a specified transition of the cesium-133 atom. It is realized by tuning an oscillator to the resonance frequency of cesium-133 atoms as they pass through a system of magnets and a resonant cavity into a detector.
Ampere–A That current which, if maintained in each of two long parallel wires separated by one meter in free space, would produce a force between the two wires (due to their magnetic fields) of 2 × 10–7 newton for each meter of length.
Kelvin–K The fraction 1/273.16 of the thermodynamic temperature of the triple point of water. The temperature 0 K is called absolute zero.
mole–mol The amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12.
candela–cd The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 (Hz) and that has a radiant intensity in that direction of 1/683 watt per steradian.
radian–rad The plane angle with its vertex as the center of a circle that is subtended by an arc equal in length to the radius.
steradian–sr The solid angle with its vertex at the center of a sphere that is subtended by the area of the spherical surface equal to that of a square with sides equal in length to the radius.
Source: NIST Special Publication 304A, August 1981 (used with permission).
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international standard. In 1960, the Eleventh General Conference on Weights and Measures redefined the meter as a length equal to 1,650,763.73 wavelengths, in a vacuum, of the orange- red radiation corresponding to the transition between the 2p10 and 5d5 levels of the krypton-86 atom. The meter so defined is identical to that previously defined, within the limits of accuracy of the various measurements involved. The new definition provided a standard for length measurement that was based on an unchanging physical constant that could be reproduced in any properly equipped laboratory in the world. The inch is defined as 0.0254 meters (Taylor and Thompson 2008).
The definition of the meter was again changed in 1983 by the General Confer- ence of Weights and Measures (Taylor and Thompson 2008). The current definition of the meter is the length of a path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second. This definition of the meter thus defines the speed of light to be exactly 299,792,458 meters/second, and with this definition the meter could be realized from the wavelength of any coherent optical source whose frequency is known. The wavelength is the speed of light divided by the frequency.
E.1.b. Physical Artifacts and Paper Standards
Physical artifacts are manufactured with high precision to embody a particu- lar quantity, dimension, or feature. These include such items as gage blocks for length, standard resistors for electrical resistance, standards for cell voltage, and so on. This class of artifacts also includes high- precision analog measurement instru- ments that can be used as masters for reference, such as mercury in glass ther- mometers and dead weight testers for pressure.
Paper standards are the many documents published by various technical soci- eties and standards- writing organizations that contain specifications or generally accepted methods for making measurements.
E.2. uncertainty in Metrology
A fundamental role of the metrology and calibration process is to assign accuracy or uncertainty statements to a measurement. This can be achieved by defining characteristics of measuring system elements as well as equipment limitations.
E.2.a. Error in Measurement
Error in measurement is the difference between the indicated value and the true value of a measured quantity. The true value of a quantity to be measured is sel- dom known. Errors are classified as random errors or systematic errors. Gosavi and Cudney (2012) present an overview of the concepts and techniques used in metrology error. While the paper is generally focused on errors, they also present topics regarding metrology in general.
Random errors are accidental in nature. They fluctuate in a way that cannot be predicted from the detailed employment of the measuring system or from knowl- edge of its functioning. Sources of errors such as hysteresis, ambient influences, or variations in the workpiece are typical but not all- inclusive in the random category.
Systematic errors are those not usually detected by repetition of the measure- ment operations. An error resulting from either faulty calibration of a local standard
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or a defect in contact configuration of an internal measuring system is typical, but not completely inclusive in the systematic class of errors (Darmody 1967).
It is important to know all the sources of errors in a measuring system, rather than merely to be aware of the details of their classification. Analysis of the causes of errors is helpful in attaining the necessary knowledge of achieved accuracy.
There are many different sources of errors that influence the precision of a measuring process in a variety of ways according to the individual situation in which such errors arise. The permutation of error sources and their effects, there- fore, is quite considerable. In general, these errors can be classified under three main headings:
1. Process environment
2. Equipment limitation
3. Operator fallibility
These factors constitute an interrelated three- element system for the measuring process as shown in Figure 4.22.
The requirement of any precision measuring instrument is that it should be able to represent, as accurately as possible, the dimension it measures. This neces- sitates that the instrument itself have a high degree of inherent accuracy. Small
Figure 4.22 Factors affecting the measuring process.
Operator fallibility
• Identification of the situation
• Analysis of alternative methods
• Selection of equipment
• Application/ measurement
Equipment limitation
• Sensitivity
• Accuracy and precision
• Consistency
• Repeatability
Process environment
• Temperature
• Vibration
• Structural instability
• Humidity
• Factors of atmospheric pollution
• Atmospheric pressure
• Gravity
Measured characteristics
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inaccuracies will exist, however, due to the tolerances permitted in the instru- ment’s manufacture. These inaccuracies will influence the degree of precision attainable in its application.
The identification of measuring situations becomes increasingly complex in modern metrology. As parts become smaller and more precise, greater attention has to be paid to geometric qualities such as roundness, concentricity, straightness, parallelism, and squareness. Deficiencies in these qualities may consume all of the permitted design tolerance, so that a simple dimensional check becomes grossly insufficient.
Operators have to be knowledgeable about what they have to measure and how satisfactorily the requirements of the situation will be met by the measuring instrument. Correct identification of the measuring situation will eliminate those methods unsuitable for the situation. Proper measuring equipment can therefore be selected from a smaller range of measuring process alternatives. Method analy- sis can then be applied to these alternatives to determine which best satisfies the situation. This usually involves examining each method for different characteris- tics and evaluating the relative accuracies between the different methods.
E.2.b. Accuracy
Accuracy is the degree of agreement of individual or average measurements with an accepted reference value or level (Montgomery 2013). Measurement science encompasses two basic approaches for determining conformity to measurement accuracy objectives: (1) an engineering analysis to determine all causes of error, and (2) a statistical evaluation of data after stripping or eliminating the errors revealed by the engineering analysis (Darmody 1967).
E.2.c. Precision
Precision is the degree of mutual agreement among individual measurements made under prescribed like conditions, or simply, how well identically performed measurements agree with each other (Montgomery 2013). This concept applies to a process or a set of measurements, not to a single measurement, because in any set of measurements the individual results will scatter about the mean. Since the means of the results from groups of measurements tend to scatter less about the overall mean than individual results, reference is commonly made to the precision of a single measurement as contrasted with the precision of groups of measure- ments, but this is a misuse of the term. What is really meant is the precision of a set of single measurements or the precision of a set of groups of measurements (McNish 1967).
E.2.d. Sensitivity and Readability
The terms “sensitivity” and “readability” often are used in discussing measure- ment, and sometimes the concepts they involve are confused with accuracy and precision (see McNish 1967). Sensitivity and readability are primarily associated with equipment, while accuracy and precision are associated with the measuring process. The most sensitive or the most readable equipment may not always lead to the most precise or the most accurate results.
Sensitivity can be defined as the least perceptible change in dimension detected by the measuring tip and shown by the indicator. Readability is the ease of reading
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the instrument scale when a dimension is being measured. It is a factor that should remain constant over the full scale range.
E.2.e. Consistency
Consistency is another characteristic of the measuring instrument. Consistency of the reading on the instrument scale when the same dimension is being measured is necessary. This property affects the performance of the measuring instrument, and, therefore, complete confidence in the accuracy of the process cannot be estab- lished in the absence of consistency.
E.3. Traceability
Traceability is a process intended to quantify a laboratory’s measurement uncer- tainty in relationship to the national standards. It is based on analyses of error contributions present in each of the measurement transfers: the calibration of the laboratory’s reference standards by NIST, the measurements made in the calibra- tion transfers within the laboratory, and the measurements made on a product. Evidence of traceability is normally required; it may be as simple as retention of certificates and reports on calibration or as complex as reproduction of the analy- ses demonstrating the uncertainties claimed for the measurements (Rice 1986).
A laboratory that maintains its own reference standards (i.e., it relies on no lab- oratory other than NIST for calibration of its standards) must continuously moni- tor its own performance. Measurements on check standards, intercomparisons of standards, and participation in measurement assurance programs sponsored by NIST are meant to quantify laboratory error sources, as well as to provide indica- tions of the causes (Rice 1986).
E.4. Calibration
Calibration refers to measurements where the individual values are reported, rather than to measurements indicating only that an instrument is functioning within prescribed limits. It also refers to the disciplines necessary to control mea- suring systems to ensure their functioning within prescribed accuracy objectives.
The general calibration provisions for a measuring system include the following:
1. Acceptance calibration of a new system
2. Periodic calibration of the system in use or when placed in use after storage
3. Availability of standards traceable to the national standard for the unit of measure under consideration
Normally, a calibration chain or pyramid of echelons is involved in the discipline of metrology control and surveillance. The levels are as follows:
Level 1. The product tolerance or measured quantity
Level 2. The calibration of the product measuring system
Level 3. The calibration of the measuring system used to calibrate the product measurement system
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Level 4. Local standards, such as gage blocks or standard cells (volts), used for calibration of level 3
Level 5. Referencing local standards of level 4 to the national standard
Each of these levels attempts to achieve an accuracy/tolerance ratio that will sat- isfy requirements of the preceding level. This achievement is, of course, subject to the limitations of the state of the art, as well as cost–accuracy trade- offs that may come into play.
The aim of all calibration activities is ascertaining that a measuring system will function to ensure attainment of its accuracy objectives.
Periodic calibration of measuring and test equipment is accepted by most as necessary for measurement accuracy. A little more controversial is the question of determining the basis of the period of recalibration. There are a number of tech- niques in use to establish calibration intervals initially and to adjust the intervals thereafter. These methods include the following:
1. The same interval for all equipment in the user’s inventory
2. The same interval for families of instruments (e.g., oscilloscopes, gage blocks)
3. The same interval for a given manufacturer and model number
Adjustments of these initial intervals are then made for the entire inventory, individual families, or manufacturer and model numbers, respectively, based on analyses or history. A study conducted for NIST in connection with a review of government laboratory practices identifies these and other methods (Vogt 1980).
E.4.a. Calibration Control System
A typical calibration program may involve all or most of the following tasks (Rice 1986):
1. Evaluation of equipment to determine its capability
2. Identification of calibration requirements
3. Selection of standards to perform calibration
4. Selection of methods/procedures to carry out the measurements necessary for the calibration
5. Establishment of the initial interval and the rules for adjusting the interval thereafter
6. Establishment of a recall system to ensure that instruments due for calibration are returned
7. Implementation of a labeling system to visually identify the instrument’s due date
8. Use of a quality assurance program to evaluate the calibration system (process, control, audit, corrective action, etc.)
Selection of the standards, methods, and procedures to carry out the calibration includes the decision of where the calibration will be performed. The recall system
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must be designed to ensure that both the calibration organization and the using organization are aware in advance that an instrument will be due for calibration. Labeling instruments to visually display their calibration due dates is a companion feature to the recall system. Labels indicate (by dates, color codes, or similar sym- bols) the date the instrument is due for its next calibration. This visual identification may be used by the quality assurance organization to ensure that the instrument is not used beyond its due date. Intervals are established in a variety of ways, as dis- cussed previously. Principal objectives of an interval adjustment program include minimizing the potential for out- of-tolerance instruments in user areas, minimiz- ing the costs of calibration, and ensuring the required accuracy of instrumentation.
E.4.b. Measurement Assurance
Measurement assurance, thought by some to relate only to methods used in the metrology or calibration laboratory to secure calibrations by NIST, is one of the more important concepts in the measurement field.
Traditionally, calibrations by NIST determine the accuracy and precision of the measuring instrument. Measurement assurance protocols (MAPs), on the other hand, are able to include not only the accuracy of the item but also the contribution to error by the metrologist/technician, laboratory environment, and practices/ procedures of the laboratory because the experiment involves measurements by participants in their own laboratories (Belanger 1980).
Measurement assurance, in addition to being a concept of importance to metrology and calibration laboratory managers, is something that should interest quality assurance personnel involved in testing and measurement. Most factory testing and measuring involves the use of equipment whose accuracy has been determined through calibration. Little, if any, consideration is given to errors that may be contributed by the test operator, by his or her instructions or procedures, or by the environments in which the equipment is operated. The application of measurement assurance can serve to reduce errors (Rice 1986).
F. MEaSurEMEnT SySTEM anaLySiS Measurement system analysis (MSA) consists of qualifying the measurement pro- cess, determining the adequacy of the measurement system for use, and identify- ing and estimating the process error. A measurement system is the entire process for obtaining measurements on some quality characteristic of interest. This process includes standards, personnel, methods of measurement, and so on.
In this section, definitions as well as the concept of gage repeatability and reproducibility (gage R&R) are introduced.
F.1. Terms and definitions
In this section we talk about types of errors as well as accuracy and precision with respect to measurement systems. Two important and common types of errors in MSA are systematic error and random error. Systematic errors can be caused by human interference, poor manufacturing methods, and measuring device imper- fections, for example. This error remains fairly constant over repeated measure- ments collected under identical conditions. The error is systematic, which results
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in values that are consistently above or consistently below the true or reference value of the quality characteristic.
Random errors vary arbitrarily over all measurements taken under identical conditions. Even when systematic errors have been identified and accounted for, normal random fluctuations will occur. If only random errors are present in the system, then increasing the number of measurements taken will provide a better estimate of the true value of the quality characteristic.
Measurement system error with respect to MSA consists of variability that can be attributed to gage bias, stability, linearity, repeatability, and reproducibil- ity. Accuracy of a measurement system is made up of bias, linearity, and stability. Repeatability and reproducibility are the components that describe precision, or measurement variation.
Accuracy is a qualitative term defined as the difference between the measure- ment taken and the actual value of the quality characteristic of interest. The three components of accuracy are bias, linearity, and stability.
Bias is defined as the difference between the observed average measurement and a reference value, and is a measure of systematic error in terms of the mea- surement system. Bias is the difference between the observed average and the ref- erence value.
The observed average measurement can be found by measuring a single part multiple times or selecting several parts at random and measuring each part mul- tiple times. The measurements should be taken under identical conditions.
ExaMpLE 4.6
Suppose three parts of different sizes are selected and the diameter of each part is mea- sured. These parts represent the normal range of part sizes for which the measurement system is used. A reference value is known for each of the parts. Suppose each part is measured five times, with the results displayed in Table 4.14. The observed averages are calculated and the resulting bias estimated for each part.
Table 4.14 Bias and average estimates for parts of different sizes.
Parts 1 2 3
Reference value 2.00 3.80 5.60
1 2.10 3.65 6.21
2 1.88 4.00 5.40
Trials 3 1.92 3.88 5.26
4 2.05 3.78 5.98
5 2.01 4.10 4.93
Average 1.992 3.882 5.556
Bias –0.008 0.082 –0.044
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Hypothesis tests can be carried out to test the significance of bias (see AIAG [2010]). If bias is found to be significant, the cause for bias should be identified. Some reasons for significant bias may include (but are not limited to) an incorrect reference value, a worn measuring device, or improper calibration of or incor- rect use of the measuring device.
Linearity measures how changes in the size of the part being measured will affect measurement system bias over the expected process range. Consider the previous example (Example 4.6) with three parts of different sizes. Notice that the bias estimates were quite different across the different sizes. There may be evidence of nonlinearity if as the part size increases the bias changes significantly. Tests can be performed to determine whether nonlinearity, if it exists, is significant.
Stability is a measure of how well the measurement system performs over time. It provides a measure of the change in bias over time when the same part is measured. Stability differs from linearity in that only one part, whose reference value is known (or assumed to be known), is measured at different points in time. This is to determine whether the measurement system has changed over time and after many uses.
In general, accuracy provides information about location, or the relationship between the measurement results and reference value of the quality characteristic.
Precision is defined as the variation encountered when the same part is mea- sured repeatedly using the same measurement system (under the same condi- tions). The two components of precision are repeatability and reproducibility. Repeatability represents the variability due to the gage or test instrument when used to measure the same part under identical conditions (i.e., same operator mea- suring the same part). Reproducibility, on the other hand, represents the variability due to different operators or setups measuring the same parts using the same mea- suring device. Reproducibility represents the variability due to the measurement system. Both repeatability and reproducibility will be discussed in more detail in this chapter.
F.2. gage repeatability and reproducibility2
Gage repeatability and reproducibility (R&R) studies are used to determine whether a measurement system is capable for its intended purpose. If the mea- surement system variation is small compared with the process variation, then the measurement system is considered capable. In general, the purposes of a gage R&R study are to:
• Determine the amount of variability in the collected data that can be attributed to the measurement system in place
• Isolate the sources of variability in the measurement system
• Determine whether the measurement system is suitable for use in a broader project
2 In the literature, you will see the terms “gage” and “gauge” used interchangeably. Both spellings are correct and acceptable.
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When conducting a gage R&R study, it is often assumed that the “parts” and the “operators” are selected at random from larger populations. The parts are typi- cally selected at random so that they represent the entire operating range of the process. Since the parts and operators are randomly selected, there is a measure of variability associated with each. There are situations where the parts or operators may be fixed. To illustrate, suppose the operators are really automatic gages and there are only three total for a particular process. If all three automatic gages are used, then we say that the factor “operator” is fixed. Assessing the capability of fixed factors is beyond the scope of this handbook, but additional information and references can be found in Burdick, Borror, and Montgomery (2005). The following link from ASTM offers additional discussion on repeatability and reproducibility: https://www.astm.org/SNEWS/MA_2009/datapoints_ma09.html.
A number of issues must be considered when designing a gage R&R experi- ment, such as the number of parts, the number of operators, and the number of replicates to include. There has been considerable debate about these issues. The “standard” experiment often included 10 parts, three operators, and two repli- cates. However, research has indicated that these recommendations may not be appropriate for many problems. Burdick and Larsen (1997) demonstrated that the lengths of confidence intervals on the variance components in a gage R&R study are significantly shortened when the number of operators is increased (see Chapter 6 for discussion on confidence intervals). They recommend at least five or six operators in a typical gage R&R study. Increasing the number of parts does not affect the confidence intervals as much as increasing the number of operators. However, it has been shown that if the practitioner has to choose between increas- ing the number of parts and increasing the number of replicates on each part, a greater benefit is obtained by increasing the number of parts. See the review paper by Burdick, Borror, and Montgomery (2003) and the references within for further discussion of these issues in gage R&R experiments.
Gage variability is a function of variance components. Let Repeatability 2σ represent
the inherent variability in the gage and Reproducibility 22σ represent the variability due
to the different operators (or setups, different time periods, etc.) using the same gage. Specifically, we can write the measurement error variability as
Measurement error 2
Gage 2
Reproducibility= = 22
Repeatability 2+σ σσσ (4.14)
Furthermore, suppose part- to-part variability is denoted by ( p 2σ ), then total vari-
ability can be written as a sum of the two variance components:
Total 2
Gage 2= + p
2σ σ 2σ (4.15)
In a gage R&R study it is important to accurately estimate these variance com- ponents for the estimation of repeatability and reproducibility. Two commonly used methods for estimating repeatability and reproducibility are (1) the tabular method (also known as the range method) and (2) the analysis of variance method. Both methods will be presented and discussed in this section.
F.2.a. The Tabular Method (Range Method)
Gage R&R studies were often conducted using a tabular method because the cal- culations were simple. Software now allows the ANOVA method (discussed in the
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next subsection) to be used more readily and is more commonly used today. The tabular method is based on information that can be obtained from control charts and from using the sample ranges to estimate variability (see Chapter 6, section A, for discussion of the sample range and Chapter 6, section F, for discussion of con- trol charts).
Estimating Reproducibility
The steps for estimating reproducibility using the tabular method are as follows:
1. Estimate the average measurement for each “operator”
2. Find the range of these averages (largest average – smallest average); this is called RO (for operator range)
3. Estimate the standard deviation for reproducibility using the relationship
=
R d
O
2
ˆ Reproducibilityσ
(4.16)
4. Estimate the variance component for reproducibility:
ˆ
Reproducibility 2 =
R d
O
2
2
e eσ
(4.17)
Estimating Repeatability
The steps for estimating repeatability are as follows:
1. Calculate the range for each part (or sample)
2. Calculate the average range across all samples; this is denoted R –
3. Estimate the standard deviation for repeatability:
= =e
R d2
ˆ Repeatabilityσ σ̂
(4.18)
4. Estimate the variance component for repeatability:
Repeatability 2 = =e
R d
2
2
2
e eσ̂σ̂
(4.19)
Estimating Part- to-Part Variability
The steps for estimating part- to-part variability are as follows:
1. Calculate the average measurement for each part (or sample)
2. Find the range of these averages (largest average – smallest average); this is denoted Rp (for part range)
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3. Estimate the standard deviation for parts:
p
pR
d =
2
σ̂
(4.20)
4. Estimate the variance component for parts:
pR
d 2
2
2
= e epσ̂
(4.21)
Complete details of determining values of d2 for each of the above quantities can be found in AIAG (2010) or Barrentine (2003). The values of d2 can also be found in Appendix B and are discussed in more detail in Chapter 6, section F. Additionally, in AIAG (2010) and Barrentine (2003) you will find details of the range method.
F.2.b. The Analysis of Variance Method
One of the reported drawbacks to using the tabular method has been the inability to estimate any possible interaction between operators and parts (or samples).
It is often assumed that the operators are well trained and as a result there should be no significant interaction between these two factors. If, however, there is a signifi- cant interaction, this effect should be quantified and taken into consideration when providing estimates of repeatability and reproducibility. Using the tabular or range method, it is not possible to estimate the interaction between operator and part.
The analysis of variance (ANOVA) method has become a common choice for practitioners conducting gage R&R studies since the computations can be easily carried out using modern statistical software. (See Chapter 6, sections D and F, for discussion of interactions, factors, and the general ANOVA method.) Before we present the ANOVA method, some basic assumptions must be discussed.
Suppose the response of interest in a gage R&R study can easily be expressed by a random two- factor model (see Chapter 6, section D, for more details on modeling a response, factors, two- way interactions, and two- way ANOVA). The factors we are interested in are “parts,” “operators,” and possibly the part- by-operator inter- action. A gage R&R study is to be carried out involving p parts, m operators, and n replicates. Suppose y represents the response of interest and can be modeled as
y P O PO
i
ijk i j ij ijk = + + + ( ) +
=
µ
for 1, 2, . . . , 1, 2, . . . , 1, 2, . . . ,p j m k n; ;= =
ε
(4.22)
where
yijk is the kth measurement of the ith part by the jth operator
μ is the overall process mean
Pi represents the effect of the ith part; we assume that Pi is a random factor that follows a normal distribution with mean zero and variance P
2σ Oj represents the effect of the jth operator; we assume that Oj is a random factor that follows a normal distribution with mean zero and variance O
2σ (PO)ij represents the part- by-operator interaction effect; we assume that (PO)ij is a random factor that follows a normal distribution with mean zero and variance PO
2σ εijk represents random error; we assume that εijk follows a normal distribution with mean zero and variance e
2σ
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The terms P 2σ , O
2σ , PO 2σ , and e
2σ are the variance components. As discussed previ- ously, gage variability is a function of these variance components. Specifically, we wrote measurement error (gage) variability in Equation (4.14). But now we will use the variance components for our random factors to determine repeatability, reproducibility, and part variability. Specifically, reproducibility and repeatability variation can be written as
2 = +PO O
2 2ˆ Reproducibilityσ σσ (4.23)
and
Repeatability 2 = e
2σσ (4.24)
Part-to-part variability is given by P 2σ . The variability of the total observed mea-
surement is given by
Total 2
Gage 2= + p
2σ σ 2σ (4.25)
The variance components will be estimated using the mean squares obtained from an ANOVA table. To begin, a standard ANOVA is conducted assuming that the two- factor standard model given earlier is valid. The reader is encouraged to see Chapter 6, section E, for complete discussion of sum of squares (SS), mean square (MS), ANOVA, p-value, and degrees of freedom (df).
The procedure is as follows:
• Treat the problem as a designed experiment (see Chapter 6, section H, for details on designed experiments)
• Conduct an ANOVA (set up an ANOVA table—see Chapter 6)
• Use the mean square values from the ANOVA table to estimate the variance components (see Chapter 6 for discussion of mean square)
Some of the quantities from the ANOVA table are given in Table 4.15. Standard sta- tistical software packages will provide these values, so it is not necessary to carry out the calculations by hand.
Table 4.15 Necessary quantities for an analysis of variance.
Source DF SS MS
Part p–1 SSP MSP
Operator m–1 SSO MSO
Part × operator (p –1)(m–1) SSPO MSPO
Error (repeatability) mp(m–1) SSE MSE
Total mpn–1 SST
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The estimates of the variance components given earlier are
Operators:
MS MS O
O PO
pn =
− 2σ̂
(4.26)
Part × operator: PO
PO== −MS MSE n
2σ̂
(4.27)
Parts: =
−MS MS P
P PO
mn 2σ̂
(4.28)
Error: = MSe EE 2σ̂ (4.29)
It is possible that one or more of the variance components could result in a negative value. Some researchers have maintained that if any variance compo- nent estimate is negative, it is set equal to zero. Other researchers recommend using different approaches for estimating these quantities so that the estimates are nonnegative (see Montgomery [2013] for more details). The variance component estimates are then used to estimate reproducibility, repeatability, and part- to-part variation, as well as the total variability, using the equations given previously.
F.2.c. Example of Gage R&R
An experiment was conducted on the thermal performance of a power module for an induction motor starter. The response was thermal performance measured in degrees C per watt. Table 4.16 displays a partial list of data collected for 20 motors by six operators. Each operator measured all parts twice. The original data have been multiplied by 100 for convenience. (The original problem statement for this example is from Houf and Berman [1988].) The specification limits are LSL = 18 and USL = 58. We assume that each motor and the operators have been selected at random from larger populations. The model of interest involves operators, parts, and the operator- by-part interaction.
Table 4.16 Typical data for the gage R&R experiment.
Operator 1 Operator 2 Operator 6
Part 1 2 1 2 … 1 2
1 44 34 43 44 … 46 46
2 21 23 20 22 … 21 21
.
.
. . . .
.
.
. . . .
.
.
. …
.
.
. . . .
.
.
. . . .
.
.
. . . .
.
.
. …
.
.
. . . .
20 29 31 31 30 … 31 29
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In this problem, p = 20, m = 6, and n = 2. We will examine results of the gage R&R study using both the tabular method and the ANOVA method. The calcula- tions for the variance components were carried out using Minitab v15 (the latest version of Minitab is v17 (2014)) for both methods.
F.2.c.i. Tabular Method Results
The results of the tabular method are given in Table 4.17. The second column in Table 4.17 provides the estimates for the variance components:
0.5678= =ˆ Repeatability 2σ ˆ e
2σ
0.2607= =ˆ Reproducibility 2σ ˆ O
2σ
P =ˆ 2σ 54.8697
Measurement error Gage Reproducibility= = +
= +
=
Repeatability
0.2607 + 0.5678 0.8285
ˆ 2σ ˆ 2σ ˆ 2σ ˆ 2σ
Total Gage= +
=
=
P
0.8285 + 54.8697 55.6982
ˆ 2σ ˆ 2σ ˆ 2σ
The last column in Table 4.17 provides the percentage of the total variability con- tributed by each source. For example, the percent contribution for “Repeatability” was found by
% contribution Repeatability= =(100%) 55.6982 0.5678
=(100%) 1.02% ˆ 2σ
Totalˆ 2σ
From Table 4.17, we see that the largest source of variability is differences between parts.
F.2.c.ii. ANOVA Method Results
Before estimating the variance components using the ANOVA method, we can determine if there is a statistically significant difference between parts
Table 4.17 Gage R&R estimates using the tabular method.
Source Variance component % contribution
Total gage R&R 0.8285 1.49
Repeatability 0.5678 1.02
Reproducibility 0.2607 0.47
Part-to-part 54.8697 98.51
Total variation 55.6982 100.00
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or between operators, and if there exists a statistically significant interaction between parts and operators. An ANOVA was carried out, with the results pro- vided in Table 4.18.
The variance components can be estimated as follows (although generally it is not necessary to calculate these by hand) using Equations (4.26) through (4.29), respectively:
Operators: MS MS
O O PO
pn =
− 13.580 – 2.060 20(2)
0.288= =2σ̂
Part × operator: PO PO==
−MS MSE n
== = 2.060 – 0.733
2 0.66352σ̂
Parts: = −MS MS
P P PO
mn = =
591.479 − 2.060 49.118
6(2) 2σ̂
Error: = MS = 0.733e EE 2σ̂
The gage R&R estimates are then:
0.6635 + 0.288 = 0.9515 = + =ˆ Reproducibility 2σ ˆ PO
2σ ˆ O 2σ
Repeatability 0.733= =e 2σ̂2σ̂
0.9515 + 0.733
+==
= 1.6845=
ˆ Reproducibility 2σˆ Gage
2σˆ Measurement error 2σ ˆ Repeatability
2σ
49.118=ˆ P 2σ
1.6845 + 49.118
+=
= 50.803=
ˆ Gage 2σˆ Total
2σ ˆ P 2σ
As with the tabular method, the variance components for the gage R&R study and the percent contribution can be found using a statistical software package such as Minitab. The results are given in Table 4.19.
Table 4.18 ANOVA for the gage R&R example.
Source DF SS MS F P
Part 19 11,238.1 591.479 287.126 0.000
Operator 5 67.9 13.580 6.592 0.000
Part × operator 95 195.7 2.060 2.810 0.000
Error (repeatability) 120 88.0 0.733
Total 239 11,589.7
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The slight differences between the estimates computed by hand and those pro- vided by the software package for the ANOVA method are strictly due to round- off error. From the results in Table 4.19, we see again that most of the total variability is due to differences in the parts. However, the ANOVA results in Table 4.18 show that there appears to be a significant difference between operators as well as a significant interaction between operators and parts (p-values are zero for all practi- cal purposes; see Chapter 6 for discussion of p-values). Since there is a significant interaction between parts and operators, this may be evidence that more operator training is necessary.
F.2.c.iii. Comparison of the Results
It is important to more fully examine and compare the results that were obtained with these two methods. The variance component estimates for both methods are repeated in Table 4.20.
The differences between the two methods are striking. The tabular method uses sample ranges to estimate the variance components, while the ANOVA method uses arguably more efficient estimates based on functions of sample variances (see
Table 4.19 Gage R&R results using the ANOVA method.
Source Variance component % contribution
Total gage R&R 1.6850 3.32
Repeatability 0.7333 1.44
Reproducibility 0.9515 1.87
Operator 0.288 0.57
Part × operator 0.6635 1.31
Part-to-part 49.1181 96.86
Total variation 50.8031 100.00
Table 4.20 Variance component estimates for both methods.
Source Tabular method ANOVA method
Total gage R&R 0.8286 1.6850
Repeatability 0.5678 0.7333
Reproducibility 0.2607 0.9515
Operator 0.2607 0.2882
Part × operator – 0.6635
Part-to-part 54.8697 49.1181
Total variation 55.6983 50.8031
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Chapter 6). In addition, when using the tabular method, the variance component for the operator- by-part interaction could not be estimated. As a result, we obtain very different estimates for reproducibility and therefore total gage R&R. From the tabular method, total gage R&R is found to be 0.8286, while for the ANOVA method it is 1.6850.
F.2.d. The Role of the Control Chart in Gage R&R Studies
Control charts (presented in Chapter 6, section F) play an integral role in gage R&R studies. Control charts display information about gage capability. Consider the x– and R charts for the thermal performance example (displayed in Figure 4.23). The x– chart shows the gage’s ability to distinguish between parts. In a gage R&R study, it is desirable for the x– chart to have many out- of-control points. Each point on the x– chart represents the average of the two measurements taken by an operator on a part. Each point on the R chart represents the range between the two measurements taken by an operator on a part. There are a total of 120 samples on each chart.
The upper and lower control limits on the x– control chart were determined using the average range (see Chapter 6). As a result, the x– control chart reflects the within- sample variability, which is related only to gage repeatability. Notice that many of the points on the x– control chart plot beyond the control limits (what we would usually consider evidence that our process is out of control). In a gage R&R study, this is actually desirable since it indicates that the gage is capable of discriminating between different parts. If most of the samples on this control chart plotted within the control limits, it would signify that it is difficult for the gage to clearly identify different parts. In this example, there are several points that
Figure 4.23 x– and R control charts for the thermal performance example.
1
20
30
40
50 6
UCL = 38.47 x– – = 36.87
LCL = 35.27
5432 x– chart by operator
S am
pl e
m ea
n
1
0
1
2
3 6
UCL = 2.777
R – = 0.85
LCL = 0
5432 R chart by operator
S am
pl e
ra ng
e
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lie within the control limits. It may be necessary to determine if these are chance occurrences or if they indicate that the gage is having difficulty discriminating between the different parts. It is not clear in this example, so further investigation is most likely needed.
The R chart can provide information about special causes of variation. For example, if many of the ranges plot beyond the control limits, this could indicate problems with operator experience, training, or fatigue, which would also result in differences among operators. It is desirable for the points on the R chart to plot within the control limits. This condition signifies that the operators exhibit consis- tency in their use of the gage. The R chart for our example has several points outside the control limits. This is not surprising since our ANOVA indicated that there was a significant interaction between operators and parts. Further investigation is needed.
F.2.e. Interpretation of Gage R&R Estimates
Since many of the variance component estimates are also used in the calculation of measures such as signal- to-noise ratios (SNR), precision- to-tolerance ratios (PTR), discrimination ratios (DR), and process capability ratios, it is imperative that these estimates be as reliable as possible. The example described in section F.2.c of this chapter illustrates that the two methods could lead to different estimates. In turn, it is possible that the two methods could lead to very different conclusions about the adequacy of the measurement system. As a simple illustration, one formula for the PTR is
PTR
USL LSL Gage= −
6σ̂
(4.30)
Note that another form uses 5.15 in place of 6. Since ˆ Gageσ is simply the square root of our variance component for total gage variability ˆ Gage
2σ , we can calculate the PTR for our example using results from both the tabular method and the ANOVA method. For the tabular method, PTR is
PTR USL LSL
= −
= ( )
− =
6 6 0 8286
58 18 0 137
. .
ˆ Gageσ
For the ANOVA method, PTR is
PTR USL LSL
= −
= ( )
− =
6 6 1 6850
58 18 0 195
. .
ˆ Gageσ
Measurement systems with PTRs less than 0.10 (10%) are generally considered to be acceptable measurement systems. PTRs between 0.10 and 0.30 (10% to 30%) may be adequate for some applications. PTRs greater than 0.30 (30%) are considered unac- ceptable (Montgomery 2013; AIAG 2010). For details on PTR, SNR, and DR, see AIAG (2010), Wheeler and Lyday (1989), or Montgomery (2013). Woodall and Bor- ror (2008) provide a discussion of the relationships between these measures as well.
F.2.f. Issues and Considerations in Gage R&R Studies
A number of assumptions are made when using either the tabular method or the ANOVA method to carry out a gage R&R study. One such assumption involves
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replication (replication is defined and discussed in Chapter 6, sections C and H). In particular, it is assumed that each measurement (replicate) is made independently of one another, where a unique setup or preparation of the measuring device is made before the next measurement is taken. Suppose an operator measures a part four times. If the setup of the measuring device is not changed or reset before the next measurement, then the measurements are not true replicates. If the measure- ments are taken consecutively without resetting the measuring device, then they are a type of repeated measure. The analysis to obtain the estimates of the variance components would have to be different from what has been presented here.
Another assumption related to replication is randomization (randomization is discussed more fully in Chapter 6, sections C and H). Randomization in a gage R&R study is understood to mean that the operator measures each part in ran- dom order. A part is selected at random and measured, and then the next part is randomly selected and measured. The operator does not randomly select a part, take four measurements, put the part back, and then select the next part. In that case, the actual randomization is a form of restricted randomization and requires estimation of repeatability and reproducibility using methods other than what has been presented. There are numerous applications where complete randomization or true replication is not practical or possible. In those situations, other methods would have to be employed to provide reliable estimates of the necessary variance components.
There are advantages and disadvantages to using either the tabular method or the ANOVA method. The tabular method is easy to carry out using ranges to estimate variance components. In addition, interpretation of the results is often intuitive for the practitioner. However, the tabular method is restricted to investi- gating a measurement system that involves only parts (with one operator) or parts and several operators. It does not lend itself to more complex measurement sys- tems that may involve more than two factors (parts and operators). Furthermore, it does not adequately lend itself to dealing with systems where randomization is restricted or true replication is not possible. In general, as long as you are inter- ested only in parts and possible operators (and not even the interaction between them), then the tabular or range method can be used—again, only if complete randomization can be guaranteed.
The ANOVA method can be more computationally intensive than the tabular method, but with modern computer software this is less of an issue. The ANOVA method is more flexible than the tabular method in that it can handle unusual experimental conditions. For example, the ANOVA method can be used if there are more factors than just parts or operators. Suppose that in addition to parts and oper- ators, location on the part is also a factor to consider. In this case, a nested design may be appropriate. The necessary variance components can be easily estimated using ANOVA for a nested design (see Burdick, Borror, and Montgomery [2005] for details on gage R&R studies for nested designs). Note that the tabular method can- not be used for this more complex experimental situation. It should also be noted that just including a third factor (not necessarily nested) and estimating variance components for the factors, all two- factor interactions, and the three- factor interac- tion is not possible using the tabular approach. A simple extension of the standard two- factor design cannot be handled using the tabular approach.
In summary, the ANOVA method for estimating repeatability and reproduc- ibility is more flexible than the tabular method. It also uses more efficient estimates than sample ranges to obtain the necessary variance components’ estimates. With
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modern computational capabilities, the ANOVA method is no more difficult to carry out than the range method.
For further details on gage R&R studies or measurement systems in general, please see AIAG (2010); Barrentine (2003); Borror, Montgomery, and Runger (1997); Burdick, Allen, and Larsen (2002); Burdick, Borror, and Montgomery (2003, 2005); Dolezal, Burdick, and Birch (1998); Engel and deVries (1997); Jensen (2002); Larsen (2002); Mader, Prins, and Lampe (1999); Majeske and Andrews (2002); Montgom- ery (2013); Montgomery and Runger (1993a, 1993b); Vardeman and VanValken- burg (1999); and Weaver et al. (2012).
For an example of an application of a gage R&R, see, for example, Erdmann, Does, and Bisgaard (2009), which presents a case study of gage R&R in a hospital. The authors describe a study to measure temperature with an ear thermometer.
F.3. additional Considerations for Measurement Systems
In this chapter, MSA has been presented for typical manufacturing situations. There are, of course, numerous applications of agreement analysis in nonman- ufacturing settings. As more QEs become involved in the service sector, it will be imperative that they understand the use of appropriate statistical methods for assessing the capability of a measurement system.
Methods for assessing the capability of a quantitative measurement system as discussed in this chapter are well documented in the literature. Sometimes, the measurement system involves attribute data. This is more common in the service and nonmanufacturing industries than in the manufacturing industry. In the case of attribute data, the standard quantitative methods are no longer appropriate. An attribute gage measurement system is appropriate when the parts or objects of interest are placed into one of two or more possible categories (Windsor 2003). The measurement of interest is the classification of the item. For example, consider tax documents, insurance appraisals, or electronic medical records. These documents are read by many people and correctness of the documents or items in the docu- ment may be assessed.
Several assessment statistics and approaches that deal with categorical mea- surements include the following:
1. Appraiser agreement statistics such as κ statistics and intraclass correlation
2. The analytic method
3. Latent-class models
These three approaches can provide some measure of reproducibility or repeat- ability, and in some cases bias. For more details on appraiser agreement statistics, the reader is encouraged to see AIAG (2010); Banerjee et al. (1999); Bloch and Krae- mer (1989); Cicchetti and Feinstein (1990); Cohen (1960); Conger (1980); de Mast and van Wieringen (2007); Feinstein and Cicchetti (1990); and Fleiss (1971). For more discussion of the analytic method or latent- class models, see AIAG (2010); Agresti (1988, 1992); Agresti and Lang (1993); Banerjee et al. (1999); Boyles (2001); de Mast and van Wieringen (2004); McCaslin and Gruska (1976); Sweet, Tjokrod- jojo, and Wijaya (2005); Uebersax and Grove (1990); and van Wieringen and van Heuvel (2005).
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Some of the most successful organizations are those in which all members of the organization believe that a part of each person’s daily job is the improvement of the processes they work with. This chapter describes tools and techniques for accomplishing this vital task and helping with continuous improvement efforts. The chapter is divided into five sections: “Quality Control Tools,” which describes the seven original problem- solving tools; “Quality Management and Planning Tools,” which discusses what have become known as the seven new tools; “Con- tinuous Improvement Techniques,” which introduces several of the broader, more systematic approaches to quality; “Corrective Action”; and “Preventive Action.” The novice practitioner may feel overwhelmed by the array of tool options. The best advice is to identify a problem and try to use one or more of the tools in its solution, rather than study the tools and memorize their individual traits. Experi- ence with these tools provides a depth of understanding not attainable from the written word alone.
a. QuaLiTy ConTroL TooLS Quality control tools as defined by ASQ and as accepted throughout the quality engineering community include the following:
• Flowcharts
• Cause-and-effect diagrams
• Check sheets
• Histograms
• Pareto charts
• Run charts/control charts
• Scatter diagrams
Collectively, these tools are commonly referred to as the seven basic tools. Kaoru Ishikawa (1985) is credited with making the following statement with respect to these tools: “As much as 95 percent of all quality- related problems in the factory can be solved with seven fundamental quantitative tools.”
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Ishikawa’s statement provides three key insights into these tools—namely, that these seven tools are:
1. Applicable in problem- solving situations most commonly encountered by certified quality engineers (CQEs)
2. Quantitative in nature and rely, with possibly the exception of flowcharts and cause- and-effect diagrams, on numerical data
3. Most commonly used in quality control—that is, as aids in tracking, monitoring, and analyzing data—as opposed to the planning functions associated with quality assurance
This section discusses six of the seven basic tools (control charts are discussed in considerable detail in Chapter 6, section F). The following subsections describe the purpose of each tool, provide information about the tool’s applications and mechanics, and give at least one illustration of the tool’s use.
a.1. Flowcharts
The purpose of a flowchart is to provide a graphical representation of the ele- ments, components, or tasks associated with a process. Flowcharts are helpful for documentation purposes and, through standardized symbols, promote a com- mon understanding of process steps and the relationships or dependencies among those process steps.
Flowcharts can be prepared for and used at a high level where users may not be familiar with process- specific jargon or terminology. In the high- level applica- tion, flowcharts are intended to help users understand what may be a complex process without providing unnecessary, and potentially confusing, details.
Likewise, flowcharts can be prepared for and used at a detail level where users have familiarity and expertise with a given process. In the detail- level application, flowcharts are intended to help users perform analyses most commonly related to optimization or process improvement.
A flowchart can be created through the following steps.
1. Select start and stop points. A flowchart, by definition, must specify start and end points. Since it is possible to have many flowcharts describing various sections, elements, or components of a process, particularly when the process is large and complex, start and end points for flowcharts are defined in terms of boundaries. Boundaries are naturally occurring breaks or division points that separate processes or systems at the macro level, or sections, elements, or components of a process at the micro level.
2. List major steps/tasks and decision points. List, in sequential order, each of the major steps or tasks and decision points that occur as part of the process between the start and stop points.
3. Use standardized graphical symbols to document the process. Using standardized symbols, document each of the steps/tasks identified above (Gilbreth and Gilbreth 1921). Placement of appropriately labeled symbols and the use of arrows define the sequence of events. Four primary flowcharting symbols are depicted in Figure 5.1. While there
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are many symbols for flowcharting, these primary flowcharting symbols are capable of and adequate for documenting any process.
4. Review results. Compare the flowchart with the process to verify that the flowchart is complete and accurately describes the process. Having more than one person independently verify the flowchart is generally considered standard protocol.
Hallock, Alper, and Karsh (2006) present a process improvement study on diag- nostic testing in an outpatient healthcare facility. The purpose of the study was to determine what factors contributed to the delay of notification of test results to patients. A general flowchart for the overall diagnostic testing process, similar to the one in Figure 5.2, was presented.
a.2. Cause- and-Effect diagrams
The purpose of a cause- and-effect diagram, also known as a fishbone diagram or Ishikawa diagram, is to graphically document the analysis of factors (causes) that relate to a single problem or opportunity (effect). Cause- and-effect dia- grams are used in problem- solving situations and in general analysis to help the problem- solving or analysis team both understand how those factors may cause the given effect and focus on “next steps” in process improvement.
A cause- and-effect diagram is most successful when created by a multidisci- plinary team of people. The team creates the cause- and-effect diagram using the following steps:
1. Select a single problem or opportunity (the effect). A cause- and-effect diagram is useful for analyzing only one problem or opportunity at a time. The problem or opportunity that is selected for analysis is documented by a keyword or short narrative description placed in a rectangle or box, generally on the right side of the diagram. When
Figure 5.1 Four primary flowcharting symbols.
Start/stop symbol The general symbol used to indicate the beginning and end of a process is an oval.
Flow line symbol A line with an arrowhead is the symbol that shows the direction of the stages in a process. The flow line connects the elements of the system.
Basic processing symbol The general symbol used to depict a processing operation is a rectangle.
Decision symbol A diamond is the symbol that denotes a decision point in the process. This includes attribute-type decisions such as pass–fail and yes–no. It also includes variable-type decisions such as which of several categories a process measurement falls into.
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analyzing more than one problem or opportunity, a separate cause- and- effect diagram is used for each problem or opportunity.
2. Identify the major causes of the problem or opportunity. Cause- and-effect diagrams have been described as fishbone diagrams, where major causes are documented as the major bones of a fish skeleton. Major causes are generally described as they relate to people, hardware/equipment, the intended operating environment, methods, and materials. Teams should be formed to brainstorm possible causes or opportunities.
3. Identify the minor causes associated with each major cause. For each major cause (i.e., people, hardware/equipment, environment, methods, measurements, and materials) associated with a problem or opportunity, minor causes are identified. Identification of minor causes may be graphically described as adding more structure to the fishbone skeleton. Minor causes appear graphically as “bones” attached to a major cause.
4. Identify additional cause structure. The analysis continues, adding detail to the fishbone structure until all causes associated with a problem or opportunity have been identified and documented. The analysis may continue until several more layers of detail have been considered and added to the diagram.
Figure 5.3 depicts a high- level cause- and-effect diagram before detailed analysis is started. As mentioned previously, a single problem or opportunity is identified
Figure 5.2 Flowchart for diagnostic testing process.
Sample delivered for testing
Test results delivered to doctor(s)
Results interpreted
Patient notified
Tests ordered
Tests run/ samples collected
Patient examined by doctor
Patient discharged
No
Yes
Prep for visit
End
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on the right side of the graphic. Major causes are normally associated with one or more of the following, sometimes referred to as the 6 Ms:
• Man (people or personnel)
• Machines (hardware/equipment)
• Materials
• Methods
• Measurements
• Mother Nature (environment)
Major causes graphically represent the major bones of a fish, while minor causes represent additional structure in the diagram. Figure 5.3 generally is the starting point for a cause- and-effect analysis and, therefore, may be used as a template to help QEs begin.
Figure 5.4 illustrates a continuation of the example shown in Figure 5.3 for a case where products of a company are damaged after shipping. In Figure 5.4, we see that the effect of interest is “damaged product after shipping.” We also see that the performance improvement team identified major causes of personnel, materials, measurements, methods, and environment but no causes related to machines. In the major cause “Methods,” for example, we see that “wrong pack- ing material” is a causal factor associated with damaged products after ship- ping. The analysis continues until each major cause has been investigated and enough supporting structure has been added to the diagram to identify all the causes associated with the problem or opportunity.
a.3. Check Sheets
The purpose of a check sheet is to summarize, and in some cases graphically depict, a tally count of event occurrences. For example, a check sheet can be used to count the number of defects. In many instances, a check sheet will summarize
Figure 5.3 Cause-and-effect diagram/template.
(Cause) Personnel
Machines (Cause)
Methods (Cause)
(Cause) Measurements
Environment (Cause)
(Cause) Material
Effect
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count data related to certain types of defects and will provide a rough graphical representation of where in a part or process defects occur.
Check sheets can be created using the following steps:
1. Design the check sheet for a given application. A check sheet is a tool designed for a specific application and must, therefore, include any and all information pertinent to the application. In general, the design of a check sheet should include enough administrative data to facilitate referencing and analysis. Administrative data frequently include identification of the product or process, duration of the data collection period, individuals responsible for the product or process, and individuals responsible for the data collection. A check sheet should also include space to record tally data for event occurrences, a rough graphical representation of where in the part, product, or process events occur, and a space to record notes.
2. Record the data. Using the space provided to record tally data, indicate each occurrence of an event with a symbol such as an “x,” check mark, or circle/dot. Each event occurrence receives one mark or symbol. Check sheets also frequently identify, through a rough graphical representation, where in the part or process events occur by highlighting that portion of the rough graphical representation provided.
3. Use the data for analysis or as input to additional graphical tools. Count data summarized on a check sheet frequently are analyzed to identify, track, or monitor defects associated with a particular area on a part or location in a process. The analysis performed on check sheet data is frequently used to trigger process improvement efforts. The data may also be used as input to other graphical tools, such as histograms and Pareto charts, discussed in the next two sections.
Figure 5.4 Cause-and-effect diagram: product damaged after shipping.
Measurements Material
Environment Methods Machines
Box size too big
Box size too small
Product does not meet specs
Material damaged prior to shipping
Wrong packing material
Too many items in package
Components defective
Courier
Packaging department
Shipping department
Too cold
Too hot
Inclement weather
Personnel
Damaged product after shipping
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Figure 5.5 depicts tally data related to specific types of problems found during the inspection of school buses (see Stevenson 2000). Each column for the tally data cor- responds to an occurrence of a problem, as follows:
• Column A: Dirty floors
• Column B: Cracked windows
• Column C: Exterior scratches
• Column D: Worn seats
• Column E: Faulty brakes
Many different types of check sheets can be created. The user should customize the check sheet by including such information as dates, shifts, and so on to allow for ease of interpretation. See Bothe (2001) or Montgomery (2013) for more details.
a.4. Histograms
The purpose of a histogram is to graphically depict the frequency of occurrence of events, where event occurrences are sorted into categories of a defined range along a continuous scale. Histograms are helpful for displaying the distribution of event occurrences among the various columns or categories of event types. Histograms are used when it is important to see and understand how a particular set of data are distributed relative to each other, and possibly to a target or tolerance. The data are recorded in each column or category as they occur, and columns are not sorted by frequency.
To create a histogram, use the following steps:
1. Determine the amount of data to be collected. As a starting point for a histogram, it is necessary to identify approximately how much data will be collected. One data point will be collected for each event occurrence.
Figure 5.5 A simple check sheet. Source: W. Stevenson, “Supercharging Your Pareto Analysis,” ASQ Quality Progress (October 2000): 51–55. Used with permission.
Check sheet
F re
q u e n cy
D E Category
CBA
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2. Determine the number of columns or bins to be used. Numerous guidelines are available for determining the number of bins. For example, one recommendation is that the number of bins be approximately equal to n, where n is the number of data points. Computer software packages use several different algorithms for determining the number of bins, including those based on Scott (1979), Freedman and Diaconis (1981), and variations of Sturges’s rule (Sturges 1926). The width of the bins should be of equal size to avoid graphing misleading results.
3. Collect and record data. As data for a histogram are collected, they are recorded in tabular or tally form.
4. Prepare the graphic. To prepare the histogram for plotting data, it is necessary to provide a descriptive title for the graphic, label each axis, provide a measurement scale for each axis, label the columns, and provide a data summary.
5. Graph the data. Using the data summary, plot the frequency or relative frequency in each column.
Note that the histogram is often considered a large- sample graphical technique and can be unreliable for small sample sizes. Some researchers argue that the histo- gram should not be used for samples with fewer than 75–100 observations (Mont- gomery 2013). For small samples, the histogram can be quite sensitive regarding the number and width of the bins chosen.
As an example, consider the use of high- strength concrete mixtures in road- way and bridge construction. With increased use of these mixtures, quality improvement and quality assurance procedures have become an important aspect of production monitoring. Reducing the use of costly, but necessary, materials while maintaining a high level of quality and meeting required specifications has become increasingly important due in part to the growing demand for materials worldwide. Quality improvement tools will aid suppliers in improving the man- ufacturing process and reducing product variation and unnecessary waste. One important quality characteristic is the compressive strength of concrete, which is directly related to the amount of Portland cement used (in addition to many other variables influencing compressive strength).
Figure 5.6 displays a histogram of compressive strengths for 133 samples col- lected for a particular product from one company. The minimum acceptable com- pressive strength in this case is 3500 psi, which is indicated on the histogram in Figure 5.6. The histogram clearly shows that quite often the strength of concrete delivered can be as much as 1500 psi to 2500 psi higher than the specified mini- mum. The amount of cement that could be saved by reducing the total cement content in the mixture is significant.
a.5. Pareto Charts
The purpose of a Pareto chart is to identify those “vital few” areas that account for the largest frequency or relative frequency in a data set and separate them from the “trivial many.”
A Pareto chart graphically depicts the “80/20 rule” originally postulated by the Italian economist Vilfredo Pareto to explain economic phenomena and later
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adapted by Juran and Gryna (1980) for quality applications. The 80/20 rule allows users to identify and focus on the approximately 20% of factors (i.e., columns or categories) that account for approximately 80% of potential problems.
To create a Pareto chart, follow the steps detailed below:
1. Rank order the columns or categories of data. Columns or categories of data displayed previously as check sheets or histograms are rank ordered from the highest frequency or relative frequency on the left to the lowest frequency or relative frequency on the right.
2. Prepare the graphic. As the data from a check sheet or histogram are rearranged for display in a Pareto chart, the title of the chart is changed, as are the column or category titles when the corresponding data are placed into different column or category locations.
3. Calculate and place a relative frequency line above the data columns or categories. A relative frequency line can be calculated and placed above the data in a Pareto chart for quick assessment of the relative contribution made by each column or category.
Figure 5.7 depicts a notional Pareto chart of problems found in the inspection of school buses based on the scenario presented in Stevenson (2000). Each column for the tally data corresponds to an occurrence of a problem, and the columns or categories have been rank ordered.
a.6. run Charts and Control Charts
While run charts are not specifically identified in the CQE BoK, they are an impor- tant tool for QEs. The purpose of a run chart is to track and monitor the number of event occurrences over time.
Figure 5.6 Histogram of compressive strength of concrete samples, where 3500 psi is the minimum allowed strength.
0
5
10
15
20 3500
3500 4000 4500 5000
Compressive strength (psi)
5500 6000
R e la
ti v e f
re q
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c y (
p e rc
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t)
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QEs use run charts to understand how a parameter or metric is behaving or per- forming over time. The run chart tracks and monitors a metric or parameter without regard to control limits or tolerances. In fact, it is the exclusion of control limits or tolerances that differentiates the run chart from various types of control charts. Con- trol charts are discussed in detail in Chapter 6, section F.
A run chart is constructed using the following steps:
1. Select a parameter or metric of interest. The run chart focuses on only one parameter or metric.
2. Set a scale for the y-axis. Once the parameter or metric has been selected, it will be graphed on the y-axis or vertical axis. A scale must be set for the y-axis that distributes the data throughout the scale.
3. Identify the time intervals for the graphic. Since the run chart displays data over time, the time frame must be meaningful for the application. Time frames such as hourly, each shift, daily, weekly, and monthly are commonly used.
4. Collect and chart the data. Having set up the graphic, collect and chart or plot the parameter or metric over the time intervals specified.
5. Calculate the average. The parameter or metric average is normally calculated for a run chart once sufficient data have been collected. A line indicating the average is plotted directly on the run chart.
Figure 5.8 continues with an extension of the data originally introduced in Fig- ure 5.5, considering the case of defects associated with school buses as originally conceived by Stevenson (2000).
Stevenson originally discussed a set of data identifying 27 defects or deficien- cies associated with a school bus safety inspection. It is reasonable to extend Ste- venson’s analysis by concluding that the inspection occurred at a specific time, say, in September at the start of the school year.
Figure 5.8 shows that 27 defects or deficiencies were recorded in September during a regular inspection of the school bus fleet. Since safety inspections are a
Figure 5.7 Typical Pareto chart.
Worn seats
Dirty floors
Cracked windows
Exterior scratches
Faulty breaks
Broken seatbelts
Brake light out
Dented rearview
mirror
Damage to bus
stop sign
Passenger door jam
0
50
100
150
F re
q u
e n
c y
C u
m u
la ti
v e P
e rc
e n
t
Problem
200
85
47
14 11 7 6 5 2 2 1
100
80
60
40
20
0
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regularly occurring event, it would be reasonable, interesting, and important to track and monitor the results of similar inspections as they occur monthly during the course of a school year. The results of such safety inspections, completed each calendar month, are summarized in Figure 5.8. As would be expected, the number of defects or discrepancies is higher in the beginning of the school year when the buses are used very frequently and decreases substantially later in the year when the buses are not used as frequently. The parameter metric average in this case is 22.67 defects or discrepancies per month.
a.7. Scatter diagrams
The purpose of a scatter diagram is to graphically display indications of a relation- ship between two quantitative variables. A QE interested in how a variable may perform or behave relative to another variable may use a scatter diagram to analyze quantitative data. The relationship being investigated is called a correlation, and Fig- ure 5.9 identifies three possible relationships as positive correlation, no correlation, and negative correlation. Correlation is discussed in detail in Chapter 6, section E.
To create a scatter diagram, use the following steps:
1. Select two variables of interest. The scatter diagram focuses on possible correlations between two variables. The two variables of interest should have the potential for a cause- and-effect relationship.
2. Set a scale for the axes. Since one variable will be plotted on the x-axis while the other variable is plotted on the y-axis, a scale must be selected for each axis such that the data use all, or nearly all, of the scale.
3. Collect and chart the data. Having set up the graphic, collect and chart or plot the data in accordance with the scale specified.
4. Evaluate the results. Using Figure 5.9, evaluate the results to identify any relationships.
Table 5.1 provides the data for Figure 5.10. The data provided are derived from a training analysis in which the number of defects produced by employees is
Figure 5.8 Run chart example.
0 Sept Oct DecNov Jan Feb AprMar May June July Aug
Average = 22.67
5
10
15
20
25
30
35
N u
m b
e r
o f
d is
c re
p a n
c ie
s 40
45
27
38 35
21
32
28
23 25
17
13
8
5
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Figure 5.9 Three possible relationships identified by scatter diagrams.
y
y
y
x
x
x
0
1. Positive correlation. An increase in y may depend on an increase in x.
0
2. No correlation. There is no demonstrated connection between x and y.
0
3. Negative correlation. A decrease in y may depend on an increase in x.
Table 5.1 Training data.
Training Hours versus Number of Defects
Observation Training
hours Defects Observation Training
hours Defects
1 1.00 33 10 3.25 23
2 1.25 33 11 3.50 20
3 1.50 32 12 3.75 17
4 1.75 31 13 4.00 14
5 2.00 30 14 4.25 12
6 2.25 28 15 4.50 9
7 2.50 27 16 4.75 8
8 2.75 27 17 5.00 8
9 3.00 25 18 5.25 7
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compared with the number of hours of employee training. The x-axis (represent- ing training hours) documents how many hours employees spent in training. The y-axis (representing defects) documents tally or count data of the number of defects produced by employees who received the training.
B. QuaLiTy ManagEMEnT and PLanning TooLS The concept of quality has existed for as long as people have existed. Qualities, defined as physical or nonphysical characteristics that constitute the basic nature of things, are readily accepted as part of the package that encompasses a good or a service. Shewhart (1980) captured the concept in the first part of the twentieth century: There are two common aspects of quality, one of these has to do with the consideration of the quality of a thing as an objective reality independent of the existence of man. The other has to do with what we think, feel, or sense as a result of the objective reality. This subjective side of quality is closely linked to value.
Shewhart and others such as Deming (1986), Juran (1989), Crosby (1979), Fei- genbaum (2004), Ishikawa (1985), Shingo (1986), and Taguchi (1986) have helped us understand the essence of quality and helped us bring it to the point of action- able issues. There have been, and continue to be, a number of approaches and ini- tiatives that advocate quality as a scientific discipline. But, on the other hand, there are also many anecdotal approaches and initiatives that treat quality as an art.
Interest in tactical, in- process approaches that stress the importance of meet- ing substitute quality characteristics (as opposed to strategic approaches that stress true quality characteristics) helped to move the quality concept upstream
Figure 5.10 Training time versus defects.
0 0.00 1.00 3.002.00 4.00 5.00 6.00
5
10
15
20
25
30
35
D e fe
c ts
Training time, hours
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from final product inspection. This evolutionary branch was eventually called kai- zen or incremental improvement (a management- by-fact related approach) and was applied primarily in production- related processes (Imai 1986). Here, evolv- ing practices were observed and eventually tools were identified, described, and adopted. Tools such as the seven basic tools—cause-and-effect diagram, flowchart, check sheet, histogram, scatter diagram, Pareto analysis, and control chart—were recognized as useful.
The Japanese further expanded the quality concept in a formal sense in the late 1970s and early 1980s with what they termed the seven “new” quality tools (Mizuno 1988). This new era was based on two fundamental requirements: (1) the creation of added value over and above consumer needs, and (2) the prevention, rather than the rectification, of failure in meeting customer needs. Hence, these tools were positioned to address strategic (as opposed to tactical) quality issues. Seven tools—relations diagram, affinity diagram, systematic diagram, matrix dia- gram, matrix data analysis, process decision program chart (PDPC), and arrow diagram—were the result of this initiative.
In the 1990s, based on Shewhart’s definition of quality, field experience/ observation, and Ishikawa’s (1985) concepts of true and substitute quality char- acteristics, Kolarik (1995) postulated a scientific framework. This framework has two major components: the experience of quality and the creation of quality. The experience of quality is a function of the fulfillment of human needs and expectations. We create quality through processes that we develop and maintain (Kolarik 1995).
The following sections describe the CQE BoK management and planning tools plus several other useful tools that help us create quality. These sections describe, position, and illustrate a selected cross section of 11 quality- related tools. The eight BoK tools are marked with an asterisk (*). The four remaining tools—process maps, process value chain diagrams, SIPOC diagrams, and benchmarking—are included to provide extended quality management/planning capabilities.
Affinity diagrams*
Force field analysis*
Matrix diagrams*
Interrelationship digraphs*
Tree diagrams* and prioritization matrices*
Process decision program charts*
Activity network diagrams*
Process maps and SIPOC diagrams
Process value chain diagrams
Benchmarking
These tools help to formulate and organize thoughts and ideas so that they can be leveraged directly toward quality/business improvement. More elaborate discus- sions of quality strategies, initiatives, and tools appear in Kolarik (1999).
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B.1. affinity diagrams
The purpose of an affinity diagram is to help people collect, organize, summa- rize, and communicate facts, opinions, and ideas. The affinity diagram is useful when we are faced with describing, organizing, and communicating the general nature of a relatively complicated situation that can be described in terms of a large number of facts, opinions, and/or ideas. It allows us to group or cluster the facts, opinions, and/or ideas into categories with some common feature so that we can locate, classify, describe, or summarize the basic issues. The affinity principle (of association and clustering) is useful in the initial stages of constructing a rela- tionship diagram or in any situation where we desire to discover, summarize, and organize a variety of facts, opinions, and/or ideas.
To create an affinity diagram:
1. Identify a general theme. The theme may be associated with a problem situation or an opportunity situation, or simply a situation in our physical and/or social environments.
2. Collect facts, opinions, and ideas. Data/information may be generated by a group of people in any number of formats; for example, we can use work teams, focus groups, or groups of experts. We can also use data/ information existing in files or archives.
3. Express and enter the data/information in a common format. Here, we might use sticky notes on a wall, cards on a table, or computer software capable of expressing each piece of data/information in a medium that can be “moved around.”
4. Identify the groups/clusters. Here, we identify, label, and describe the groups or clusters regarding the common attributes or summary characteristics that apply.
5. Cluster the data/information pieces. At this point we cluster or organize our data/information into cohesive groups.
6. Repeat steps 4 and 5 to form supergroups/superclusters. It may be possible to relate two or more of the initial groups/clusters and develop a supergroup or supercluster. Supergrouping can be repeated until the facts, opinions, or ideas are suitably classified/organized.
7. Present the results. The final product is an organized set of facts, opinions, and ideas that make sense in terms of providing help in understanding the nature of the situation or theme from step 1.
Figure 5.11 depicts the results of a student focus group session. The goal of the focus group was to communicate issues that are important to undergraduate stu- dents in their college program. Here we can see that a number of concerns were voiced, in no particular order, and that the affinity principle was used to sort, organize, and isolate/label relevant issues for further action. For more infor- mation on affinity diagrams see Mizuno (1988), Brassard (1989), and Kolarik (1995, 1999).
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B.2. Force Field analysis
While affinity diagrams are useful for organizing ideas into relationships, the pur- pose of force field analysis is to take these ideas a step further by identifying driv- ing and opposing forces associated with a desired change in an organization. A force field analysis is created by a group environment and is done by completing the following steps:
1. Identify a desired change. Ideally, if this change is accomplished, the results would provide positive outcomes within an organization. The driving force should be written at the top of a document, and a vertical line should be drawn under it.
2. Brainstorm forces for the change.
a. Driving forces should be identified and written below the desired change on the left side of the vertical line.
b. Opposing forces should be identified and written below the desired change on the right side of the vertical line.
3. Estimate the strength of the forces identified. For all driving and opposing forces, the team should decide on the strength of the
Figure 5.11 Student focus group affinity diagram.
Computer facilities and support issue
Curriculum design issue
Student–faculty communication issue
Student involvement issue
Flexibility in of�ce hours/coordinated
with TA hours issue
Early student awareness issue
Theme: Undergraduate program improvement issues Source of comments: Undergradute focus group
Programmatic
Sr. design requires advanced level knowledge from last semester classes
Students want more electives and choices
Students want more computer languages
Interaction
Students are sensitive to harsh criticism
Students feel intimidated by some instructors
Some instructors are difficult to approach
Some instructors treat students as if they were stupid
Recognition
Students are not recognized for good work
Courses and feedback
Students do not get continuous feedback from courses
Students want to know how they are doing in class
Students want level point loading in a course throughout the semester
Computing support
Students need software that works (info software)
Students need computing support at all hours (lack of computers in dept)
Students need unlimited printing privileges
Students don't like to hunt down which lab contains which software
Involvement
Freshmen and sophomores want to get to know upper class persons
Students want to be informed of activities and events
Students want to interact with other students
Students want to get involved with organizations
Pre-professional practice
Students are not aware of the value of internships/co-ops
Students have difficulty understanding what is expected in professional practice
Instructor availability
Instructors are out of town often
Lack of availability of instructors/TA
TA/faculty hours/schedules create problems for interaction
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associated force. Arrows can be drawn next to the force pointing toward the vertical line. The strength of the force can be represented by the length or size of the arrow. While this can be quite subjective, it will provide a quick visual reference for identifying the strongest forces discouraging and encouraging the change.
4. Discuss methods for reducing opposing forces. The opposing forces hinder a desired change. If there are methods for reducing the impact or eliminating the forces, they should be identified and targeted.
5. Discuss methods for enhancing/encouraging supporting forces. Supporting forces that help create positive change should be identified and encouraged.
Force field analysis highlights weaknesses and strengths and is useful in deter- mining whether a change is feasible. Force field analysis is sometimes used after the construction of a fishbone diagram, because each cause identified can be related to a desired change (Tague 2005). In the affinity diagram in Figure 5.11, computing support is identified as a computer facilities and support issue. Improved computing support is a desired change of the undergraduate pro- gram. Figure 5.12 illustrates a simplified example result of a force field analysis for this change.
B.3. Matrix diagrams
The purpose of a matrix diagram is to help people discover, visualize, and com- municate relationships within a single set of factors or between two or more sets of factors.
A matrix diagram is typically used to display relationships between two sets of characteristics or factors. However, it can be used to display interrelationships within one set of characteristics or factors. The typical layout is a two- dimensional matrix with the vertical dimension used to lay out one set of factors and the hori- zontal dimension used to lay out the other set. In the case of displaying inter- relationships within one set of factors, the same factors are laid out in both the
Figure 5.12 Force field analysis for computer support.
Driving Forces
Improved Computing Support
Opposing Forces
Computing tech support resources are expensiveSoftware that works is
critical to education
Students and instructors need access to printing
Fast, reliable responses from computing tech improve
education
Poorly trained tech support
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horizontal and vertical dimensions. We typically identify and document relation- ships within each set and between the two sets at intersection points in our graphic.
The concept of a matrix diagram is relatively simple; essentially, we develop it to help us relate sets of factors or characteristics, usually in a qualitative fashion. The actual development of a matrix diagram, however, is rather involved in terms of defining the level of detail, completeness, and association. Quantification and prioritization are addressed in section B.5.
In quality- related work, a primary application of the matrix diagram is to relate customer needs, demands, and expectations in the customer’s language to technical characteristics of the product/process expressed in the producer’s lan- guage. Figure 5.13 illustrates this particular application of a matrix diagram. This illustration contains interrelationships in the triangular appendages at the left side and the top of the matrix. Here, we use “+” and “–” symbols to represent positive and negative relationships, respectively. We use the bull’s-eye, open circle, and triangle symbols to represent, respectively, very strong, strong, and weak relation- ships between characteristics of the two sets. In this particular matrix diagram, we have included customer needs, demands, and expectations; technical definition characteristics; and competitor characteristics together.
Matrix diagrams differ in scope and detail, as well as in layout format. See Mizuno (1988), Akao (1990), Day (1993), and Kolarik (1995, 1999) for details regard- ing the matrix diagram in general and specific QFD applications in particular.
Figure 5.13 provides a simplified matrix diagram regarding a laundry ser- vice. Ignoring the quantification numbers in the matrix for now, we see customer demands on the left and technical quality characteristics on the top. Two interrela- tionship matrices appear at the left and top. Customer degrees of importance and laundry sales points appear in vertical columns. Here, critical laundry sales points include clean clothes, good- looking clothes, friendly service, and return of pocket contents. This type of matrix diagram is commonly found in QFD work.
B.4. interrelationship digraphs
The purpose of an interrelationship digraph is to help people discover, visual- ize, and communicate high- level sequential and/or cause- and-effect relationships. Constructing an interrelationship digraph is best addressed in a team environ- ment, so as to capture a diversity of perspectives regarding sequences, effects, and causes. Typical starting points include effects or symptoms, both undesirable and desirable. Logical development from these effects back to potential causes is common to most relations diagramming efforts. Clustering and sequencing of causes are common to all interrelationship digraphs. Boxes, circles, ovals, loops, and directional arrows are used to depict cause- to-effect flows.
In general, the interrelationship digraph helps us to identify and isolate rel- evant causal factors concerning a situation, a problem, or an opportunity. Ulti- mately, it helps us understand and communicate the essence of causal or sequential relationships regarding a situation in our physical and/or social environments. It is a graphical aid in cause- and-effect discovery as well as relationship determi- nation and expression. Interrelationship digraphs express basic causal sequences, introduce assertions, assess or project resulting effects, and communicate critical
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Figure 5.13 Quality function deployment matrix diagram example. Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 150.
Laundry service
Quality characteristics (technical language)
Brightness 73
*
38 73
*
38 64
*
56 56 97
*
27 35 76
*
33 18 39 61
*
33
Sm ell
Spot rem oval
Press
Search pockets
Buttons
Alterations
Cleaning cycle tim e
H om
e pickup/delivery
Custom er greetings
Custom er relations
Building location
Building access
Business hours
Tim e in line
Paym ent m
ethods
Sales points— ours
Com petitor A
Com petitor B
Cleaning and cust. ser. edge
Low cost edge
Location edge
R elative w
eights— ours
Degree of im portance to custom
er
Demanded quality (customer language)
Clean clothes 7
7
5
7
3
3
5
7
3
3
35 35 35 35
25
25
9 9 99 15
15
21
25
35
25
15 15
15
21
15
15
15 15 15
15 15
35 35
35 35 35 21
15 15
7
5
33
3
3333 3 3
5
7 7Good looking clothes
Fast service
Friendly service
Convenience
Handy location
Fix clothes
Return pocket contents
Inexpensive
Easy to pay
Priority scores
* Priority quality characteristics
++ + +––
––––– +
– –
+
+
+
+
+
++
– –
–
– –
–
– + +
+ ++–
+– +
+ +
+– +–
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relationships. The interrelationship digraph is one form of relationship diagram (see Brassard [1989] for details).
The mechanics of constructing an interrelationship digraph generally follow the same lines as in the affinity diagram, but extend the affinity diagramming pro- cess into cause–effect and/or sequential ordering, generally indicated by arrows that connect the “boxes” or statements.
Steps in constructing an interrelationship digraph are as follows:
1. Identify a general situation. The situation may be associated with a problem or an opportunity in our physical, economic, and/or social environments.
2. Collect facts, opinions, and ideas. Data/information may be generated from a group of people in any number of formats; for example, we can use work teams, focus groups, or groups of experts.
3. Express and enter the data/information in a common format. Here, we might use sticky notes on a wall, cards on a table, or computer software capable of expressing each piece of data/information in a medium that can be “moved around.”
4. Identify the groups/clusters. Here, we identify, label, and describe the groups or clusters regarding the common attribute(s) or summary characteristics that apply and describe their relationship (as a group) to the situation at hand.
5. Cluster the data/information pieces. At this point, we cluster or organize our data/information into cohesive groups.
6. Identify relations/sequences. Once we have basic descriptions and clusters/groups, we express the relationships between these entities with arrows.
7. Repeat steps 4, 5, and 6 to form supergroups/superclusters. It may be possible to relate two or more of the initial groups/clusters and develop a supergroup or supercluster. Supergrouping can be repeated until the facts, opinions, and ideas are suitably classified/organized. The result here is a supergroup and its description/relationship to the situation.
8. Present the results. The final product is an organized set of facts, opinions, and ideas that make sense in terms of providing help in understanding the nature of the situation from step 1, and summarizing the situation in a problem or opportunity format that flows logically from the facts and figures.
The illustration in Figure 5.14 provides a relatively simple interrelationship digraph where the situational descriptions are grouped and labeled using the affinity principle. Arrows are then used to indicate convergence toward a logi- cal, actionable conclusion. For more information on interrelationship digraphs, see Mizuno (1988), Brassard (1989), Kolarik (1995, 1999), and Tague (2005).
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B.5. Tree diagrams and Prioritization Matrices
The purpose of a tree diagram is to help people discover, visualize, and commu- nicate logical hierarchical relationships between critical events or goals/objectives and means. Tree diagrams are useful in situations where we want to define a hier- archical relationship between events, both desirable and undesirable. A fault tree (FT) can be constructed to relate an undesirable “top event” or failure to a sequence of events that led to the top event. In other words, the FT depicts logical pathways from sets of basic causal events to a single undesirable result or top event. We typi- cally use logical operators, such as AND or OR gates, to connect lower- level events with higher events. Hence, once the logic has been described, quantification can take place and risk level can be assessed.
Several steps are involved in the development of the FT:
1. Identify the top event. The top event is an undesirable event that we are motivated to prevent.
2. Identify the next- level events. The second- level events represent events that could lead to the top event.
3. Develop logical relationships between the top and next- level events. Here we use logical operators, for example, AND or OR gates, to connect the second- level events to the top event.
Figure 5.14 Line support subprocess interrelationship digraph. Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 450.
Some vendors furnish raw materials to
High Lift High Lift is a small
account to some vendors
Some vendors furnish finished assemblies
to High Lift
Some vendors furnish generic parts/supplies
to High Lift
Everything coming through receiving goes to storage and back out to the line eventually
High Lift uses centralized storage facilities
High Lift does not certify vendors
Some vendors have certificates with other customers
Vendors are not aware of line shortages at High Lift
New line flow is faster than expected by master scheduling
About 10% of "on-time" orders to the line are actually late
All received items are palletized for storage
All palletized items are shrink-wrapped before storage
High Lift is a large account to
some vendors About 35% of late deliveries to the line were also late
deliveries to receiving
About 53% of inbound deliveries to receiving are delivered on time
About 80% of line shortages were also late deliveries
from vendors Hot/shortage items
go directly to the line from receiving
Line shortages lead to lost schedule time and
additional costs
Obsolete/damaged/deteriorated materials/parts are purged from storage semiannually—the total
runs about 5% of total purchases
About 69% of line orders are delivered on or before
scheduled date
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4. Identify and link lower- level events. Now, we develop the logic tree down to the lowest level desired by repeating steps 2 and 3, moving down through event sequences one level at a time.
5. Quantify the FT (optional). Here we develop probability of occurrence estimates for the events in the FT and then develop a probability statement and estimate for the top event.
Figure 5.15 presents an FT focused on unintended line shutdowns. This illustra- tion contains OR gates that connect lower- level events with higher- level events.
An FT does not contain all possible failure modes or all possible fault events that could cause system failure. However, an FT is capable of considering or mod- eling human error, hardware and software failures, and acts of nature. It finds widespread use in the fields of reliability, safety, and risk analysis. The FT is a more focused tool than the FMEA. FMEA is sometimes used to help determine the top event in an FT. FTs work well for independent events; common cause is difficult to model, especially in terms of quantification.
Other tree diagram formats include event trees, systematic diagrams, and goal trees, as well as concept fans. Event trees are simply tree diagrams that start with an event and work backward from the event by defining binomial response (yes or no) branches. The response branches form a hierarchy of responses that eventually
Figure 5.15 Simplified line shutdown fault tree. Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 469.
Unintended line shutdown
Part/supply shortage online
Part/supply failure at line
Line equipment failure
Production error/defect
(Not developed) (Not developed)
(Not developed)
Part/supply out of stock
Vendor late delivery
Vendor out of stock
Part/supply order failure
Order issue— failure to outside
Error in schedule
Lost passback card
Error in communication
Late order
Line oversight
Wrong location Racked/binned
improperly
Order passback— failure within plant
Rack/bin unavailable
Wrong part/supply
Part/supply stocking failure
Defective parts/supplies
A B
C
Certified vendor
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lead to an outcome. A systematic diagram depicts a sequence of goals or objectives and their respective means chained together so that we can visualize our possible alternatives with respect to the accomplishment of the high- level goal/objective. The goal tree is very similar to the systematic diagram in that it is built around a high- level goal that we want to accomplish. It is also similar to the FT in that it links lower- level subgoals, functions, and success trees together with logic sym- bols or gates that lead up to the top goal.
A concept fan is built in a tree format but differs from the other formats sig- nificantly. The concept fan is a creativity- based tool, where we start with a purpose or functional requirement in a generic sense and expand it backward to provide alternative concepts that can accomplish the purpose or functional requirement. It is simple to construct and allows us to visualize possibilities for accomplishing our purpose early in the creative process.
A partial goal tree is illustrated in Figure 5.16. This tree structure uses AND gates to connect goals, subgoals, and functions. Success trees are then hooked into the functions using OR gates. The essence of the goal tree is to support strategic and tactical planning by depicting paths of goal accomplishment. For more infor- mation on treelike diagrams, see Mizuno (1988), Brassard (1989), Kolarik (1995, 1999), and Tague (2005).
Once we develop or identify relationships and options or alternatives that might be developed through a relationship matrix, a relations diagram, a tree dia- gram, or some other means, we typically move into decision mode. The purpose of a prioritization matrix is to help people measure/evaluate relationships from a matrix or tree analysis relative to a weighting scheme and decision criteria in order to set implementation priorities for the decisions at hand. The prioritization matrix allows us to make relative comparisons and present our information in an organized manner so that we can support our decisions with consistent, objective, quantitative evaluation.
Prioritization typically requires two things: (1) decision criteria and (2) a means of structuring relative comparisons. Decision criteria stem from our perception of what is important. For example, economics, timeliness, physical performance, and customer service form basic categories from which to develop decision criteria. Once developed, these criteria must be assessed as to their importance within the judgment of each decision maker and collectively between decision makers. This assessment may be carried out subjectively or objectively.
In the subjective case, we as individuals draw on our past experiences and perceptions of the future and collectively use some sort of consensus/voting/ ranking-based process. Various types of rating/voting schemes include the Delphi method and the nominal group technique. The matrix diagram illustration in Fig- ure 5.13 contains two sets of rankings, one for the strength of the relationships in the body and one for the relative importance of our sales points. Together, these two sets allow us to quantify the body of the matrix and develop quality charac- teristic scores. Hence, we can prioritize our thinking/action in terms of the more critical quality characteristics. In this case, criticality is indicated by the asterisk symbol (*), and a total score of 60 (selected subjectively) was used as the criterion for selection.
In the objective case, we assign relative weighting values and quantita- tively manipulate these values to converge on a relative priority number. Several
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Figure 5.16 Partial manufacturing cost improvement goal tree. Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 269.
Reduce manufacturing costs, product XX
Improve manufacturing practices
Improve design for manufacturing
Improve system design
Improve material handling
Milling center
Maintenance Setup Run TQM tools Product training Process training Experimental
program training
Improve fabrication
Improve assembly
Improve packaging
Improve parameter design
Improve tolerance design
Improve production planning
Improve processes Improve training
and knowledge Improve
incentives
Improve technical knowledge
Improve TQM training
Improve employee effectiveness and efficiency
Goal
Subgoals
Functions
Success trees
Speed/feed
Cutting fluid
Tooling
Fixtures
Load/unload
Tooling
Fixtures
Machines
SPC
Cause/effect diagram
Scatter diagram
Process flowchart
Basic statistics
Experiment design
Robust design
Materials
Product specs
Assembly technicians
Control chart
Machines
Tools
Process methods
Process specs
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techniques exist for objectively establishing prioritization criteria. The analytical hierarchy process (AHP) is widely described as a quantitative technique (Saaty 1982). The AHP allows a number of decision makers to integrate their priorities into a priority matrix where the decision criteria are compared as to relative impor- tance in a pairwise fashion. The results include a decision criteria priority matrix and a corresponding alternative priority- weighted matrix. Hence, a quantitative group consensus analysis matrix emerges. From this analysis, the alternatives can be selected with the confidence that all criteria—economic, technological, and intangible factors—are integrated into the decision process. For more information on prioritization matrices, see Brassard (1989).
B.6. Process decision Program Charts
The purpose of a process decision program chart (PDPC) is to help people orga- nize and evaluate process- related events and contingencies with respect to imple- mentation and/or early operations. The PDPC is useful in helping us proactively evaluate or assess process implementation at a high level, early in the planning stage, or in the initial start- up phases of process operations. We may use the PDPC to argue or work our way through implementation, including events that might occur and possibly disrupt our process and/or its implementation. Or we might use the PDPC to guide early operations in case of deviations from plan. The key use of the PDPC is to help us both anticipate deviations from expected events and provide effective contingencies for these deviations.
The PDPC can take several general formats. One format resembles an annotated tree diagram. Another format resembles an annotated process flowchart. In either case, the distinguishing mark of a PDPC is its ability to offer the user an overview of possible contingencies regarding process implementation and/or operations.
Although the PDPC can take several general formats, several steps are com- mon to all formats:
1. Identify the purpose of the process. Understanding the purpose of the process is critical to building and using the PDPC. This purpose will guide development of the PDPC from the standpoint of possible contingencies and their resulting impacts relative to the desired outcome.
2. Identify the basic activities and related events associated with the process. Here, we use a tree or process flow format to place the activities in the expected sequence. This step should present a graphical depiction of activities that are part of the plan to be implemented and/or the basic operation.
3. Annotate the basic activities and related events. Working from step 2, we provide summarized descriptions of activities and events relative to what we normally expect to happen.
4. Superimpose the possible (conceivable) deviations. At this point, we add branches/events that represent identified deviations from the expected activities/events.
5. Annotate the possible deviations. We provide summarized descriptions relative to the deviations that have been mapped onto our chart in step 4.
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6. Identify and annotate contingency activities. This step provides a description of the contingencies that we identify in order to avoid or counter the mapped deviations.
7. Weight the possible contingencies. At this final step, we examine the PDPC as a whole, consider the purpose, and select/mark the most appropriate contingencies. At this point we have a contingency plan complete with our priorities for avoiding and/or dealing with possible deviations from our original implementation and/or operational plan.
A receiving/storage/stocking subprocesses PDPC is depicted in Figure 5.17. This depiction provides a basic look at the existing process, with several deviations indi- cated: damage, shortage, salvage, expedition, and line delay. It provides a number of facts and figures. Contingency- related issues are discussed in Table 5.2 relative
Figure 5.17 Receiving/storage/stocking subprocesses PDPC. Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 446.
Inventory/ receiving
entry
Excess back to storage
Hot order
Expedited delivery
Unloading
From procurement/ outside vendors
To procurement
To fabrication To paint To assembly To product repair
To fabrication To paint To assembly To product repair
From production scheduling
To procurement
Receiving subprocess
Receiving/stocking team
Storage subprocess
Stocking subprocess
Unpacking/ inspection
Storage entry Storage
Storage retrieval
Order assembly
Order delivery/ tracking
Line order
Inventory update
Line delay?
Shortage?
Expedite?
Salvage?
Expedite?
Damage?
Shortage?
Supply-side performance metrics
Inbound deliveries: 137 Rush/airfreight deliveries: 43 Damaged shipments: 3 Short shipments: 5 On-time delivery: 53%
Primary suppliers
Procurement/ outside vendors
Production scheduling
Supply-side demands (summarized)
Right stuff: materials, assemblies, parts, supplies—meet all technical specifications, right amount, on time
Assembly, part, supply orders: right item, right amount, right lead time, right due date to line
Primary customers
Fabrication, paint, assembly, repair
Procurement
Customer-side outcomes (summarized)
Right stuff: materials, assemblies, parts, supplies—meet technical need, right amount, on time, in right place
Receiving report: right purchase order number, accurate count, condition, timely
Customer-side performance metrics
Outbound deliveries: 324 Expedited orders: 112 Line shortages: 63 Line delays (for lack of materials, assemblies, parts, supplies): 17 On-time delivery to line: 69%
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Table 5.2 Issues, possible root causes, and general impact summary for receiving/storage/ stocking PDPC.
Issue Possible root causes General impact
1. Our vendors are not aware of, or responsive to, our line shortage problems—why?
We do not communicate as well as we should with our vendors.
Our vendors do not see High Lift as a large account.
Our vendors are not capable of providing better service to us under current conditions.
More prompt deliveries from our vendors could decrease our airfreight costs and reduce our line shortages, speeding up/smoothing out our assembly subprocess.
Estimated savings potential: $1.1 million per year.
2. Line flow is faster than the master schedule algorithm reflects in order issuance—why?
Our manufacturing time estimates that drive several parts of our master scheduling system were made using time estimates from our former/ pre-redefinition product/ production processes.
Our redefined product/ production processes flow better than we anticipated/ estimated—provided materials, assemblies, parts, and supplies are readily available.
A lack of current reality of our present redefined processes within our scheduling algorithm is holding our production process back from realizing its full potential.
Present mismatches are putting brakes on potential assembly improvement on the lines.
Estimated savings potential: $1 million to $10 million per year.
3. Everything that enters receiving goes through the storage area, with the exception of “hot” items that are needed to resolve a line shortage—why?
High Lift supplies centralized storage/inventory system solutions to its customers. This concept is a part of High Lift culture and reflected in current operations.
Centralized storage for all items is questionable.
Centralized storage capital as well as operational costs are running about $1.2 million per year.
Material, assembly, part, and supply obsolescence costs are running at about 5% of purchased part costs or about $2 million dollars per year.
Potential customers are brought in to observe the technical operations of the High Lift storage system. This demonstration is viewed as a decisive element in customers choosing High Lift. Such observation is involved with about 45% of system sales.
Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 451.
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to possible root causes and impact. In this case, general contingencies are process improvement, process redefinition, or the status quo subprocess. For more infor- mation on PDPCs, see Mizuno (1988), Brassard (1989), and Kolarik (1995, 1999).
B.7. activity network diagrams
The purpose of activity network diagramming is to help people sequentially define, organize, and manage a complex set of activities and events with respect to time schedule planning and implementation.
The Japanese scheduling/planning tool known as an arrow diagram is a hybrid derived from Gantt chart technology and a simplified extraction from the PERT and CPM technologies (see Kolarik [1995] and Mizuno [1988]). An arrow diagram is a network planning method that displays activities on the “arrows” as opposed to on the “nodes.” An activity network diagram is a derivation of PERT, CPM, and the arrow diagram (Brassard 1989). We will describe a useful simplified version of CPM with activities on the nodes.
Complex processes are typically made up of a number of activities that must be carried out in a defined sequence in order to accomplish the desired result. We use an activity/sequence list to identify and organize a set of activities as to sequence and estimated duration.
In general, each activity involved with an endeavor will be sequential, paral- lel, or coupled with other activities. Sequential activities require that a predeces- sor activity be completed before its successor can begin. Parallel activities can be undertaken and executed simultaneously. Coupled activities are executed together and hence their progression is linked together in some manner. The activity/sequence list addresses these relationships. First, each activity on the list is uniquely identi- fied. Then, the sequence as to predecessor and successor activities is established. Finally, we estimate duration for each activity. An example activity/sequence list appears in Table 5.3.
From the activity/sequence list, we construct CPM- like networks of our planned activities, allowing us to organize and display a schedule of project activities/ events with regard to starting and finishing time estimates, both as a whole project and as individual activities.
In order to develop a CPM network for a project, we first identify activities and events. An activity is something that requires action of some type, such as shingling a roof. An event happens at a specific time, for example, the beginning or ending point of an activity. Our critical events represent milestones, points at which we reassess our progress. The activity/sequence list is a helpful tool to sum- marize activities, sequences, and time estimates.
A CPM- like network diagram is depicted in Figure 5.18. Here, we have taken the symbol, sequence, and duration information from our activity/sequence list in Table 5.3. The network flows from left to right in a time sequence. Each activity is represented by a node, that is, a circle. Within each circle, we list the activity’s symbol and its estimated time duration. Other information developed includes earliest start time (ES), earliest completion time (EC), latest start time (LS), and latest completion time (LC). These estimates are provided for each node/activity on the network.
The critical path is defined as the path that determines the minimum comple- tion time for the entire project. Boldface arrows usually depict the critical path. If
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Table 5.3 Line support improvement process activities, sequences, and durations.
Activity description Activity symbol Predecessor
Duration, days
Explain change/plan to affected areas Identify rackable/binnable items Design racks/bins/storage facility modifications Build/test racks/bins* Identify/inform affected vendors
A B C D* E
A B C A
2 7
21 14 4
Prepare High Lift and vendor training/certification materials
Gain vendors’ cooperation Certify/train vendors* Review/modify High Lift team needs Train High Lift people*
F
G H* I J*
A
E G, F F I
15
8 10 2 5
Modify staging facilities Modify in/out facilities Develop procurement scheduling/card system* Rack/bin existing bulk inventory*
K L M* N*
C C F D, K, L
12 14 25 14
Stage racks/bins to line Remove/salvage old storage area Limited-scale operation, test/tune/mistakeproof* Full-scale operations
O P Q* R
N O H, J, M, O Q
5 20 30 –
Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 474. *Indicates milestone activities; milestone occurs at the end of the marked activity.
a delay occurs on any activity on the critical path, then the project duration will be increased. Hence, we watch the activities on the critical path very carefully with respect to time duration violations.
The ES and EC estimates are developed on a forward pass through the net- work of activities and durations. We develop the network using a start event and a finish event. We usually start at time zero and finish at the shortest time possible, considering our time/duration estimates. On the forward pass, we begin at the start node and develop our ESj estimates for each node. Usually, we assume the ESStart node is equal to zero. However, we could assume some posi- tive value. Then, we develop ESj estimates for each activity as we move from left to right (across time) through the network. Each ESj is equal to the maximum of the ECi estimates taken from the set of all immediate predecessor activities. Each ESj is estimated by summing its ESj and its duration, tj. The ECFinish node is equal to the maximum of the ECi estimates taken from the set of all immediate predecessor activities.
The LC and LS estimates are developed on a backward pass through the net- work of activities. Starting at the finish node, we set the LCFinish node equal to the ECFinish node. We set the LSFinish node equal to the LCFinish node. We estimate LCj as the minimum of the LSi estimates taken from the set of all immediate successor activities. Each LSj is equal to its LCj minus its activity duration, tj.
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Tables such as Table 5.4 are constructed in order to both facilitate our network development and summarize our results. We usually repeat our activity descrip- tions, symbols, and durations. We list our ES, EC, LS, and LC estimates, which match those in our CPM network. Additionally, we include estimates of total slack (TS) and free slack (FS). In the CPM network method, we define TS as the amount of time activity j may be delayed from its earliest starting time without delaying the latest completion time of the project.
TSj = LCj – ECj = LSj – ESj (5.1)
Whenever the TSj equals zero, we have a critical path activity. FS is defined as the amount of time activity j may be delayed from its earliest
starting time without delaying the starting time of any of its immediate successor activities.
FSj = Min {(ESi = 1 – ECj), (ESi = 2 – ECj), . . . , (5.2) (ESi = Last successor activity – ECj)}
where i corresponds to the index for all successor activities, i = 1, 2, . . . , last suc- cessor for activity j.
We can use updated CPM graphics and tables to update our plan as activi- ties are completed. Additionally, we can project changes in subsequent activity
Figure 5.18 Simplified CPM schedule network–line support improvement implementation. Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 475.
Start
0 0
0 0
Finish 93
93
93 93 A 2
0 2
0 2
B 7
2 9
2 9
E 4
2 6
41 45
F 15
2 17
23 38
G 8
6 14
45 53
H 10
17 27
53 63
I 2
17 19
56 58
M 25
17 42
38 63
J 5
19 24
58 63
L 14
30 44
30 44
C 21
9 30
9 30
A t
ES EC A: Activity
t : Activity duration (days)
ES: Earliest starting time
EC: Earliest completion time
LS: Latest starting time
LC: Latest completion time
Critical path
Milestone activity
LS LC
K 12
30 42
32 44
D 14
30 44
30 44
N 14
44 58
44 58
O 5
58 63
58 63
63 83
73 93
63 93
63 93 93 93
93 93
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Table 5.4 Line support improvement scheduling details.
Activity description Activity symbol
Duration, days ES EC LS LC TS FS Critical?
Explain change/plan to affected areas
A 2 0 2 0 2 0 0 Yes
Identify rackable/binnable items
B 7 2 9 2 9 0 0 Yes
Design racks/bins/storage facility modifications
C 21 9 30 9 30 0 0 Yes
Build/test racks/bins* D* 14 30 44 30 44 0 0 Yes
Identify/inform affected vendors
E 4 2 6 41 45 39 0 No
Prepare High Lift and vendor training/certification materials
F 15 2 17 21 36 19 0 No
Gain vendors’ cooperation G 8 6 14 45 53 39 3 No
Certify/train vendors* H* 10 17 27 53 63 36 36 No
Review/modify High Lift team needs
I 2 17 19 56 58 39 0 No
Train High Lift people* J* 5 19 24 58 63 39 39 No
Modify staging facilities K 12 30 42 32 44 2 2 No
Modify in/out facilities L 14 30 44 30 44 0 0 Yes
Develop procurement scheduling/card system*
M* 25 17 42 38 63 21 21 No
Rack/bin existing bulk inventory*
N* 14 44 58 44 58 0 0 Yes
Stage racks/bins to line O 5 58 63 58 63 0 0 Yes
Remove/salvage old storage area
P 20 63 83 73 93 10 10 No
Limited-scale operation, test/tune/mistake-proof*
Q* 30 63 93 63 93 0 0 Yes
Full-scale operations R 93 93 93 93 0 0 Yes
Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 476. *Indicates milestone activities; milestone occurs at the end of the marked activity.
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estimates. Here we use the same basic rules that we used to develop the initial CPM network, but begin at the end of the completed event. Hence, we can gener- ate updated ES, EC, LS, and LC estimates for the remaining activities, as well as redevelop our slack estimates. We also can determine whether our critical path has changed as a result of our changes. Additional details pertaining to project planning and implementation are available in project management texts, such as Badiru and Pulat (1995).
B.8. Process Maps and SiPoC diagrams
The purpose of a process map is to help people discover, understand, and commu- nicate the input- to-transformation-to-output characteristics of a process. Process flowcharts are used to map processes at any level of detail. Gross- level maps are useful in high- level planning work, while minute- level maps are useful in process control work. A flowchart depicts process flow by using a sequence of symbols and words to represent process flow components, all connected with directional lines/arrows to indicate flow paths. A wide variety of processes are charted, and hence a wide variety of symbols are used. In some cases, simple box or rectangular symbols are used that are self- descriptive or annotated near the symbol. In other cases, the symbols are iconic in the sense that the symbol shape is indicative of the process element. Usually, a legend is provided to define specialized symbols. Typi- cally, the more focused the flowchart, the more specialized the symbols. Process mapping is performed by teams and individuals such as operators, technicians, engineers, specialists, and/or managers. Diverse perspectives are gained through process mapping when an interdisciplinary team is involved with the mapping. See Kolarik (1995) for general details, Barnes (1980) for specialized charting tech- niques relative to classical industrial engineering, and Hughes (1995) for auto- matic process control–related flowcharting basics.
We map processes to help us understand how processes work or how they are expected to work. Process flow mapping usually involves several steps:
1. Establish flowchart/map purpose. Initially, we clearly state the purpose for our charting efforts. This purpose will dictate the level of detail we need in our map.
2. Define map boundaries. We determine the starting and ending points for the mapping effort, relative to purpose and necessary observations.
3. Observe process. Next, provided the process is in operation, direct process observation/experience is necessary to develop the process map. We may also observe/map processes in other organizations through benchmarking activities.
4. Establish gross process flow. Here we develop/chart a process overview depicting the production system or process in terms of major components, for example, processes or subprocesses, respectively.
5. Develop map details. Once we have obtained and captured the general essence of the process flow, we focus on details, cascading the level of detail down to the point where it is compatible with our purpose. Details are sequenced to represent the order/position that they occupy in the actual process.
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6. Check for validity/completeness. Finally, we move from level to level in our maps, examining them for validity and completeness. Validity checks typically involve map review as to accuracy of inputs, transformation, output, and sequence. Completeness extends to the level of detail within the target process as well as interactions with other processes.
Figure 5.19 provides an illustration of a macro- level process map, broken out by the seven fundamental processes: market/definition, design/development, pro- duction, distribution/marketing/sales/service, use/support, disposal/recycle, and business integration. Here, we can see a global depiction of the essential pro- cesses involved in an enterprise. We can drill down through these fundamental processes and build more detailed process maps, sometimes resembling a PDPC in nature. Figure 5.20 provides an illustration of such a map for a visual manufactur- ing alternative subprocess plan. Process maps may be layered to depict a process hierarchy; see Kolarik (1999) for details.
The SIPOC diagram is a high- level process map used to identify the important aspects of the current process, such as the process outputs and customers, in order to capture the voice of the customer. It is a useful tool in the early stages of the DMAIC process to determine the critical- to-quality factors (see section C of this
Figure 5.19 Enterprise-level process map. Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 441.
Process Type
Market/ definition
Design/ development
Production
Distribution/ marketing/sales/ customer service
Use/support Disposal/ recycle
Business integration
Processes/Subprocesses
Customer solutions definition— customer process characterization
Customer solutions/preliminary alternatives/modeling
Receiving
Promotion/ product
information
Sales and post-sales
service
Information reporting,
record keeping
Business planning and
budgeting
Safety/health/ environmental
compliance
Storage/materials/ assemblies/parts/supplies
Customer training
Modular process Modular processes are characterized by primarily self-contained operations, provided clear input/output requirements are defined
Integrated process Integrated processes are characterized by cross-functional or service operations, where multiple sets of customers and input/output requirements are defined
Public relations
Procurement
Field support parts/maintenance
Accounts payable, receivables
Disposal/recycle information
Line stocking
Customer solutions assessment—customer need/demands/expectations determination
Customer solutions/system concepts/specifications
Customer solutions/field engineering
Product test/checkout
Product shipping
Product repair
Financial and cash management
Payroll/human resources/benefits
Fabrication Paint Assembly
Customer account services
Facility and equipment maintenance
Customer solutions/final design/model demonstration
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chapter for a discussion of the DMAIC methodology). SIPOC is an acronym for suppliers, inputs, process, outputs, and customers—which are defined as follows:
• Suppliers. Those who provide inputs to the process, including materials, resources, services, information, and so on.
• Inputs. Materials, services, resources, information, and so on.
• Process. Process description and listing of all key process steps.
• Outputs. Products, information, services, and so on.
• Customers. Those who receive the outputs. Customers may be internal or external.
Figure 5.21 displays a simple SIPOC diagram for the process used to report and investigate work- related injuries at a manufacturing firm.
Figure 5.20 Visual alternative–improved subprocess map/PDPC. Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 468.
Unloading
Loading
Verify arrivals
Line order/ cards
Empty return
Order delivery
Document defectives
Production scheduling
Order staging at line
Vendor order/cards
Line orders
Qualify/certify vendors
Temporary rack/bin storage
Line delay?
Expedite?
Damage?
Shortage?
Salvage?
Temporary rack/bin storage
Inventory/ defective
entry
Inventory/ receiving
entry
At fabrication At paint At assembly At product repair
From fabrication From paint From assembly From product repair
To business integration
To business integration
From/to outside
vendors Stocking
subprocess
Procurement subprocess
Receiving/storage subprocess
Receiving/stocking team
Procurement team
Supply-side performance metrics Targets
Present Improved Inbound deliveries: 137 Rush/airfreight deliveries: 43 0 Damaged shipments: 3 0 Short shipments: 5 0 On-time delivery: 53% 100%
Customer-side performance metrics Targets
Present Improved Outbound deliveries: 324 Expedited orders: 112 0 Line shortages: 63 0 Line delays (for lack of materials, assemblies, parts, supplies): 17 0 On-time delivery to line: 69% 100%
Control metrics Targets
Downstream Expedited orders 0 Line shortages 0 Line delays 0 On-time delivery to line 100% Obsolete inventory parts/dollars 0.05%
Upstream Deviation to due date (by vendor/stocking) ±1 Overtime hours in receiving/ stocking 0 Defectives returned to vendors 0 Vendor certificates/progress/ maintenance 100% Design change/inventory coordination 100%
To vendors
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B.9. Process value Chain diagrams
The purpose of a process value chain (PVC) diagram is to help people depict and understand a sequence of cause- to-effect and effect- to-cause relationships between business results/outcomes and basic physical, economic, and social vari- ables. PVC analysis links basic physical and social variables with business results so that value- added process sequences are clearly depicted. This linkage is not pre- cise, because each basic variable has its own natural/technical units of measure (e.g., length, pressure, volume, or composition), while process/business results are expressed in their own units or unitless ratios (e.g., production units, percent confor- mance, scrap rate, efficiency, cost, revenue, profit, and return on investment [ROI]). Hence, PVCs have discontinuities where unitary incompatibility presents gaps and challenges. The point is to link variables related to specific process decisions and process control points to business results and vice versa as best we can. Hence, understanding as to cause–effect and time lags in moving from cause to effect become more obvious for all concerned (e.g., operators, engineers, and managers).
The PVC diagram connects the business world to the technical world through a logical, sequential linkage that cascades up and down all processes and their respective subprocesses. PVC diagrams are useful for operators to see how opera- tional decisions in the technical world ultimately impact business results. They are useful for managers/leaders to clearly see that business targets are met through a sequence of operational decisions. An efficient and effective PVC adds value to products throughout the chain.
A generic PVC is depicted in Figure 5.22. Across the top we see basic busi- ness outputs on the right- hand side and basic inputs in the form of controlled and uncontrolled variables on the left- hand side. Transformations in the form of
Figure 5.21 SIPOC diagram for work-related injuries.
Accident occurs
Process
Submit report within 48 hours
Detailed investigation
Submit final report
Complete follow-up
Develop corrective action plan
Implement corrective action plan
Inputs Accident information Work history Causes Equipment information Site information Site history
Suppliers Person injured Witnesses Coworkers Medical personnel Technicians/engineers Management
Customers Employees Management Health/safety personnel
Outputs Complete report Indentification of cause(s) Corrective action plan Solutions implemented
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processes and subprocesses are depicted in the middle. The oval cycle on the bot- tom half of the figure shows that we develop a PVC working from one of sev- eral starting points. We may start somewhere in our outputs (the business results) and work toward inputs (the basic variables). Or, we may start somewhere in our input variables and work toward our business outputs. The focus is to understand how the processes work and how they impact the business objectives. See Kolarik (1999) for more details regarding PVC.
B.10. Benchmarking
The purpose of benchmarking is to help people learn from the work of others— seek out, study, and emulate the best practices associated with high performance/ results—so as to enhance or better their own performance.
We have a tendency to perceive our organization as the “best” through rather subjective arguments, for example, exhortations of various types. In reality, our perceptions may not be accurate. We may lack insight as to what is happening around us: what others are doing and the results they obtain. In essence, we lack
Figure 5.22 Generic production system process value chain diagram. (a) Analytical view. (b) General systems view.
Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 54.
Time delay Results targets
(a)
(b)
Subprocess (Subprocess purpose)
Control, stability Subprocess results
Targets and benchmarks
Process (Process purpose)
Control, stability Process results
Targets and benchmarks
Production system (Vision, mission, core
values)
Business results Targets and benchmarks
Uncontrollable variables
Process action
Process counteraction—Feedback/feedforward
Response variables
Customer demands/expectations
Controllable variables
Time delay
Process action
P rocess counteraction—Feedback/feedforward
Uncontrollable variables
Controllable variables
Cause–effect relationships
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outside standards/benchmarks from which to judge our own performance. Bench- marking helps us to both gain an awareness of shortfalls in our own performance and plan and implement countermeasures to enhance our performance.
Informal benchmarking is a matter of natural curiosity and has always been practiced; however, formal benchmarking was positioned as an organizational ini- tiative at Xerox. Kerns and Nadler (1992) define benchmarking as “the continuous process of measuring products, services, and practices against the toughest com- petitor or those companies recognized as industry leaders.” Camp (1989, 1995) defines benchmarking as “the search for and implementation of best practices.” Benchmarking encompasses four aspects: (1) analyze the operation, (2) know the competition and industry leaders, (3) incorporate the best of the best, and (4) gain superiority. The formal scope of benchmarking includes products, processes, and performance metrics.
Camp (1989, 1995) cites four types of benchmarking: internal, competitive, functional, and generic. Internal benchmarking focuses on best practices within our own organization. Competitive benchmarking provides a comparison between direct competitors. Functional benchmarking refers to comparisons of methods across organizations executing the same basic functions outside our industry. Generic process benchmarking focuses on innovative work processes in general, wherever they occur.
Benchmarking, which was also discussed in Chapter 1, is a broad initiative. Watson (1993) describes the evolution of benchmarking in terms of generations. He cites reverse engineering as the first generation. Here, we see essentially a rote copy- ing strategy. The second generation is termed competitive benchmarking, which focuses on direct competitors. As the third generation, he cites process benchmark- ing, where processes common to different industries are assessed for best practices. The fourth generation is termed strategic benchmarking. Here, the focus is on the strategies that a competitor or noncompetitor uses to guide their organization. The fourth generation is used to feed process reengineering initiatives. A futuristic fifth generation is cited as global benchmarking. Here, the focus is international in scope and deals with trade, cultural, and business process distinctions among companies. In all cases, the driving force is “profit oriented,” as addressed through three param- eters: (1) quality beyond that of competitors, (2) technology before that of competi- tors, and (3) costs below those of competitors.
The benchmarking initiative focuses on two basic issues: (1) best practices and (2) metrics or measurement. We recognize performance gaps and address them with improvement plans. Management commitment, communication, and employee participation are all critical elements in a benchmarking initiative. For more information on benchmarking, see Camp (1989, 1995), Kolarik (1995, 1999), and Watson (1993).
Steps in the benchmarking process are as follows:
1. Preplan the benchmarking initiative. Assess and understand customer needs and the business results/outcomes desired.
2. Plan and execute the initiative. Identify comparative organizations and what is to be benchmarked. Determine data collection methods and collect data.
3. Analyze the data and information collected. Determine the current performance gap. Project future performance levels/goals.
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4. Integrate the information into actionable issues. Communicate the findings and gain acceptance within your organization. Establish functional goals that are actionable.
5. Prepare for action and act. Develop action plans, implement specific actions, monitor progress, and recalibrate the benchmarks.
6. Gain maturity in benchmarking. Attain a leadership position and integrate benchmarking practices into processes.
Benchmarking clearly is an invaluable asset in quality improvement work. It pro- vides a perspective of how things are done in other organizations, leads to the identification of best practices, and encourages adoption of the same. However, a best practice today will undoubtedly be eclipsed by a better practice in the near future.
In many cases, we inject creative elements within/beyond current practices. These extensions require creative thinking or breakthrough thinking. Break- through thinking typically is approached very differently from benchmarking. Here, we are encouraged to think “out of the box,” as opposed to thinking “in the box” (e.g., finding an existing best practice). Creative thinking is the only tool available that allows us to move beyond best practices.
Creativity has received considerable attention in quality improvement work (Kolarik 1995, 1999). Nadler and Hibino (1994) proposed seven principles of break- through thinking: (1) the uniqueness principle, (2) the purposes principle, (3) the solution- after-next principle, (4) the systems principle, (5) the limited information collection principle, (6) the people design principle, and (7) the betterment time- line principle. DeBono (1992) encourages the use of hats in creative thinking: the white hat (data and information), the red hat (feelings, intuition, hunches, and emotions), the black hat (pessimistic perspective), the yellow hat (optimistic per- spective), the green hat (creative effort), and the blue hat (thinking process control).
Figure 5.23 depicts an overview of the integration of breakthrough thinking into a benchmarking model. From this depiction, we can see that breaking through
Figure 5.23 Benchmarking and breakthrough thinking. Source: Reproduced by permission from W. J. Kolarik, Creating Quality: Process Design for Results (New York: McGraw-Hill, 1999), 164.
Benchmarking Our operation
Improvement need
Short-term focus
Long-term focus
Noncompetitor’s operation
Competitor’s operation
Improvement
Breaking through
Product
Service Technology
Focusing
Strategy
Awareness Purpose Planning Acting Following through
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focuses on the essence of the product, technologies (relative to both products and process), and services. The long- term focus of breakthrough thinking tends to complement the shorter- term focus of benchmarking, yielding a broad view of improvement efforts.
C. ConTinuouS iMProvEMEnT METHodoLogiES Quality improvement is achieved by continuously improving the production and business processes of an organization (Besterfield 1999). It is optimized by:
• Viewing all work as a process, whether it is associated with production or business activities
• Making all processes effective, efficient, and adaptable
• Anticipating changing customer needs
• Controlling in- process performance using metrics such as scrap and cycle time, and monitoring tools such as control charts
• Maintaining constructive dissatisfaction with the present level of performance
• Eliminating waste and rework wherever they occur
• Investigating activities that do not add value to the product or service, with the aim of eliminating those activities
• Eliminating nonconformities in all phases of everyone’s work, even if the increment of improvement is small
• Using benchmarking to improve competitive advantage
• Innovating to achieve breakthroughs
• Holding gains so there is no regression
• Incorporating lessons learned into future activities
• Using technical tools such as SPC, experimental design, benchmarking, QFD, etc.
Continuous process improvement is designed to utilize the resources of the organization to achieve a quality- driven culture. Individuals must think, act, and speak quality. An organization attempts to reach a single- minded link between quality and work execution by educating its constituents to continu- ously analyze and improve their own work, the processes, and their work group (Langdon 1994).
Process improvement achieves the greatest results when it operates within the framework of the problem- solving method. In the initial stages of a program, quick results are frequently obtained because the solution is obvious or an indi- vidual has a brilliant idea. There are a number of models for quality improve- ment. We discuss total quality management ( TQM), kaizen, PDSA (sometimes known as PDCA), theory of constraints (TOC), and Six Sigma in the following sections.
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C.1. Total Quality Management
TQM is based on the principles of Feigenbaum, Deming, and Juran. The exact ori- gin of TQM has been debated, but TQM as a process improvement methodology received the most use and attention in the mid-1980s and early 1990s before being mostly replaced by lean and Six Sigma efforts, which focus on more rigorous statis- tical methods. The goal of TQM is to manage quality improvement methods across the entire organization. Some sources claim that the TQM name was developed by the United States Navy and originally used at the Naval Air Systems Command (A. Houston and S. L. Dockstader, Total Quality Leadership: A Primer, http://www. balancedscorecard.org/portals/0/pdf/primer.pdf ). TQM encompasses the follow- ing ideas:
• Customer focus. The customer determines whether a product or service is good enough.
• Employee empowerment. All employees must understand that continuous improvement is a part of everyone’s job.
• Leadership. Upper management must provide the impetus and motivation for the quality programs. Many TQM organizations have quality councils that lead strategic quality initiatives.
This model, when properly implemented, often results in an enterprise that is more productive and more competitive. Customer loyalty will improve and stakeholder value will increase. As Montgomery (2013) argues, many organizations focused on training their workforce in quality and basic methods but did not emphasize tools used to reduce variability or how to implement these methods in practice.
C.2. Kaizen
Kaizen is a Japanese word for the philosophy that defines management’s role in continuously encouraging and implementing small improvements involving everyone in an organization. It is a method of continuous improvement in small increments that makes processes more efficient, effective, under control, and adapt- able. Improvements are usually accomplished at little or no expense and without sophisticated techniques or expensive equipment.
Kaizen focuses on simplification by breaking down complex processes into their subprocesses and then improving them. Its application is not limited to qual- ity, but quality professionals have effectively applied it. When done correctly, it humanizes the workplace, eliminates hard work (both mental and physical), and teaches people how to use the scientific method and to detect waste. Some com- panies have created a spin- off called kaizen blitz. This is a carefully orchestrated intensive activity designed to produce a significant improvement quickly.
The kaizen improvement involves the following activities and assessment:
1. Value-added and non- value-added work activities (see section D of this chapter for details).
2. Muda, which refers to the seven classes of waste: overproduction, delay, transportation, processing, inventory, wasted motion, and defective parts (see section D of this chapter for details).
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3. Principles of motion study.
4. Principles of materials handling.
5. Documentation of standard operating procedures.
6. The five S’s for workplace organization, which are five Japanese words that mean proper arrangement (seiri), orderliness (seiton), personal cleanliness (seiso), cleanup (seiketsu), and discipline (shitsuke). Various authors have translated them slightly differently. NIST, through the Manufacturing Extension Partnership, uses sort, set in order, shine, standardize, and sustain (refer to section D of this chapter for more information).
7. Visual management by means of visual displays that everyone in the plant can use for better communications.
8. Just-in-time principles to produce only the right units in the right quantities, at the right time, and with the right resources.
9. Poka-yoke to prevent or detect errors.
10. Team dynamics, which include problem solving, communication skills, and conflict resolution (Gee, McGrath, and Izadi 1996).
Kaizen relies heavily on a culture that encourages suggestions by operators who continually try to incrementally improve their job or process. An example of a kaizen- type improvement would be changing the color of a welding booth from black to white to improve operator visibility. This change would result in a small improvement in weld quality and a substantial improvement in operator satis- faction. The PDSA cycle, described next, may be used to help implement kaizen concepts.
C.3. Plan- do-Study-act (PdSa)
The basic plan- do-study-act (PDSA) cycle, sometimes known as the plan- do- check-act (PDCA) cycle, was developed by Shewhart and is an effective improve- ment technique. It is sometimes called the Shewhart cycle or the Deming cycle (see Figure 5.24).
The four steps in the cycle are exactly as stated. First, plan carefully what is to be done. Next, carry out the plan (do it). Third, study the results: Did the plan work as intended or were the results unexpected? Finally, act on the results by identifying what worked as planned and what did not. Using the knowledge learned, develop an improved plan and repeat the cycle. The PDSA cycle is a sim- ple adaptation of the more elaborate Six Sigma process, discussed in section C.5 of this chapter.
PDSA is still widely used as an effective tool in industry. For example, two PDSA cycles were used in a quality improvement project for the Cancer Genetics Service clinic at the National Cancer Centre in Singapore. The PDSA cycles identi- fied the need for a genetic counselor. The results of the quality improvement proj- ect were improved patient access and reduced costs (Tan et al. 2016).
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C.4. Theory of Constraints
TOC is a problem- solving methodology that focuses on the weakest link in a chain of processes. Usually the constraint is the process that is slowest. Flow rate through the system cannot increase unless the rate at the constraint increases. TOC lists five steps to system improvement:
1. Identify. Find the process that limits the effectiveness of the system. If throughput is the concern, then the constraint often will have work in process (WIP) awaiting action.
2. Exploit. Use kaizen or other methods to improve the rate of the constraining process.
3. Subordinate. Adjust (or subordinate) the rates of other processes in the chain to match that of the constraint.
4. Elevate. If the system rate needs further improvement, the constraint may require extensive revision (or elevation). This could mean investment in additional equipment or new technology.
5. Repeat. If these steps have improved the process to the point where it is no longer the constraint, the system rate can be further improved by repeating these steps with the new constraint.
The strength of TOC is that it employs a systems approach, emphasizing that improvements to individual processes will not improve the rate of the system
Figure 5.24 Basic plan-do-study-act cycle.
Feedback assessment and corrective action
Product and process
validation
Plan and define
Product design and development
Process design and development
A
C T PL
A N
S T
U
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O P
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conceptim pr
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unless they improve the constraining process. Creasy and Ramey (2013) describe a performance improvement project at Clinch Valley Medical Center where they combined Lean- Six Sigma (LSS) and TOC to identify and eliminate bottlenecks in the preadmission testing process of the hospital. The average unnecessary patient wait time before the improvement project was 20 minutes, with some patients waiting more than one hour. The project team used a SIPOC diagram, a pro- cess flowchart, and TOC and identified five bottlenecks in the process. After process improvements using several LSS tools, the wait time dropped to under six minutes, a 70% reduction.
C.5. Six Sigma
The average half- life of companies is about 10 years, regardless of past performance or the type of product produced (Daepp et al. 2015). Companies are embracing Six Sigma not only as a method to reduce defects but also as a catalyst to change their culture and impact how employees engage in their everyday work.
Utilizing a Six Sigma business strategy, organizations can understand threats and recognize new opportunities for growth, not only to survive but also to thrive within competitive environments.
Quality practitioners often note that the tools of Six Sigma are not unique. It is true that most Six Sigma techniques are familiar; however, the power of properly integrating them as a total system is new. Six Sigma creates a road map for chang- ing data into knowledge, resulting in process- focused change and bottom- line benefits for organizations. Not all organizations have achieved success with Six Sigma, which depends on the successful integration of two components: strat- egy and metrics.
The strategy of Six Sigma relates to how the methodology (tools and tech- niques) is integrated into an organization through key projects, yielding substan- tial benefits to the organization’s bottom line. Companies experiencing success with Six Sigma have created an effective infrastructure for selecting, supporting, and executing projects. These projects are focused on achieving strategic business goals, as well as addressing the voice of the customer.
The success of Six Sigma also depends on the wise application of metrics. Unfortunately, much confusion exists relative to the metrics of Six Sigma. There is no “one size fits all” metric applicable to every project. Effective metrics are cross- functional, providing a holistic view of the process and contributing insight to the project team. A lot of resources will be wasted if Six Sigma metrics are not applied wisely and subsequently used to orchestrate improvement activities. “Fire prevention” is preferred to “firefighting.”
This section details the two components previously mentioned, as well as other important aspects of a successful Six Sigma implementation, including the following:
• Six Sigma needs assessment
• Six Sigma as a business strategy
• Implementing Six Sigma
• The metrics of Six Sigma
• Sustaining and communicating change
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Organizations often become overwhelmed with day- to-day activities and lose sight of what needs to be done to make process- focused improvements or changes in order to survive the “long haul.” Individuals within organizations might be aware of Six Sigma and think that the techniques could be useful to reduce the amount of firefighting activities that occur; however, they may have trouble determining where it applies and where the benefits are achievable. This type of organization requires a simple and quick approach to make a Six Sigma needs assessment.
For this situation, we suggest that people within the organization respond to the Six Sigma needs checklist shown in Table 5.5. Upon completion of this survey, the additional question can then be asked, “How much money are the affirmative responses costing the business annually?” An improvement opportu- nity often can be accurately quantified if the amount is initially determined as a percentage of the gross revenue of the organization.
Monetary estimates from this survey could be considered the perceived “cost of doing nothing” within the organization—that is, the cost to the business of not “doing Six Sigma.” When this survey is conducted during a meeting of informed individuals, an even more accurate estimate of this cost can be obtained. We sug- gest that during this meeting, individuals describe the logic used for their vote. Consensus might then be achieved for an overall monetary estimate for the group. When consensus does not seem possible, an average of the responses can give a very good estimate.
Estimated projected benefits from Six Sigma could then be determined as the projected monetary “cost of doing nothing.” Full- time Six Sigma Black Belts (i.e.,
Table 5.5 Six Sigma needs checklist.
Six Sigma needs checklist Answer
yes or no
Do you have multiple “fix-it” projects in a critical process area that seem to have limited or lasting impact?
Are you aware of a problem that management or employees are encountering?
Are you aware of any problem that a customer is having with the products/ services your organization offers?
Do you believe that primary customers might take their business elsewhere?
Is the quality from competitive products/services better?
Are your cycle times too long in certain process areas?
Are your costs too high in certain process areas?
Do you have concerns that you might be “downsized” from your organization?
Do you have a persistent problem that you have attempted to fix in the past with limited success?
Do you have regulatory/compliance problems?
Source: F. W. Breyfogle III, J. M. Cupello, and B. Meadows, Managing Six Sigma (New York: John Wiley & Sons, 2000). Adapted by permission of John Wiley & Sons, Inc.
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Six Sigma practitioners) can save an organization large amounts of money (direct impact on company finances) annually, depending on the following:
• Executive-level support
• Process focus area (i.e., some areas have more room for improvement than others)
• Team motivation
• Six Sigma Black Belt proficiency
A question we frequently hear from executives is, “How does Six Sigma fit with other corporate initiatives?” Six Sigma should not be considered just another ini- tiative but should be integrated with other programs (e.g., lean manufacturing and kaizen) at a higher level as part of an overall business strategy. Six Sigma should not replace other initiatives but instead create an infrastructure that offers a tacti- cal approach to determine the best solution for a given process/situation.
Successful implementation should be viewed as an ongoing process of infus- ing the Six Sigma methodology into the way your employees approach their every- day work. It requires a proactive view and the commitment to evolve into a more process- oriented culture and reduce the amount of daily firefighting on strategic processes. The implementation process requires up- front work to develop aware- ness and generate buy- in before projects are selected. This process often displays unique characteristics in each organization; however, two elements are essential for success: executive leadership and customer focus.
To date, companies achieving significant results with Six Sigma have the commitment of their executive management. Executive leadership is the founda- tion of any successful Six Sigma business strategy. Upper- level managers need to develop an infrastructure to support the changes that implementing Six Sigma will create, not only to strategic business processes but also, as previously dis- cussed, to the culture of the organization. Past quality programs resulted in vary- ing success because they typically did not have an infrastructure that supported change.
The results received from a Six Sigma business strategy are highly dependent on how well leaders understand the value of wise implementation of the method- ology and sincerely promote it within their organization. An executive retreat can help identify true champions who will promote change and can also prioritize the actions necessary to establish a road map to successful implementation. Through discussion and the careful planning of the process of successfully implementing Six Sigma, employees will have an easier journey to success in applying the meth- odology to their projects.
Establishing a customer focus mind- set within an organization goes hand in hand with creating a successful Six Sigma business strategy. The factors that are critical to your customers’ success are necessary to a process improvement team’s success. Therefore, evaluating customers’ perception of quality should be at the forefront of the implementation process.
Every complaint from a customer should be viewed as an opportunity for growth and increased market share, a spotlight on areas needing process improve- ment focus. The key to success in this initial step is to make it easy for your cus- tomers’ comments to be heard. Various methods exist to obtain this valuable input, including walking the customer process, performing customer surveys,
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conducting personal interviews with key customers, establishing feedback com- plaint systems, and developing customer panels.
Depending on the size of your organization and its core values, the word “customer” can take on many different definitions. When collecting feedback, care should be taken to maintain a comprehensive view of your customers. By combin- ing external feedback with such things as internal business strategies, employee needs, and government regulations, your organization will obtain a balanced list of customer needs.
Through customer feedback, learning about what works and what does not will help to establish a mind- set of continual process improvement within your organization. Jack Welch, former CEO of GE and advocate of Six Sigma, has been quoted as saying that a business strategy alone will not generate higher quality throughout an organization.
C.5.a. Implementing Six Sigma
Six Sigma can be either a great success or a failure, depending on how it is imple- mented. Implementation strategies can vary significantly between organizations, depending on their distinct culture and strategic business goals. After completing a needs assessment and deciding to implement Six Sigma, an organization has two basic options:
• Implement a Six Sigma program or initiative
• Create a Six Sigma infrastructure
C.5.a.i. Option 1: Implement a Six Sigma Program or Initiative
The traditional approach to deploying statistical tools within an organization has not been very effective. With this approach, certain employees (practitioners) are taught the statistical tools from time to time and asked to apply a tool on the job when needed. The practitioners might then consult a statistician if they need help. Successes within an organization might occur; however, these successes do not build on each other to encourage additional and better use of the tools and overall methodology.
When organizations implement Six Sigma as a program or initiative, it often appears that they have only added, in an unstructured fashion, a few new tools to their toolbox through training classes. A possible extension of this approach is to apply the tools as needed to assigned projects. However, the selection, manage- ment, and execution of projects are not typically an integral part of the organiza- tion. These projects, which often are created at a low level within the organization, do not have the blessing of upper management; hence, resistance is often encoun- tered when the best solution directly affects another group that does not have buy- in to the project. In addition, there typically is no one assigned to champion projects across organizational boundaries and facilitate change.
A program or initiative does not usually create an infrastructure that leads to bottom- line benefits through projects tied to the strategic goals of the organization. As a program or initiative, Six Sigma risks becoming the “flavor of the month” and will not capture the buy- in necessary to reap a large return on the investment in training. With this approach, employees may end up viewing Six Sigma as a
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program similar to TQM and other quality programs that may have experienced only limited success within their organization.
Even if great accomplishments occur through the individual use of statistical tools within organizations, there is often a lack of visibility of the benefits to upper management. A typical missing element for success with this approach is manage- ment buy- in. Because of this lack of visibility, practitioners often have to fight for funds, and these funds may be eliminated whenever the organization becomes financially strained. Effective use of statistical tools often does not get recognized and the overall company culture is not affected. For true success, executive- level support is needed that asks the right questions and leads to the wise applica- tion of statistical tools and other Six Sigma methodologies across organizational boundaries.
C.5.a.ii. Option 2: Create a Six Sigma Infrastructure
Instead of focusing on the individual tools, it is best when Six Sigma training provides a process- oriented approach that teaches practitioners a methodology to select the right tool, at the right time, for a predefined project. Training of Six Sigma practitioners (Black Belts) utilizing this approach typically consists of four weeks of training over four months, where students work on their projects during the three weeks between training sessions.
Deploying Six Sigma as a business strategy through projects instead of tools is the more effective way to benefit from the time and money invested in Six Sigma training. Consider the following benefits of Six Sigma deployment via projects that have executive management support:
• Offers bigger impact through projects tied to bottom- line results
• Utilizes the tools in a more focused and productive way
• Provides a process/strategy for project management that can be studied and improved
• Increases communications between management and practitioners via project presentations
• Facilitates the detailed understanding of critical business processes
• Gives employees and management views of how statistical tools can be of significant value to organizations
• Allows Black Belts to receive feedback on their project approach during training
• Deploys Six Sigma with a closed- loop approach, creating time for auditing and incorporating lessons learned into an overall business strategy
A project- based approach relies heavily on a sound project selection process. Proj- ects should be selected that meet the goals of an organization’s business strategy. Six Sigma can then be utilized as a road map to effectively meet those goals. Once a strategic project has been selected, many practitioners (Black Belts) find a “21-step integration of tools” road map helpful in developing a plan for the specific project.
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Initially, companies might choose projects that are too large, or may not choose certain projects because of their lack of strategic impact to the bottom line. Frustra- tion with the first set of projects can be vital experience that motivates improve- ment in project selection in the next phase of implementing Six Sigma. Six Sigma is a long- term commitment. Treating deployment as a process allows objective anal- ysis of all aspects of the process, including project selection and scoping. Utilizing lessons learned and incorporating them into subsequent waves of an implementa- tion plan creates a closed feedback loop and real opportunities for improvement. Deploying Six Sigma through projects can lead to dramatic bottom- line benefits if the organization invests the time and executive energy necessary to implement Six Sigma as a business strategy.
C.5.b. The Metrics of Six Sigma
Much confusion exists relative to the metrics of Six Sigma. The sigma level (i.e., sigma- quality level) sometimes used as a measurement within a Six Sigma pro- gram includes a ±1.5σ value to account for “typical” shifts and drifts of the mean, where σ is the standard deviation of the process. This sigma- quality-level relation- ship is not linear. In other words, a percentage unit improvement in parts per mil- lion (ppm) defect rate (or defects per million opportunities [DPMO] rate) does not equate to the same percentage improvement in the sigma- quality level.
Figure 5.25 shows the sigma- quality level associated with various services (con- sidering the 1.5σ shift of the mean). From this figure, we note that the sigma- quality level of most services is about four sigma, while “world class” is considered six.
Figures 5.26, 5.27, and 5.28 illustrate various aspects of a normal distribution as it applies to Six Sigma program measures and the implication of the 1.5σ shift.
Figure 5.25 Implication of sigma-quality level. The ppm rate for part or process step considers a 1.5σ shift of the mean where only 3.4 ppm fail to meet specifications at a six sigma quality level.
Source: F. W. Breyfogle III, Implementing Six Sigma (New York: John Wiley & Sons, 1999). Adapted by permission of John Wiley & Sons, Inc.
1 2 3 4 5 6 7
1,000,000
100,000
10,000
1,000 Average company
Best in class
Domestic airline flight fatality rate (0.43 ppm)
100
10
1
0.1
0.01
p p
m r
a te
Sigma level (with ± 1.5σ shift)
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Figure 5.26 Normal distribution curve illustrates three sigma and six sigma parametric conformance.
Source: Copyright of Motorola, used with permission.
Customer needs
± 3σ
LSL
1350 ppm
0.001 ppm
1350 ppm
0.001 ppm
USL
Specification range
± 6σ
Figure 5.27 With a centered normal distribution between six sigma limits, only two devices per billion fail to meet the specification target.
Source: Copyright of Motorola, used with permission.
Normal distribution centered
µ +1σ +2σ +3σ +6σ–6σ
Specification limit Percent Defective ppm
± 1 sigma 68.27 317300 ± 2 sigma 95.45 45500 ± 3 sigma 99.73 2700 ± 4 sigma 99.9937 63 ± 5 sigma 99.999943 0.57 ± 6 sigma 99.9999998 0.002
–3σ –2σ –1σ
LSL USL
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Figure 5.26 illustrates the basic measurement concept of Six Sigma, where parts are to be manufactured consistently and well within their specification range. Fig- ure 5.27 shows the number of ppm that would be outside the specification limits if the data were centered within these limits and had various standard deviations. Figure 5.28 extends Figure 5.26 to noncentral data relative to specification limits, where the mean of the data is shifted by 1.5σ. Figure 5.29 shows the relationship of ppm defect rates versus sigma- quality level for a centered and 1.5σ shifted process, along with a quantification for the amount of improvement needed to change a sigma level. Refer to Appendix D for additional Z-values.
To achieve this basic goal of a Six Sigma program might then be to produce at least 99.99966% “quality” at the “process step” and part level within an assembly (i.e., no more than 3.4 defects per million parts or process steps if the process mean were to shift by as much as 1.5σ). If, for example, there was on average one defect for an assembly that contained 40 parts and four process steps, practitioners might consider the assembly to be at a four sigma quality level from Figure 5.29, since the number of defects in ppm is (1 ÷ 160)(106) = 6250.
Problems that can occur using the sigma- quality-level metric include the following:
• The improvement from 4.1 to 4.2 sigma- quality level is not the same as improvement from 5.1 to 5.2 sigma- quality level.
• Determining the number of opportunities for any given process can be dramatically different between individuals.
Figure 5.28 Effects of a 1.5σ shift where only 3.4 ppm fail to meet specifications. Source: Copyright of Motorola, used with permission.
Normal distribution shifted 1.5σ
World class
µ +1σ +2σ +3σ +6σ–6σ
Specification limit Percent Defective ppm
± 1 sigma 30.23 697700 ± 2 sigma 69.13 308700 ± 3 sigma 93.32 66810 ± 4 sigma 99.3790 6210 ± 5 sigma 99.97670 233 ± 6 sigma 99.9996600 3.4
–3σ –2σ –1σ
LSL USL
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• A sigma- quality-level metric can be deceiving. For example, one process might have a 50% defective unit rate and a sigma- quality level much greater than six, while another process might have a 0.01% defective unit rate and a sigma- quality level much worse than six. To illustrate this, first consider the counting of opportunities for failure within a computer chip as junctions and “components.” The sigma- quality- level metric for this situation typically leads to a very large number of opportunities for failure for a given computer chip; hence, a very high sigma- quality level is possible even when the defective rate per unit is high. Compare this situation with another situation in which only a very small number of components or steps are required for a process. The sigma- quality-level metric for this situation typically leads to a very low number of opportunities for failure; hence, a very low sigma- quality-level metric is possible even when the defective rate per unit is low.
• The sigma- quality-level metric can only be determined when there are specifications. Service/transactional applications typically do not have specifications as manufacturing does. When a sigma- quality level is forced on a service/transactional situation, this can lead to the fabrication of specifications and alterations of these “specifications” to “make the numbers look good.”
Another Six Sigma metric that describes how well a process meets requirements is process capability. A six sigma–quality level process is said to translate to process capability index values for Cp and Cpk requirements of 2.0 and 1.5, respectively.
Figure 5.29 Defect rates (ppm) versus sigma-quality level. Source: F. W. Breyfogle III, Implementing Six Sigma (New York: John Wiley & Sons, 1999). Adapted by permission of John Wiley & Sons, Inc.
2 3 4 5 6
1.5σ shift: Centered:
100,000 66,810 ppm
6210 ppm
10× improvement
30× improvement
70× improvement
233 ppm
3.4 ppm
10,000
1,000
100
10
1
0.1
0.01
0.001
D e fe
c t
ra te
( p
p m
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Sigma-quality level
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Unfortunately, there is much confusion with these values, even though the follow- ing basic equations for these metrics are simple:
C
USL LSL p =
− 6σ
(5.3)
C min
USL 3
LSL 3pk
= − −µ σ
µ σ
,e e
(5.4)
where USL is the upper specification limit, LSL is the lower specification limit, and σ is the standard deviation. Computer programs often will not even give the same answer for a given set of data. Some programs consider the standard deviation to be short term, while others consider standard deviation to be long term. There are many ways to estimate standard deviation. Breyfogle (1999) describes eight dif- ferent approaches. Process capability indices are discussed in detail in Chapter 6, section G.
Another metric is rolled throughput yield (RTY). Reworks within an operation make up what is termed the hidden factory. RTY measurements can give visibility to process steps that have high defect rates and/or rework needs. One way to find RTY is to first determine the yield for all process operations. Then, multiply these process operation yields together. A cumulative throughput yield up through a process step can be determined by multiplying the yield of the current step by the yields of previous steps.
RTY can be calculated from the number of defects per unit (DPU) using the relationship:
RTY = e–DPU (5.5)
To understand this relationship, consider that the probability of observing exactly x events in the Poisson situation is given by the Poisson probability density func- tion (pdf):
P X x e
x e np
x x
x np x
=( ) = = ( ) = − −
! ! 0, 1, 2, 3 . . .
λλ
(5.6)
where e is a constant approximately equal to 2.71828, x is the number of occur- rences, and λ can be equated to a sample size multiplied by the probability of occurrence (i.e., np). It then follows that
PRTY X e e DPU= =( ) = =− −0 λ (5.7)
It is best not to force a sigma- quality metric on the various groups and/or projects within an organization. It is most important to use the right metric for any given situation. However, we believe that the sigma- quality-level metric should be included, along with the other Six Sigma metrics, in all Six Sigma training. The positive, negative, and controversial aspects of each Six Sigma metric should be covered within the training so that organizations can more effectively communi- cate with their customers and suppliers. Often, customers and suppliers ask the wrong questions relative to Six Sigma and other metrics. When people understand
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the pluses and minuses of each metric, they can work with their customers and/or suppliers to direct their efforts toward the best metric for a given situation, rather than react to issues that result from mandated metrics that make no sense.
The training people receive in Six Sigma should lead them to the right metric for a given situation. As depicted in Figure 5.30, in addition to devising a business strategy, organizations wanting success with Six Sigma must be able to under- stand, select, and communicate Six Sigma metrics, including sigma- quality levels; Cp, Cpk, Pp, and Ppk; RTY; DPMO; cost of poor quality (COPQ); and “30,000-foot level” control charts (Breyfogle 2003).
Care must be taken that the training an organization receives in Six Sigma metrics is not sugarcoated or avoided altogether. In addition to the careful selec- tion of metrics, Six Sigma training should also address the effective use of statisti- cal methodologies, providing insight into how one can best determine what truly is causing a problem.
C.5.c. The DMAIC Process
DMAIC is a data- driven quality strategy used to improve processes. It is an integral part of a Six Sigma initiative, but in general it can be implemented as a stand- alone quality improvement procedure or as part of other process improvement initia- tives such as lean (which is discussed in the next section). DMAIC is an acronym for the five phases that make up the process:
• Define the problem, improvement activity, opportunity for improvement, project goals, and customer (internal and external) requirements
• Measure process performance
• Analyze the process to determine root causes of variation, poor performance (defects)
• Improve process performance by addressing and eliminating the root causes
• Control the improved process and future process performance
Figure 5.30 Six Sigma metrics and implementation strategy.
6σ
Metrics
Cp, Cpk, Pp, Ppk
Sigma–quality level Provides insight to business
Provides insight to process
Leads to the ‘‘right’’ activity
Driven from the top
Selecting the right players
Effective project selection
DPMO
RTY
COPQ
‘‘30,000-foot level’’ metrics
Program
Initiative
Business strategy
Strategy
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The DMAIC process easily lends itself to the project approach to quality improve- ment encouraged and promoted by Juran. There are many tools used at each step of the process, most of which are described in this handbook. The reader is encour- aged to consult additional resources for detailed discussion of the DMAIC process, Six Sigma quality initiatives, and the numerous tools used. See, for example, the ASQ website (http://www.asq.org); Britz and Emerling (2000); Hahn, Doganaksoy, and Stanard (2001); Hoerl (2001); Hoerl and Snee (2012); Snee and Hoerl (2003, 2004); Montgomery (2013); and the numerous references within these sources.
Additional data- driven process improvement methods have been proposed since the inception of DMAIC. Anderson- Cook and Lu (2015) introduced a process called DMRCS (define, measure, reduce, combine, select), which is similar to the DMAIC process in that DMRCS is a structured approach for identifying and com- paring alternative solutions for multi- objective problems. DMRCS is especially helpful as a decision- making tool. See additional details in Anderson- Cook (2017) and Anderson- Cook and Lu (2015).
C.5.d. Sustaining and Communicating Change
Many companies attempt to improve products with numerous small changes or tweaks to their current processes; however, changes frequently are not documented and the associated results not reported. Substantial results are rarely obtained with this half- hearted method of change. When employees in this type of corporate cul- ture hear of a new initiative such as Six Sigma, they wonder what will be different.
In today’s constantly changing marketplace, companies that are able to embrace change in a focused and proactive manner are leaders in their field. Companies that not only master the technical side of Six Sigma but also overcome the cultural challenges associated with change can realize significant bottom- line benefits.
Launching a Six Sigma business strategy is an excellent opportunity to assess current culture in an organization. Consider the following questions:
• How has your company historically dealt with change initiatives?
• Does your company often make changes that do not last?
• How effective are your project teams?
• Are you frequently focusing on the same problem?
• How do your employees attack problems and conduct their daily work?
• What is required within your company culture to make continual process improvement a lasting change?
• What will prevent your company from achieving success with Six Sigma?
By evaluating the key cultural drivers and restraints to embracing Six Sigma, orga- nizations can develop plans that enhance the key drivers and mitigate the critical restraints.
A common key driver of sustaining Six Sigma change that is often over- looked is a communication plan. Company leaders usually implement Six Sigma because they possess a clear vision of what their company can achieve. Frequently,
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however, they do not realize the power behind effectively communicating this vision throughout the corporation. Executives need to get everyone engaged in and speaking the language of Six Sigma. A shared vision of how Six Sigma fits the strategic needs of the business should be created. A communication plan should be carefully considered and executed with enthusiasm. If successful, it will be your biggest ally in key stakeholder buy- in.
Implementing Six Sigma does not guarantee tangible benefits within an orga- nization. However, when Six Sigma is implemented wisely as a business strategy accompanied by effective metrics, as illustrated in Figure 5.30, organizations can yield significant bottom- line benefits. Through the wise implementation of Six Sigma, the successes of individual projects can build on each other, gaining the sustained attention of executive management and resulting in a corporate culture change from a reactive or firefighting environment to a learning organization.
d. LEan Achieving what is known as a lean enterprise requires a change in attitudes, pro- cedures, processes, and systems. It is necessary to “zoom out” and look at the flow of information, knowledge, and material throughout the organization. In any organization there are multiple paths through which products, documents, and ideas flow.
d.1. Continuous Flow Manufacturing
The traditional manufacturing strategy is to study the marketplace to obtain a forecast of sales of various products. This forecast is used as a basis for orders that are issued to suppliers and to departments responsible for fabrication and assem- bly. This is referred to as a push system. One major problem with this strategy is that if the forecast is imperfect, products are produced that are not wanted by customers and/or products that customers want are not available. A second major problem with the forecast- based strategy is the increasing expectation of custom- ers for exactly the product they want and exactly when they want it. These two problems have led to a response by manufacturers that is sometimes called mass customization. As illustrated by the automotive industry, a customer order of a vehicle with choices among dozens of options with perhaps hundreds of possible combinations cannot be accurately forecasted. Instead, the customer order initiates the authorization to build the product. This is referred to as a pull system because the pull of the customer instead of the push of the forecast activates the system.
Rather than producing batches of identical products, a pull- oriented organiza- tion produces a mix of products with the mix of features that customers order. In the ideal pull system, the receipt of the customer order initiates orders for the com- ponent parts to be delivered to the assembly line at scheduled times. The mixture of features of the components as they continuously flow to and through the line results in exactly the product the customer needs. Making this happen in a reason- able amount of time would have been unthinkable only a few years ago.
When a pull system is in a state of perfection, each activity moves a compo- nent through the value stream so that it arrives at the next activity at the time it is needed. Achieving and maintaining this may require a great deal of flexibility in
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allocating resources to various activities. Cross- training of personnel is essential. The resulting flexibility and system nimbleness permits reduction of WIP.
d.2. non- value-added activities
Some functions perform activities that do not change the form or function of the product or service. The customer is not willing to pay for these activities. These activities are labeled non- value-added. A classic example is rework. The customer expects to pay for the printing of a document, for instance, but does not want to pay for corrections that are needed because of supplier error. A key step in making an organization more lean is the detection and elimination of non- value-added activities.
In searching for non- value-added activities, the operative guideline should be “question everything.” Steps that are assumed to be necessary are often rife with opportunities for improvement. Team members not associated with a process will often provide a fresh eye and ask the impertinent questions.
Some authors list seven or eight categories of waste, or muda as it is referred to in some sources. These lists usually include overproduction, excess motion, wait- ing, inventory, excess movement of material, defect correction, excess processing, and lost creativity. The following paragraphs examine the causes and results of each of these wastes.
Overproduction is defined as making more than is needed or making it ear- lier or faster than is needed by the next process. The principal symptom of over- production is excess WIP. Companies adopt overproduction for various reasons, including long setup times, unbalanced workload, and a just- in-case philosophy. One company maintains a six- month supply of a particular small part because the machine that produces it is unreliable. In some cases, accounting methods have dictated that machines overproduce to amortize their capital costs. All WIP should be continuously scrutinized for possible reduction or elimination.
Excess motion can be caused by poor workplace layout, including awkward positioning of supplies and equipment. This often results in ergonomic problems, time wasted searching for or moving supplies or equipment, and reduced quality levels. Kaizen events, discussed in section C.2, have been effectively used to focus a small, short- term team on improvements in a particular work area. The team must include personnel with experience at the positions involved as well as those with similar functions elsewhere. In addition, it is essential to include people with the authority to make decisions. Such teams have made startling changes in two to five days of intense activity.
Waiting typically is caused by such events as delayed shipments, long setup time, or missing people. This results in a waste of resources and, perhaps more importantly, demoralization of personnel. Setup time reduction efforts and TPM are partial answers to this problem. Cross- training of personnel so that they can be effectively moved to other positions is also helpful in some cases. Most important, of course, is carefully laid and executed scheduling.
Inventory is wasteful when inventories of raw materials, finished goods, or WIP are maintained; costs are incurred for environmental control, record keeping, storage and retrieval, and so on. These functions add no value for the customer. Of course, some inventory may be necessary; however, if a competitor finds ways to reduce costs by reducing inventory, business may be lost. One of the most tempting
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times to let inventory levels rise is when a business cycle is in the economic recov- ery phase. Instead of increasing inventories based on forecasts, the proper strategy is to synchronize production to increase with actual demand. Similarly, production or administrative functions that use more space or other resources than necessary increase costs without adding value. The overused analogy of the sea of inventory shown in Figure 5.31 illustrates how excess inventory makes it possible to avoid solving other problems. As the level of inventory is lowered, some problems will rear their ugly heads and need to be solved before further progress is possible.
There are various methods for sorting inventory. Toyota created the heijunka box tool, which is a visual scheduling tool used to improve the flow of production. Figure 5.32 provides an example of a typical heijunka box. The rows in the box represent product or part numbers (in this case, part number), and the columns (slots) represent the material and information flow timing (Lean Enterprise Insti- tute, “Heijunka Box,” https://www.lean.org/lexicon/heijunka-box). The vertical columns should be organized into equally spaced time intervals and include cards that describe the material location and quantity to obtain from a kanban system.
Excess movement of material, as indicated by large conveyor systems, huge fleets of forklifts, and so on, makes production more costly and complex, often reducing
Figure 5.31 A sea of inventory often hides unresolved problems.
Unbalanced workload
(a)
(b)
Long setup times
Poor quality
Poor maintenance And so on
Order
Unbalanced workload
Long setup times
Poor quality
Poor maintenance And so on
In ve
n to
ry le
ve l
In ve
n to
ry le
ve l
Order
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quality through handling and storing. Poor plant layout is usually to blame. Plants with function- oriented departments (such as all lathes together and all presses together) require excessive material movement. A better plan is to gather together equipment that is used for one product or product family. This may mean having a manufacturing cell contain several types of equipment requiring personnel with multiple skills. Many companies have had success with cells that form a C shape (as shown in Figure 5.33) because they can be staffed in several ways. If demand
Figure 5.32 Example of a heijunka box. Source: Photo taken in the Toyota Production Systems Laboratory in the Industrial and Systems Engineering Department, Rochester Institute of Technology.
Figure 5.33 C-shaped manufacturing cell.
Machine #1
Machine #2
M a ch
in e #
3 M
a ch
in e
# 4
Machine #5 Machine #6
Material flow
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for the cell’s output is high, six people could be assigned there, one per machine. If demand is very low, one person could move from machine to machine producing parts one at a time.
Defect correction is non- value-added because the effort required to fix the defec- tive part is wasted. Typical causes of defects are poor equipment maintenance, poor quality system, poor training and/or work instructions, and poor product design. Lean thinking demands a vigorous look at these and other causes in order to continually reduce defect levels.
Excess processing is often difficult to recognize. Sometimes entire steps in the value chain are non- value-added. A steel stamping operation produces a large volume of parts before they are scheduled for painting, which necessitates the practice of dipping parts in an oil solution to prevent rust as they wait to be painted. As the paint schedule permits, the parts are degreased and painted. The customer is unwilling to pay for the dip/degrease activities because they do not enhance the product. The best solution in this case is to schedule the pre- paint activities so that the parts are painted immediately upon production. This solu- tion may require smaller batch sizes and improved communication procedures, among other things. The purpose of the grinding step that often follows a welding operation is to remove some of the weld imperfections. Improving the welding process may reduce or eliminate the need for grinding. In this case, the unneces- sary grinding would be classified as excessive processing. Excessive processing can occur in the office as well as on the plant floor. Information from customer purchase orders is sometimes entered into a database and the order itself is filed as a backup hard copy to resolve any later disagreements. One company revealed that the hard copies, although they are occasionally pulled from files and initialed, stamped, stapled, and so on, really serve no useful purpose. The company now discards the purchase order once the information has been entered. The processes of filing, storing, and maintaining these records required one person performing non- value-added activity for half the time.
Lost creativity is perhaps the most unfortunate waste. Most manufactur- ing employees have ideas that would improve processes if implemented. Stan- dard organizational structures sometimes seem designed to suppress such ideas. Union/management divides seem almost impossible to bridge. Lean thinking rec- ognizes the need to involve employees in teams that welcome and reward their input. These teams must be empowered to make changes in an atmosphere that accepts mistakes as learning experiences. The resulting improved morale and reduced personnel turnover impact the bottom line in ways that no accountant has calculated.
There are, of course, gray areas where the line between valued- added and non- value-added may not be obvious. One such area is inspection and testing. A process may be so incapable that its output needs to be inspected to prevent defective parts from entering downstream processes. It could be argued that this inspection is a value- added activity because the customer does not want defec- tive products. The obvious solution is to work on the process, making it capable and rendering the inspection activity unnecessary. Most authorities would agree that this inspection is non- value-added. On the other hand, a gas furnace manu- facturer must fire test every furnace in order to comply with Canadian Standards
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Association (CSA) requirements. Customers are willing to pay for the CSA listing, so this test step is a value- added activity.
Studies have shown that an overwhelming percentage of lead time is non- value-added, much of it spent waiting for the next step. Yet over the years, efforts to decrease lead time often have focused on accelerating value- added func- tions rather than reducing or eliminating non- value-added functions.
d.3. Lean Tools
The tools shown in the following list are associated with lean. The eight tools with an asterisk (*) are the lean tools defined in the BoK. The eight tools with a caret (^) are the lean tools discussed in this section.
*Visual control^
*5S^
*Kanban^
Poka-yoke^
*Standardized work^
*Single minute exchange of die (SMED)^
Total productive maintenance^
*Value stream mapping^
*Takt time
*Waste (muda)
Kaizen, which was discussed in section C.2 of this chapter, is a term often associ- ated with, but not limited to, lean and quality principles. Its focus is on continuous improvement of functions within an organization, and it places a heavy emphasis on employee involvement. The goals of kaizen include the elimination of waste (defined as activities that add cost but do not add value), just- in-time delivery, production load leveling of amount and types, standardized work, paced moving lines, right- sized equipment, and others. Some of these goals are accomplished with lean tools such as muda (a Japanese word meaning wastefulness and a pro- cess associated with the removal of waste). We discussed muda in section D.2, the non- value-added activity section of lean. Takt time is listed in the BoK as a lean tool, but we include it in section D.4, which is focused on two methods of evalua- tion: cycle time and takt time.
D.3.a. Visual Control and 5S
In a visual factory, locations for tools, inventory, safety equipment, and so on, are clearly marked and identified. Signs and floor paint designate traffic patterns and storage locations. Information needed by personnel to perform their functions is readily available. Monitors display current information about the activity. An example of visual control in a manufacturing laboratory is shown in Figure 5.34. In this system, a red light can be turned on to signify that a restock is required. In addition, the digital displays (not in the photo) indicate the quantity of the items.
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For example, in bin 74, the display would read that there are three cereal boxes remaining.
5S is viewed as both an element of visual control and a methodology to help develop a better work environment, both physically and mentally. 5S is based on five Japanese words all starting with the letter s: seiri, seiton, seiso, seiketsu, and shitsuke. In English, these words can be translated to sort, straighten, shine, stan- dardize, and sustain. The 5S methodology helps improve the layout of a space based on efficiency and effectiveness. Each of these elements helps improve the work area. The purpose of sorting is to eliminate all unnecessary items and keep only what is required. Straightening is done to arrange the necessary items and to have a designated place for everything; this makes the work area more efficient because less time is required to search for tools, documents, equipment, and so on. Shine means to keep the work area clean. Standardizing enforces work to be done in a consistent manner and may include implementing regular cleaning and maintenance schedules. This element makes it easy to determine when problems occur. Finally, similar to many improvement methodologies, the goal of sustain is to maintain all of these elements and make 5S a way of life in the work area. As new equipment, new products, or new employees join the work area, 5S must adapt to remain effective (ReVelle 2004).
Figure 5.34 Example of visual control. Source: Photo taken in the Toyota Production Systems Laboratory in the Industrial and Systems Engineering Department, Rochester Institute of Technology.
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ReVelle (2004) lists several benefits gained by implementing 5S:
• Improved safety
• Higher equipment availability
• Lower defect rates
• Reduced costs
• Increased production agility and flexibility
• Improved employee morale
• Better asset utilization
• Enhanced enterprise image to customers, suppliers, employees, and management
Van Patten (2006) emphasizes that 5S is not simply “cleaning up a shop floor.” 5S can provide many benefits, one of which includes cleaning. 5S should change the way the workforce thinks about their work area and can therefore provide a foun- dation for all future improvement. “Within a culture of 5S, employees are expected to be organized, neat, clean, standardized, and disciplined in everything they do” (Van Patten 2006).
D.3.b. Kanban
Kanban is a pull- based inventory control system that is used in lean manufac- turing and just- in-time initiatives. A kanban system can be used to simplify and improve resupply procedures for inventory. The idea of the kanban system is to incorporate organization by setting visual and sometimes online queues that pro- vide a more efficient and cost- effective method for keeping track of and restock- ing inventory.
While kanban systems can be arranged in different ways, most kanban sys- tems use a two- bin arrangement. In a typical two- bin kanban arrangement, when the first bin is emptied, the user uses the second bin to restock the first bin. Then the user signals resupply personnel to restock the second bin and continues to periodically check the bins. The indication that inventory is needed is the “pull” part of the system: inventory is restocked only when needed. The signal for needed inventory is usually visual and may be placement of a card that came with the bin, turning on a light, or just displaying the empty bin. The resupply employee gath- ers the information on supplies needed and replenishes the bins. Sometimes the bins are resupplied from a stockroom, although it is often from a closer supply point, sometimes referred to as the supermarket. In some cases, bins are replen- ished directly by an outside vendor. The entire string of events occurs routinely, often with no paperwork. The result is smoother flow and less inventory.
An example of a kanban system from a manufacturing laboratory is shown in Figure 5.35. As seen in the top shelf of Figure 5.35, the angled bins are the first bin in the two- bin kanban system, and the second bins are located behind them.
In some of the newer kanban systems, the process is integrated into the Enter- prise Resource Planning (ERP) system so that the replenishment of supplies can be done online and in a more seamless manner. Automated material handling systems or conveyors can also be used to deliver materials throughout an organization.
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D.3.c. Poka- Yoke
A poka-yoke device is designed to prevent errors. Suppose several people placed documents in four separate trays, depending on the type of document. A kaizen team discovered that a person sorts the trays at the end of the day because about 5% of the documents are in the wrong tray, even though signs clearly state docu- ment type. The team recommended printing the documents on different colored paper and also printing the signs on the corresponding paper color. This reduced the percentage of misplaced documents to 0.7% and made the sorting job much easier. Figure 5.36 illustrates a poka- yoke device used to ensure that round and square tubing items are placed in the correct containers.
Figure 5.35 Example of a kanban system. Source: Photo taken in the Toyota Production Systems Laboratory in the Industrial and Systems Engineering Department, Rochester Institute of Technology.
Figure 5.36 A poka-yoke technique example.
The technique ensures that the round and square parts are placed in the correct containers. Neither part will fit through the hole in the top of the incorrect container.
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Poka-yoke methods are helpful in both reducing the occurrence of rare events and serving as a preventive action tool. In one example, a manufacturer found that about 1 in 2000 of its assemblies shipped was missing 1 of its 165 components. A poka- yoke technique was used to eliminate this defect. The manufacturer now bar- codes each component and scans serial number and component bar codes as each component is added to the assembly. The software is written so that the printer at the shipping department will not print a shipping label if any component is missing. Another example of poka- yoke involves the selection of the correct part from several bins with similar contents. As the product reaches the workstation, its barcode is read. Light beams crisscross the front of the bins. If the operator reaches into the wrong bin, as determined from the bar code, the conveyor stops until appropriate corrections have been made.
D.3.d. Standard Work
The lean tool called standard work states that each activity should be performed the same way every time. The application of this principle can help reduce variation in cycle time, can produce a better, more consistent product or service, and can also simplify downstream activities. The best procedure is to have the people involved with the activity reach a consensus regarding the standard method and agree to use it. The agreed- upon method should be documented and readily available to all involved. Charts and posters in the work area are often used to reinforce the method. These documents must be updated as continuous improvements are made.
An example of a standard work chart is shown in Figure 5.37. This standard work chart is a printed document that is hung above workstation 5 in a skate- board production line. Workstation 5 is the station in the production line where the wheels are added to the skateboard. The standard work chart provides a sequence of steps for the operator at the station to follow. The pictures provide additional clarity for the operator.
Figure 5.37 An example of a standard work chart for a skateboard assembly production line. Source: Photo taken in the Toyota Production Systems Laboratory in the Industrial and Systems Engineering Department , Rochester Institute of Technology.
Station 5 1. On the close side of one of the axles, place a Washer, a Wheel, another Washer, and then the Wheel Nut.
2. Tighten the nut with a Wrench no more than three half-turns. Wheels should spin freely.
3. Repeat steps 1 and 2 for the close side of the other truck assembly.
4. Gently push the fixture toward the next station.
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D.3.e. Single Minute Exchange of Die
Lean thinking is built on timely satisfaction of customer demand, which means there must be a system for quickly responding to changes in customer require- ments. In metal- forming industries, it was common practice to produce thousands of parts of a particular type before changing the machine’s dies and producing thousands of another part. This often produced vast inventories of WIP and the associated waste. These procedures were justified because changing machine dies took several hours. In a process value chain map of a given process, the time required to change over from one part to another is displayed below. The system used to reduce changeover time and improve timely response to demand is called single minute exchange of dies (SMED). Shigeo Shingo is given credit for develop- ing the SMED concept and using it in the Toyota Production System. The goal is to reduce the time from the last good part of one type to the first good part of the successive run. The initial application of SMED often requires considerable resources in special staging tables and die storage areas, among others. Activities done while the machine is down are referred to as internal activities versus the external activities performed in preparation for or follow- up to the die change. Shingo’s method is to move as many activities from internal to external as possible. A useful technique is to make a video recording of a typical changeover and have involved personnel use it to identify internal activities that can be converted to external activities. Positioning correct tooling, equipment, and manpower should all be done in external time. Activities that don’t involve die changes also need to be nimble in their response to changing customer requirements. This can be achieved through analysis of the changeover process.
As an example, consider the application of SMED to a photography operation. The current procedure for changing cameras requires several steps:
1. Shoot last good picture with camera A.
2. Remove camera A and its power supply and place in storage cupboard.
3. Remove type A tripod.
4. Remove type A lighting and reflectors.
5. Install type B lighting and reflectors.
6. Install type B tripod. Measure distance to subject with tape measure.
7. Locate camera B in cupboard and install it and its power supply.
8. Shoot first good picture with camera B.
A team working to reduce changeover time designed a fitting so both cameras could use the same tripod. Purchasing extra cables made it possible to avoid mov- ing power supplies. More flexible lighting reflectors were designed so one set would work with both cameras. Taped marks on the floor now show where to locate tripod feet to avoid the necessity of using a tape measure. Another alterna- tive would be to obtain a more versatile camera that would not need to be changed.
Another potential application of SMED is in an assembly line of a factory. An assembly department that produced three different models spent consider- able time converting the assembly line from one model to another. They found that three different assembly lines worked best for them. They now switch models
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by walking across the room. Opportunities to apply SMED concepts abound in many businesses and industries. Recognizing and developing these opportunities depend on the creativity and perseverance of the people involved.
D.3.f. Total Productive Maintenance
In order for lean systems to work, all equipment must be ready to quickly respond to customer needs. This requires a system that foresees maintenance needs and takes appropriate action. A total productive maintenance (TPM) system uses histori- cal data, manufacturer’s recommendations, reports by alert operators, diagnos- tic tests, and other techniques to schedule maintenance activity so that machine downtime can be minimized. TPM goes beyond keeping everything running, however. A TPM system includes continuous improvement initiatives as it seeks more effective and more efficient ways to predict and diagnose problems.
D.3.g. Value Stream Mapping
A value stream map (VSM) is similar to a flowchart but includes additional infor- mation about various activities that occur at each step of the process. Value stream mapping is a powerful tool based on the principles of lean and is used to identify opportunities for improvement of a process and track performance. Current- state VSMs provide information about the process as it is currently defined. Future- state VSMs provide information about the process as it could look once it has been redefined. Figure 5.38 displays a generic VSM for a manufacturing process. Detailed examples can be found in Manos (2006) and Montgomery (2013).
Figure 5.38 Value stream map for a manufacturing process.
Inventory Inventory
Refinery Orders Orders
Expeller Refinery Cleaning process
Supplier Customer
1 day 60–80 days 10–20 days
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Figure 5.39 Value stream map.
Coil inventory: 10 days
Production control
Process A
Cycle time: 55 sec Changeover: 23 min
Process E
C/T: 72 sec C/O: 45 min
Process G
C/T: 64 sec C/O: 29 min
Process H
Shipping
C/T: 37 sec C/O: 14 min
Process B
C/T: 18 sec C/O: 7 min
Daily orders
Schedule for each shift
Orders and forecasts
Distribution center
Inventory 1 shift
Inventory 1 shift
Inventory 1 shift
Inventory 1.5 shifts
Shaft inventory
5 days
Daily
Suppliers
Hourly
The process of applying lean thinking to such a path can be divided into the following steps:
1. Produce a VSM (also referred to as a value chain diagram). Rother and Shook (1999) describe this diagram in detail. It has boxes labeled with each step in the process. Information about timing and inventory is provided near each process box. Figure 5.39 shows an example of a VSM. Some symbols used on VSMs include the following:
= inventory—originally a tombstone shape indicating dead
material.
= supermarket where employees can pick needed parts.
Supermarkets are usually replenished by stockroom staff.
= kanban post where cards or other visual signals are
displayed.
= visual signal used to make stocking decisions.
275 sec. 315 sec.
7 se
c.
3 se
c.
= graph of value- added versus non- value-added times.
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2. Analyze all inventory notations with an eye toward reduction or elimination. Inventory tends to increase costs because:
a. Storage space may be expensive (rubber awaiting use in a tire factory is stored at 120°F; wood inventory may need to have humidity control).
b. Quality may deteriorate (rust, spoilage, etc.).
c. Design changes may be delayed as they work their way through the inventory.
d. Money invested in inventory could be used more productively elsewhere.
e. Quality problems that are not detected until a later stage in the process will be more expensive to correct if an inventory of defective products has accumulated.
One company refers to its racks of safety stock as the “wall of shame.”
3. Analyze the entire value stream for unneeded steps. These steps are called non- value-added activities, as discussed in section D.2 of this chapter.
4. Determine how the flow is driven. Strive to move toward value streams in which production decisions are based on the pull of customer demand. In a process where pull- based flow has reached perfection, a customer order for an item would trigger the production of all the component parts for that item. These components would arrive, be assembled, and be delivered in a time interval that would satisfy the customer. In many situations, this ideal has not been reached and the customer order will be filled from finished goods inventory. The order will, however, trigger activities back through the value chain that produce a replacement part in finished goods inventory before it is needed by a customer.
5. Extend the VSM upstream into suppliers’ plants. New challenges occur regarding compatibility of communication systems. The flows of information, material, knowledge, and money are all potential targets for lean improvements.
When beginning the process, pick a narrow focus—do not try to boil the ocean, as the saying goes.
d.4. Cycle Time
Cycle time and takt time are metrics associated with the evaluation of a pro- cess. Cycle time, displayed below each process in Figure 5.39, is defined as the amount of time required to complete the named activity for one product or service. If the cycle time is variable, it is useful to show a range and average on the VSM.
Reducing variation in cycle time makes a system more predictable. Sometimes the cycle time variation can be reduced by using the cycle times of sub- activities instead. For example, suppose the activity consists of using a word processor to
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modify a standard bid form. Sub- activities might include inserting client informa- tion, listing proposed budget, detailing alternatives, and so on. The total time to prepare the bid might vary a great deal, while the time required to accomplish each sub- activity should show less variation. The activities performed should be continually studied in an effort to eliminate non- value-added components and find better and faster ways to complete the value- added components.
Some techniques that have been successfully applied to accomplish these goals are kaizen methods, kaizen blitz, and rapid continuous improvement (RCI). The usual procedure is to form a small team that is given a process to improve and a limited time frame, often only a few days. The team should include the people who perform the targeted activity, outsiders who can provide a fresh perspective, and people who are authorized to approve changes.
The team observes the process and raises questions about its various parts. Typical questions might include:
• Why is that stored there? Is there a better place to put it?
• Why do things in that order?
• Would a different table height work better?
• Could your supplier (internal or external) provide a better service? Does your supplier know what you need?
• Are you providing your customer, whether internal or external, with the best possible services?
• Do you know what your customer needs?
• Should parts of this activity be performed by the customer or the supplier?
• Are there steps that can be eliminated?
• Is there enough light, fresh air, and so on, to do the job efficiently?
• Would another tool, software package, or other material be more helpful?
• Are tools conveniently and consistently stored?
• Can the distance the person and/or product moves be reduced?
• Should this activity be moved closer to the supplier or customer?
• How many of these items should be kept on hand?
• Would it help to do this activity in less space?
In other words, the team questions everything about the process and its environ- ment. Kaizen activity usually results in making several small improvements. In many situations, the team actually implements a change and studies the result before making a recommendation.
Cycle time must not be confused with takt time. Takt time is determined by customer demand. Its formula is time available divided by units required. For example, if 284 units are to be produced in a shift consisting of 27,000 seconds,
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then takt time = 27,000 ÷ 284 ≅ 95 seconds. That is, the system must average one unit every 95 seconds. To meet this demand rate, the cycle time for each process must be less than 95 seconds. The basic relationship between cycle time and takt time is that cycle time is less than or equal to takt time. If cycle time exceeds takt time, more than one person is needed. To approximate the number of people required, use the following formula: cycle time divided by takt time.
Takt time is recalculated whenever the production schedule is changed. In the previous example, if 312 units are scheduled, the takt time is reduced to 87 sec- onds. Adjustments to cycle times, possible by adding people or equipment, may be necessary.
E. CorrECTivE aCTion Corrective and preventive actions often are best taken using a problem- solving method. Problem- solving methods (also called the scientific method) have many variations, depending, to some extent, on the use; however, they are all similar. The seven phases of corrective action are shown in Figure 5.40, which also shows the relationship to the PDSA cycle. The phases are integrated in that they are all dependent on the previous phase. Continuous improvement is the objective, and these phases are the framework to achieving that objective.
Another method, described by Duffy (2014), is called the eight disciplines (8D) model. The 8D model is a problem- solving approach typically employed by QEs for establishing permanent corrective action. The disciplines as stated in Duffy (2014) are as follows:
• D0: Plan
– Plan for solving the problem
– Determine the prerequisites
– Identify and prioritize opportunities for improvement
Figure 5.40 The seven phases of corrective action.
Act Plan
Study Do
Phase 1 Identify the opportunity
Phase 2 Analyze the process
Phase 3 Develop the optimal solution(s)
Phase 4 Implement
Phase 6 Standardize the solution
Phase 7 Plan for the future
Phase 5 Study the results
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• D1: Use a team
– Establish a team of people
– Team members should have product and process knowledge
• D2: Define and describe the problem
– Specify the problem by identifying it in quantifiable terms
– Answer the questions of who, what, where, when, why, how, and how many (5W2H) for the problem
• D3: Develop interim containment plan
– Implement and verify interim actions
– Define and implement containment actions to isolate the problem from any customer
• D4: Determine, identify, and verify root causes and escape points
– Identify all applicable causes that could explain why the problem occurred
– Identify why the problem was not noticed at the time it occurred
– Cause and effect diagrams can be used to map causes against the problem identified
• D5: Choose and verify permanent corrections (PCs) for problem/ nonconformity
– Through pre- production programs, quantitatively confirm that the selected correction will resolve the problem for the customer
• D6: Implement and validate corrective actions
– Define and implement the best corrective actions
• D7: Take preventive measures
– Modify the management systems, operation systems, practices, and procedures to prevent recurrence of this and all similar problems
• D8: Congratulate your team
– Recognize the collective efforts of the team
– The team needs to be formally thanked by the organization
Continuous improvement means not being satisfied with merely doing a good job or having a good process but striving to improve that job or process. It is accom- plished by incorporating process measurement and team problem solving into all work activities. TQM tools and techniques are used to improve quality, deliv- ery, and cost. We must continuously strive for excellence by reducing complex- ity, variation, and out- of-control processes. Lessons learned in problem solving, communications, and group dynamics, as well as technical know- how, must be transferred to appropriate activities within the organization.
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The most common elements of the corrective action process are problem iden- tification, failure and root cause analysis, problem correction, recurrence control, and verification of effectiveness. Each of these items is described in the subsequent subsections.
E.1. Problem identification
The objective of the planning phase is to identify and prioritize opportunities for improvement, which is done by the team. If the team is a natural work group or one where members already work together, then this part is complete. If the prob- lem is of a multifunctional nature, then the team should be selected and directed by the quality council to address the improvement of a specific process. The team leader is then selected and becomes the owner of the process improvement. Goals and milestones are established. If the improvement strategy is the repair or refine- ment of an existing process, an individual, rather than a team, may be assigned.
Defining the problem involves identification and scope. Problem identifica- tion answers the questions of who, what, where, when, and why, but specifically drives at the question, “What are the problems?” The answer leads to those prob- lems that have the greatest potential for improvement and have the greatest need for solution. Problems can be identified from a variety of inputs, such as:
• Pareto analysis of repetitive external alarm signals, such as field failures, complaints, returns, and others
• Pareto analysis of repetitive internal alarm signals (e.g., scrap, rework, sorting, and the 100% test)
• Proposals from key insiders (managers, supervisors, professionals, and union stewards)
• Proposals from suggestion schemes
• Field study of users’ needs
• Data on performance of competitors (from users and from laboratory tests)
• Comments of key people outside the organization (customers, suppliers, journalists, and critics)
• Findings and comments of government regulators and independent laboratories
• Customer surveys
• Employee surveys
• Brainstorming by work groups, for example, through fishbone diagrams
Problems identified provide opportunities for improvement. For a condition to qualify as a problem, it must meet the following three criteria:
1. Variable performance from an established standard
2. Deviation from the perception and the facts
3. The cause is unknown; if we know the cause, there is no problem
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Identifying problems for improvement is not difficult, as there often are many more than can be analyzed. The quality council or work group must prioritize them using the following selection criteria:
1. Is the problem important and not superficial? Why?
2. Will problem solution contribute to the attainment of goals?
3. Can the problem be defined clearly using objective measures?
In selecting its initial improvement opportunity, a work group should find one that gives the maximum benefit for the minimum amount of effort. Problem iden- tification should include scope. Failure in problem solving is frequently caused by poor definition of the problem. A problem well stated is half solved. Criteria for a good problem statement are as follows:
• It clearly describes the problem as it currently exists and is easily understood
• It states the effect: what is wrong, when it happens, and where it is occurring, not why it is wrong or who is responsible
• It focuses on what is known, what is unknown, and what needs to be done
• It uses facts and is free of judgment
• It emphasizes the impact on the customer
An example of a well- written problem statement is the following:
As a result of a customer satisfaction survey, a sample of 150 billing invoices showed that 18 had errors that required one hour to correct.
This example statement describes the current state. We might also wish to describe the desired state, such as “Reduce billing errors by 75%.”
In addition to the problem statement, this phase requires a comprehensive charter for the team. The charter specifies the following:
1. Authority. Who authorized the team?
2. Objective and scope. What are the expected outputs and specific areas to be improved?
3. Composition. Who are the team members and process and subprocess owners?
4. Direction and control. What are the guidelines for the internal operation of the team?
5. General. What are the methods to be used, the resources, and the specific milestones?
E.2. Failure and root Cause analysis
After problem identification, if possible, the team should implement and verify interim actions. Containment actions can be used to isolate the problem from any
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customer. After containment is established, it is critical to understand the process and how it is currently performed. Key activities are to define process boundaries, outputs and customers, inputs and suppliers, and process flow; determine levels of customer satisfaction and measurements needed; gather data; and identify root causes. Identification of why the problem was not noticed at the time it occurred can be helpful. All causes shall be verified or proved, not determined by fuzzy brainstorming. Cause- and-effect diagrams and flow diagrams are useful for root cause analysis.
A flow diagram translates complex work into an easily understood graphic description. This activity often is an eye- opening experience for the team because it is rare that all members of the team understand the entire process.
Next, the target performance measures are defined. Measurement is funda- mental to meaningful process improvements. If something cannot be measured, it cannot be improved. There is an old saying that what gets measured gets done. The team will determine whether the measurements needed to understand and improve the process are presently being used; if new ones are needed, the team will:
• Establish performance measures with respect to customer requirements
• Determine data needed to manage the process
• Establish regular feedback with customers and suppliers
• Establish measures for quality/cost/timelines of inputs and outputs
Once the target performance measures are established, the team can collect all available data and information. If these data are not enough, then additional new information is obtained. Gathering data (1) helps confirm that a problem exists, (2) enables the team to work with facts, (3) makes it possible to establish measure- ment criteria for a baseline, and (4) enables the team to measure the effectiveness of an implemented solution. It is important to collect only needed data and to get the right data for the problem. The team should develop a plan that includes input from internal and external customers and answers the following questions:
1. What problem or operation do we wish to learn about?
2. What are the data used for?
3. How much data do we need?
4. What conclusions can be drawn from the collected data?
5. What action should be taken as a result of the conclusion?
Data can be collected by a number of methods, such as check sheets, comput- ers with application software, data- collection devices like hand- held gages, or an online system.
The team will identify the customers and their requirements and expectations as well as the inputs, outputs, and interfaces of the process. Also, the team will systematically review the procedures currently being used. Common items of data and information are as follows:
• Customer information, such as complaints and surveys
• Design information, such as specifications, drawings, function, bills of materials, costs, design reviews, field data, service, and maintainability
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• Process information, such as routing, equipment, operators, raw material, and component parts and supplies
• Statistical information, such as average, median, range, standard deviation, skewness, kurtosis, and frequency distribution
• Quality information, such as Pareto diagrams, cause- and-effect diagrams, check sheets, scatter diagrams, control charts, histograms, process capability, acceptance sampling, run charts, life testing, inspection steps, and operator and equipment matrix analysis
• Supplier information, such as process variation, on- time delivery, and technical competency
The cause- and-effect diagram is particularly effective in this phase. Determining all of the causes requires experience, brainstorming, and a thorough knowledge of the process. It is an excellent starting point for the project team. One word of cau- tion: the objective is to seek causes, not solutions. Therefore, only possible causes, no matter how trivial, should be listed.
It is important to identify the root cause. This activity can sometimes be deter- mined by voting. It is a good idea to verify the most likely cause because a mistake here can lead to the unnecessary waste of time and money by investigating pos- sible solutions to the wrong cause.
Some verification techniques are the following:
1. Examine the most likely cause in regard to the problem statement
2. Recheck all data that support the most likely cause
3. Check the process when it is performing satisfactorily versus when it is not by using the who, where, when, how, what, and why approach
4. Utilize an outside authority who plays devil’s advocate with the data, information, and reasoning
5. Use experimental design and other advanced techniques to determine the critical factors and their levels
6. Save a portion of the data used in the analysis to confirm during verification
Once the root cause is determined, the next phase can begin.
E.3. Problem Correction
The objectives of corrective action are establishing potential and feasible solutions and recommending the best solution to improve the process. Once all the infor- mation is available, the project team begins its search for possible solutions. Fre- quently, more than one solution is required to remedy a situation. Sometimes the solutions are quite evident from a cursory analysis of the data.
In this phase, creativity plays a major role and brainstorming is the principal technique. Brainstorming on possible solutions requires not only a knowledge of the problem but also innovation and creativity.
There are three types of creativity: (1) create new processes, (2) combine dif- ferent processes, or (3) modify the existing process. The first type is innovation in
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its highest form, such as the invention of the transistor. Combining two or more processes is a synthesis activity to create a better process. It is a unique combina- tion of what already exists. This type of creativity relies heavily on benchmarking. Modification involves altering a process that already exists so that it does a better job. It succeeds when managers utilize the experience, education, and energy of empowered work groups or project teams. There is not a distinct line between the three types; they overlap (Rother and Shook 1999).
Creativity is a unique quality that separates mankind from the rest of the ani- mal kingdom. Most of the problems that cause inefficiency and ineffectiveness in organizations are simple problems. There is a vast pool of creative potential avail- able to solve these problems. Quality is greatly improved because of the finding and fixing of a large number of problems, and morale is greatly increased because it is enormously satisfying to be allowed to create (Mallette 1993).
Areas for possible change include the number and length of delays, bottle- necks, equipment, timing and number of inspections, rework, cycle time, and materials handling. Consideration should be given to simultaneously combining, eliminating, rearranging, and executing the process steps.
Once possible solutions have been determined, evaluation or testing of the solutions comes next. As mentioned, more than one solution can contribute to the situation. Evaluation or testing determines which of the possible solutions has the greatest potential for success and the advantages and disadvantages of these solutions. Criteria for judging the possible solutions include such things as cost, feasibility, resistance to change, consequences, and training. Solutions also may be categorized as short range and long range. At a minimum, the solution must prevent reoccurrence.
Control charts give us the ability to evaluate possible solutions. Whether the idea is good, poor, or has no effect is evident from the chart.
Once the best solution is selected, it can be implemented. Although the project team usually has some authority to institute remedial action, more often than not the approval of the quality council or other appropriate authority is required. If such is the case, a written and/or oral report is given. The contents of the imple- mentation plan report must fully describe:
• Why it will be done
• How it will be done
• When it will be done
• Who will do it
• Where it will be done
The report will designate required actions, assign responsibility, and establish implementation milestones. The length of the report is determined by the com- plexity of the change. Simple changes may require only an oral report, whereas others may require a detailed written report. After approval by the quality council, it is desirable to obtain the advice and consent of departments, functional areas, teams, and individuals that may be affected by the change. A presentation to these groups will help gain support from those involved in the process and provide an opportunity for feedback with improvement suggestions.
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The final element of the implementation plan is the monitoring activity, which answers the following:
• What information will be monitored or observed, and what resources are required?
• Who will be responsible for taking the measurements?
• Where will the measurements be taken?
• How will the measurements be taken?
• When will the measurements be taken?
Another step of problem correction is monitoring and evaluating the change by tracking and studying the effectiveness of the improvement efforts through data collection and review of progress. It is vital to institutionalize meaningful change and ensure ongoing measurement and evaluation efforts to achieve continuous improvement. Measurement tools such as run charts, control charts, Pareto dia- grams, histograms, check sheets, and questionnaires are used to monitor and eval- uate the process change.
The team should meet periodically during this phase to evaluate the results to see if the problem has been solved or if fine- tuning is required. In addition, they will want to see if any unforeseen problems have developed as a result of the changes. If the team is not satisfied, some of the phases will need to be repeated.
E.4. recurrence Control
Once the team is satisfied with the change, it must be institutionalized by positive control (positrol) of the process, process certification, and operator certification. Positrol ensures that important variables are kept under control. It specifies the what, who, how, where, and when of the process and is an updating of the moni- toring activity. Standardizing the solution prevents backsliding. Table 5.6 illus- trates a few variables of a wave soldering process.
In addition, the quality peripherals (the system, environment, and supervi- sion) must be certified. The partial checklist in Table 5.7 provides the means to
Table 5.6 Positrol of a wave soldering process.
What Specs Who How Where When
An 880 flux 0.864 g ± 0.0008
Lab technician Specific gravity meter
Lab Daily
Belt speed ft/min ± 10%
Process technician
Counter Board feed Each change
Preheat temperature
220˚± 5˚ Automatic Thermocouple Chamber entrance
Continuous
Source: Reprinted from World Class Quality by Kiki Bhote. ©1991 AMACOM, a division of the American Management Association International. Reprinted by permission of AMACOM, a division of American Management Association International, New York, NY. All rights reserved. http://www.amanet.org.
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initially evaluate the peripherals and periodically audit them to ensure that the process will meet or exceed customer requirements for the product or service.
Finally, operators must be certified to know what to do and how to do it for a particular process. Also needed is cross- training in other jobs within the pro- cess to ensure next- customer knowledge and job rotation. Total product knowl- edge is also desirable. Operator certification is an ongoing process that must occur periodically.
E.5. verification of Effectiveness
This final stage of corrective action has the objective of achieving improved lev- els of process performance. Regardless of how successful initial improvement efforts are, the improvement process must continue. Everyone in the organization is involved in a systematic long- term endeavor to constantly improve quality by developing processes that are customer oriented, flexible, and responsive.
A key activity is to conduct regularly scheduled reviews of progress by the quality council and/or work group. Management must establish the systems to identify areas for future improvement and to track performance with respect to internal and external customers. They also must track changing customer require- ments. Durivage (2017) notes that good verification of effectiveness is specific, measurable, achievable, relevant, and time bound. Corrective action is only use- ful if it is effective. Incorporating the five questions that the verification of effec- tiveness is specific, measurable, achievable, relevant, and time bound can help ensure that the corrective action plan is effective and has not caused unintended consequences.
F. PrEvEnTivE aCTion The problem- solving method discussed in the previous section often may be use- ful for preventive actions. The function of QEs has moved from that of detection of defects to prevention of defects. The concept of preventive action has been around for many years and has been practiced extensively in Japan, where it has the name “poka-yoke,” which we discussed in section D.3 of this chapter.
Table 5.7 Checklist for process certification.
Quality system Environment Supervision
Authority to shut down line Water/air purity Coach, not boss
Preventive maintenance Dust/chemical control Clear instructions
Visible, audible alarm signals Temperature/humidity control Combining tasks
Foolproof inspection Electrostatic discharge Encourage suggestions
Neighbor and self-inspection Storage/inventory control Feedback of results
Source: Reprinted from World Class Quality by Kiki Bhote. ©1991 AMACOM, a division of the American Management Association International. Reprinted by permission of AMACOM, a division of American Management Association International, New York, NY. All rights reserved. http://www.amanet.org.
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There are five error- proofing principles: elimination, replacement, facilitation, detection, and mitigation. Elimination of the possible error occurs when the pro- cess or product is redesigned so that the error can no longer occur. Replacement is a change to a more reliable process. Facilitation occurs when the process is made easier to perform and, therefore, more reliable. Detection occurs when the error is found before the next operation. Mitigation minimizes the effect of the error. Each of the error- proofing principles can be applied with the following meth- ods of preventive action: fail- safe devices, magnification of senses, redundancy, countdown, and special checking and control devices. Each of these methods is discussed in turn below. Prevention can also be considered an error- proofing principle. One method for prevention is robust design, which we discuss at the end of this section.
Fail-safe devices are used to ensure that problems or abnormalities in pro- cesses will be discovered in a manner that will maintain a safe working environ- ment and ensure that quality is not compromised. See Table 5.8.
Magnification of senses is used to increase the power of human seeing, hear- ing, smelling, feeling, tasting, and muscle power. Some examples are optical magnification, multiple visual and audio signals, remote- controlled viewing of a hazardous process, robotic placement of parts or tools, and use of pictures rather than words.
Redundancy is the use of additional activities as a quality safeguard. Multiple- identity codes, such as bar and color codes, are used to prevent product mix- ups. Redundant actions and approvals require two individuals working independently. Audit review and checking procedures ensure that plans are being followed. Design for verification utilizes special designs, such as holes for viewing, to deter- mine whether the product or process is performing satisfactorily. Multiple test sta- tions may check a number of attributes, such as those that occur on a high- speed production line.
Countdown, which structures sensing and information procedures to paral- lel the operating procedures in order to check each step, is another category of fail- safe devices. The most familiar example of this category of error- proofing is the launching of a space vehicle. It also has been effectively used in surgical opera- tions and in welding.
Special checking and control devices are another method of preventive action. A familiar example is the computer checking of credit card numbers whereby invalid numbers are rejected and instant feedback provided.
Table 5.8 Types of fail-safe devices.
Type of fail-safe device Device function
Interlocking sequences Ensure that the next operation cannot start until the previous operation is successfully completed
Alarms and cutoffs Activate if there are any abnormalities in the process
All-clear signals Activate when all remedial steps have been taken
Foolproof work-holding devices Ensure that a part can be located in only one position
Limiting mechanisms Ensure that a tool cannot exceed a certain position or amount
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It is not enough to merely plan and execute preventive actions. In accordance with the PDSA and PDCA models, the effectiveness of preventive actions must be verified. Such verification can be difficult, however, since preventive actions that are effective eliminate problems before they occur. In this respect, verification of preventive actions is completed by ensuring that problems for which preventive actions have been planned and executed have not, in fact, recurred. Creating a robust process or product is another method of preventive action. A product or process is called robust if its function is relatively unaffected by variation in the environment in which it operates, as shown in the following examples:
• A laundry appliance must operate correctly with a variety of water chemistry and cleaning products as well as variations in temperature, humidity, and other factors.
• An automobile must function correctly under various weather conditions as well as variation in operator techniques.
• The process of assembling a bid must be performed in an area where frequent interruptions, phone calls, and so on, occur. In order to make certain that all 23 required elements of the bid document are included, 23 color- coded trays are set out with the appropriate form in each tray before the document is assembled.
• The raw material for a punching process has a wide variation in thickness; in other words, the process operates in an environment of thickness variation. This results in unacceptable burrs on some parts. One solution is to impose a tighter thickness specification on the raw material supplier. The robust design solution might be a new die that would prevent burrs regardless of the thickness.
g. ConTinuouS iMProvEMEnT CaSE STudiES In this chapter, we discussed several graphical tools and continuous improvement methodologies. In Chapter 6, we discuss the statistical methods used to support these improvement techniques. For example, robust design for design of experi- ments is discussed in Chapter 6, section H. Over the years, a number of continu- ous improvement efforts have been documented in case studies. Vining (2011) discusses basic components for quality improvement strategies. Many of these com- ponents are highlighted in quality improvement case studies. Some of these case studies are briefly described below.
Wijma et al. (2009) present the implementation of a structured LSS approach for improving the efficiency of the nursing department within a hospital. The case study presents a number of tools that were discussed in this chapter, including a SIPOC diagram, Pareto chart, and flow diagram.
Nepal, Mohanty, and Kay (2013) use a Six Sigma–based quality improvement framework for a cardiovascular wire drawing process. The wire drawing requires careful attention to detail, and a high- quality end product is critical. The DMAIC procedure is discussed in this case study, but particular attention is given to the use of ANOVA and regression in the analyze phase of DMAIC. The authors note
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that the main contribution of the case study is in the process- driven approach to quality improvement.
Zwetsloot and Does (2015) describe the use of the DMAIC approach for improving the sales of a consulting company that provides courses and training in the Netherlands. The use of LSS helped improve sales by using a targeted approach to attract additional clients (an approximate 10% increase) through the company’s website. The structured DMAIC approach and the focus on data- driven results are discussed throughout the case study.
For additional details of the case studies discussed, the reader can refer to the references. All three case studies were published in Quality Engineering, a Taylor & Francis published journal. Quality and Reliability Engineering International (Wiley) and the Journal of Quality Technology (ASQ) are also excellent peer- reviewed jour- nal resources that can be used for finding new research theory, application, and case studies with regard to quality engineering and continuous improvement. For other LSS projects detailed in case study reports, see Garrett and Lee (2011); Schall (2012); Schoonhoven, Lubbers, and Does (2013); Anderson and Kovach (2014); and Filho et al. (2015).
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This chapter covers eight topics in data analysis that CQEs must understand and routinely employ: collecting and summarizing data, quantitative concepts, prob- ability distributions, statistical decision making, relationships between variables, statistical process control, process and performance capability, and the design and analysis of experiments.
The methods and tools presented in this chapter, although complex in their details, provide a basis for what is sometimes referred to as management by fact. The proper use of this content will permit the user to determine how best to collect and analyze data so that sound decisions are possible.
a. CoLLECTing and SuMMarizing daTa This section covers six aspects related to collecting and summarizing data: types of data, measurement scales, data collection methods, data accuracy and integrity, graphical methods for depicting relationships, and descriptive statistics.
a.1. Types of data
Two types of data are encountered in practice: discrete data and continuous data. Discrete (count) data are obtained when the characteristic being studied can
only take on certain values and is countable—for example, number of noncon- forming units in a lot, pass/fail data, or number of successes per trial. Another example would be the number of scratches on an object. In this case, the possible values are 0, 1, 2, . . . , a so- called countably infinite set. In quality control, discrete data are referred to as attribute data.
Continuous (variables) data are obtained when the characteristic being stud- ied can take on any value over an interval of numbers. For example, the length of a part can be any value above zero. Between each two values on a continuous scale there are infinitely many other values. For example, between 2.350 inches and 2.351 inches are the values 2.3502 inches, 2.35078 inches, and so on.
a.2. Measurement Scales
There are four types of measurement scales: nominal, ordinal, interval, and ratio. Nominal scales classify data into categories with no order implied, such as an
equipment list of presses, drills, and so on. Sometimes we assign zero and one to
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represent, say, a conforming item and a nonconforming item; however, the num- bers have no meaning in terms of order.
Ordinal scales refer to positions in a series where order is important, but precise differences between values are not defined. For example, on the Mohs hardness scale of 10 minerals, talc has a hardness of one, fluorite has a hardness of four, and topaz has a hardness of eight. However, topaz is harder than fluorite, but not twice as hard. Another example of an ordinal scale is survey responses, such as strongly dissatisfied, dissatisfied, neutral, satisfied, and strongly satisfied, which can be scaled as 1, 2, 3, 4, and 5.
Interval scales have meaningful differences but no absolute zero. In this case, ratios are not meaningful. An example is temperature measured in degrees Fahr- enheit (°F). In this case, 20°F is not twice as warm as 10°F. Although the Fahrenheit scale has a zero, it is not an absolute zero. That is, the zero value does not signify that there is an absence of temperature. Data on an interval scale can be added and subtracted but cannot be multiplied or divided.
Ratio scales have meaningful differences and an absolute zero exists. One example of a ratio scale is length in inches because zero length is defined as hav- ing no length, and 20 inches is twice as long as 10 inches. Heat in degrees kelvin (K) is another example of a ratio scale because zero degrees K is defined as having no heat and 10 degrees K has twice as much heat as 5 degrees K.
a.3. data Collection Methods
In this subsection we discuss check sheets, automatic gauging, and data coding as ways of tallying.
A tally or check sheet consists of a column of potential values usually shown from smallest to largest. As measurements are read, a tally mark is placed next to the appropriate value. Although no sophisticated analysis is provided, the tally sheet is very simple to use and understand. An illustration of such a sheet is shown in Figure 6.1. The data represent a sample of diameters from a drilling operation.
Data also may be collected by automatic gauging equipment. Potential advan- tages of this approach include improved precision as well as reduction of labor, time, error rates, and costs. When considering automated inspection, CQEs must pay attention to the possibility of high initial costs, including the possibility of
Raw data: 0.127 0.125 0.123 0.123 0.120 0.124 0.126 0.122 0.123 0.125 0.121 0.123 0.122 0.125 0.124 0.122 0.123 0.123 0.126 0.121 0.124 0.121 0.124 0.122 0.126 0.125 0.123
Value Tally 0.120 | 0.121 ||| 0.122 |||| 0.123 ||||||| 0.124 |||| 0.125 |||| 0.126 ||| 0.127 |
Figure 6.1 Example of tally or check sheet.
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part redesign to adapt the part to the constraints of the measurement system. If the measured values are fed directly into a database, care must be taken to make certain that the communication link is reliable and free of noise.
ExaMpLE 6.1
A dimension has values that range from 1.031 to 1.039. For convenience, these numbers may be coded using digits from 1 to 9 so that:
1 → 1.031
2 → 1.032 and so on
Coding data can simplify recording and analysis. Sometimes it is useful to code data using an algebraic transformation. Suppose a set of data has mean μ and standard deviation σ (see section A.6 of this chapter for more details on μ and σ). A new set of data may be formed using the formula y = ax + b. That is, each element of the new set is formed by multiplying an element of the original set by a, then adding b. The mean μy, standard deviation σy, and variance σy
2 of the new set are
μy = aμ + b (6.1)
σy = |a|σ (6.2)
σy 2 = a2σ 2 (6.3)
For further information on the affect of algebraic transformations, see Hogg, Tanis, and Zimmerman (2014).
a.4. data accuracy and integrity
In this subsection we discuss data collection errors and sampling methods.
A.4a. Data Collection
The best data collection and analysis techniques can be defeated if the data have errors. Common causes of errors include the following:
• Units of measure that are not defined (e.g., feet or meters?).
• Similarity of handwritten characters (e.g., 2 or Z?).
• Inadequate measurement system.
• Rounding (generally should only be done at last stage of computation).
• Batching input versus real- time input.
• Inadequate use of validation techniques.
• Multiple points of data entry.
• Poor instructions or training.
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• Ambiguous terminology (e.g., calendar or fiscal year? day ends at 3 pm or midnight?). For example, the NASA team working with the Mars rovers uses the term “sol” to designate a Martian day to avoid confusion with earth days (thus,“yestersol” refers to the previous Martian day).
Use strategies like these to minimize error:
• Have a carefully constructed data collection plan
• Maintain a calibration schedule for data collection equipment
• Conduct gage R&R studies on data collection equipment
• Record appropriate auxiliary information regarding units, time of collection, conditions, measurement equipment used, name of the data recorder, and so on
• Use appropriate statistical tests to identify potential outliers
• If data are transmitted or stored digitally, use an appropriate redundant error- correction system
• Provide clear and complete instruction and training
If data are obtained through sampling, the sampling procedure must be appropri- ately designed. Some of the techniques that can be used to establish a well- designed sampling strategy are discussed in the following subsection.
A.4.b. Sampling Methods
Simple random sampling is a procedure by which each item has an equal probability of being selected as part of the sample. One way to do this is to assign each item a number and create a set of numbered tags so that each tag number corresponds to exactly one item. The tags are thoroughly mixed in a container and one is drawn out. The number on the tag identifies which item is selected as part of the sample. If the population size is quite large, the use of tags may not be feasible. In this situ- ation, random numbers generated by calculators or computer software such as Microsoft Excel can be used to select the elements of the sample.
If the population of parts to be sampled is naturally divided into groups, it may be desirable to use stratified sampling. For example, suppose 300 parts came from Cleveland, 600 came from Chicago, and 100 came from Green Mountain. A stratified sample of size 50 could be formed by randomly selecting 15 items from the Cleveland batch, 30 from the Chicago batch, and 5 from Green Mountain. In other words, each group makes up a proportional part of the stratified sample.
Sample homogeneity refers to the need to select a sample so that it represents just one population. Sample homogeneity is desirable regardless of the type of sampling. In the case of the stratified sampling procedure, the population consists of the original 1000 parts, and stratification is used to help ensure that the sample represents the various strata.
When selecting the data collection scheme for time- related data, the entire sample should be collected at the same time in the process so that it comes from the population being produced at 9:00 am, not that produced at 9:15 am, which may be a different population. In fact, the purpose of a control chart is to use sam- pling to determine whether the population produced at one time is different from the other populations sampled.
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For additional information on data collection methods see Vining (2013) and Doganaksoy and Hahn (2012). Doganaksoy and Hahn provide a structured pro- cess to gather data.
a.5. graphical Methods for depicting relationships
Stem-and-leaf plots and box plots are two common ways to summarize data graphically. Both are discussed in this subsection.
A stem- and-leaf plot is constructed much like the tally column shown in Fig- ure 6.1, except that the last digit of the data value is recorded instead of the tally mark. This kind of diagram often is used when the data are grouped. Consider the example shown in Figure 6.2.
The stem- and-leaf diagram conveys more information than the tally column or the associated histogram. Note that the ordered stem- and-leaf sorts the data and permits easy determination of the median. Histograms were discussed in Chapter 5 and will be discussed further in A.6.
A sorted data set may be divided into four approximately equal subsets sepa- rated by three boundary points called quartiles. The quartiles are denoted Q1, Q2, and Q3. Q2 is defined as the median. Q1 is usually defined as the median of the values less than or equal to Q2. Q3 is the median of the values greater than or equal to Q2. The interquartile range, or IQR, is Q3–Q1. The box plot (also called a box- and- whisker diagram), developed by John Tukey of Princeton University, uses the high and low values of the data as well as the quartiles to graphically summarize a data set. This is illustrated in Figure 6.3. Some software packages, rather than extending the “whiskers” to the maximum and minimum values, terminate them at 1.5(IQR) above Q3 and 1.5(IQR) below Q1. Values beyond these whiskers are designated “potential outliers.”
Data: 0.18 0.24 0.21 0.17 0.36 0.34 0.19 0.25 0.18 0.22
0.37 0.24 0.42 0.33 0.48 0.56 0.47 0.55 0.26 0.38 0.54
0.19 0.24 0.42 0.44 0.11 0.39
Measurement Tally
0.10–0.19 ||||||
0.20–0.29 |||||||
0.30–0.39 ||||||
0.40–0.49 |||||
0.50–0.59 |||
Stem and leaf
1 8 7 9 8 9 1
2 4 1 5 2 4 6 4
3 6 4 7 3 8 9
4 2 8 7 2 4
5 6 5 4
Ordered stem and leaf
1 1 7 8 8 9 9
2 1 2 4 4 4 5 6
3 3 4 6 7 8 9
4 2 2 4 7 8
5 4 5 6
Leaf unit = 0.01
Figure 6.2 Stem-and-leaf diagrams.
62 63 64 65 6766 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85
Figure 6.3 Box-and-whisker diagram.
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The data depicted in the box plot in Figure 6.3 (after sorting) are as follows: 63, 65, 67, 69, 71, 71, 75, 76, 76, 76, 81, 85
The low value is 63 and the high value is 85, with Q1 = 68, Q2 = 73, and Q3 = 76
Note that quartiles need not be values in the data set itself. The box plot of these data is shown in Figure 6.3. Figure 6.4 shows how the shape of the dot plot is reflected in the box plot.
Box plots can be used to mine information from a database. In this hypotheti- cal example, a stainless steel casting has a tight tolerance on the machined inside diameter. The quality team has heard a number of proposed fixes. Some people believe the problem is caused by a slightly out- of-round condition on a cross sec- tion of the casting. Others feel there is a taper, and still others insist the problem is too much part- to-part variation. The question is, “Which type of variation is giv- ing the most trouble?” The team decides to measure the inside diameter at three angles (12 o’clock, 2 o’clock, and 4 o’clock) and at three locations along the bore (top, middle, and bottom) on five pieces. The resultant data and box plots are shown in Figure 6.5.
The box plots in Figure 6.5 show that the largest source of variation is part- to-part. The Pareto principle says that the part- to-part variation should be attacked first. Furthermore, any improvements in out- of-round or taper may be masked by the large part- to-part variation. How would the box plot have looked if out- of-round or taper had been the principal source of variation?
a.6. descriptive Statistics
The two principal types of statistical studies are descriptive and inferential. The purpose of descriptive statistics is to present data in a way that will facilitate understanding. Some important statistics that can be used to describe a set of data include:
• A measure of the center of the population or a sample
• A measure of the variability (measure of the spread of the data) for the population or a sample
• A graphical display (displaying overall shape) of the data
Center, spread, and shape are key to understanding data and the process that generated them. The next few paragraphs discuss these attributes. (Complete definitions and discussion of population, sample, parameters, and statistics are provided in section C.)
a) Approximately symmetric b) Increased variability c) Left-skewed
Figure 6.4 Box plots.
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A.6.a. Measures of Central Tendency
Three ways to quantify the center of a data set are the mean (or average), median, and mode.
The mean is the arithmetic average of a set of data or observations. The mean is also referred to as a “balancing point” for the set of observations. Suppose the data in a sample of size n are denoted by x1, x2, x3, . . . , xn. The sample mean, denoted x
– (read “x-bar”), is given by
x
x x x x n
x
n n
i i
n
= + + + +
= = ∑
1 2 3 1...
(6.4)
Part #1 T M B
0.998 0.992 0.99612 0.994 0.996 0.994 0.996 0.994 0.995
2 4
Part #2 T M B
0.984 0.982 0.981 0.982 0.980 0.982 0.984 0.983 0.980
Part #3 T M B
0.998 0.998 0.997 0.999 0.998 0.997 0.996 0.996 0.996
Part #4 T M B
0.986 0.987 0.986 0.985 0.986 0.986 0.984 0.985 0.984
Part #5 T M B
0.975 0.980 0.976 0.975 0.976 0.974 0.978 0.980 0.974
0.998 0.997 0.996 0.995 0.994 0.993 0.992 0.991 0.990 0.989 0.988 0.987 0.986 0.985 0.984 0.983 0.982 0.981 0.980 0.979 0.978 0.977 0.976 0.975 0.974
Figure 6.5 Multiple box plot example.
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ExaMpLE 6.2
Emergency room waiting times are continually increasing. One factor that was identified as affecting wait time was turnaround time for basic blood analysis. Turnaround times (in minutes) for 10 such tests on one particular day are:
62 68 72 60 50 58 58 49 66 70
The average turnaround time is
x = 6 + + + + + + + + +
=
2 68 72 60 50 58 58 49 66 70 10
61 3. min
If the data represent the entire population of interest, then the average is the popu- lation mean and commonly denoted by μ. Suppose there are N observations in the population. The population mean is
µ = + + + +
= = ∑
x x x x N
x
N
N
i i
N
1 2 3
1
...
(6.5)
ExaMpLE 6.3
An accident investigator has been contracted by a large tire company to investigate accidents where the company’s tire may have been at fault. The investigator was con- tracted by this company for a total of six months before the company determined that his work was unacceptable. At the time his contract was terminated, the investigator had submitted a total of eight invoices for time spent at accident scenes. The amounts for each invoice are:
$4390 $3285 $1582 $725 $3001 $2971 $463 $8923
Since these invoice amounts are the only amounts for this investigator’s work for the tire company, they represent the entire population of amounts. Therefore, the average invoice amount will be the population mean:
µ = + + + +
= + + + + +
x x x x N
N1 2 3
4390 3285 1582 725 3001
...
22971 463 8923 8
3167 50 + +
= $ .
The median is the value that divides ordered data into two equal parts—half of the data lie at or below that value and half of the data lie above that value. Suppose we have a sample of size n. If the sample contains an odd number of observations, the sample median is the central value. If there is an even number
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of observations, the median is the average of the two central values. The loca- tion of the median for n observations is (n + 1)/2. The sample median is often denoted by M.
ExaMpLE 6.4
Consider the turnaround times for blood analysis given in Example 6.2, now written in increasing order:
49 50 58 58 60 62 66 68 70 72
Since there are n = 10 observations in the data set, the location of the median is (10 + 1)/2 = 5.5. Therefore, the median is the average of the fifth and sixth observations from the smallest in the data set:
M = +
=
60 62 2
61 min
The sample mode is the observation that occurs most often in the sample. There can be more than one mode for a set of data. For example, the mode for the blood analysis turnaround times is 58 minutes, since it occurs more often than any other observation.
As with the mean, the population median and population mode can be deter- mined if the entire population is known. The mean and median are the most com- monly used measures of the center of a data set.
One final note on measures of the center: the median is known as a resistant measure of the center, while the mean is not a resistant measure. A resistant mea- sure is one that is not highly influenced by extreme observations. For example, the median is often used as the measure of the center for data that involve prices or salaries, or for any data that may naturally contain extreme observations.
ExaMpLE 6.5
Housing prices in Glendale, Arizona, vary over a wide range. Suppose five houses on the market in May 2008 were listed at the following prices:
$54,900 $75,000 $79,000 $101,500 $386,000
The average house price for this set of data is $139,280. Does the average appear to represent the sample of data itself? The price of $139,280 lies above all but one house price. The median house price for this set of data is $79,000. The average was pulled toward the extreme value of $386,000 while the median was not influenced by this par- ticular value.
Now suppose we discovered that the house priced $386,000 was reduced to $345,000. All other housing prices remained constant at the time of the data collection. With this reduction, the sample mean house price is now $131,080; the sample median house price remains at $79,000. Therefore, the median was resistant to the change in price while the mean was not resistant.
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A.6.b. Measures of Variability (Spread)
Measures of variability describe the spread of the data around the center or central point of the distribution of the data. Three common measures of variation are the range, variance, and standard deviation.
The sample range is the difference between the maximum value (xmax) and the minimum value (xmin) in the sample. The sample range is often denoted by R and given by
R = xmax – xmin (6.6)
For example, the sample range for the blood analysis turnaround times given pre- viously is R = 72 – 49 = 23 minutes.
The sample variance is a measure of the variability based on the deviations of the actual observations from the mean. Suppose we have a sample of size n with observations x1, x2, . . . , xn. The sample variance is
s
x x
n
i i
n
2
2
1
1 =
−( ) −
= ∑
(6.7)
ExaMpLE 6.6
Consider the blood analysis turnaround times (in minutes) given in Example 6.2:
49 50 58 58 60 62 66 68 70 72
The sample average was found to be x– = 61.3 minutes. The sample variance is
s x x
n
i i
n
2
2
1
2 2
1
49 61 3 50 61 3
= −( )
−
= − + − +
= ∑
( . ) ( . ) .... ( . )+ − −
=
72 61 3 10 1
62.23
2
2min
The population variance, denoted by σ 2, can be determined if the data from the entire population are given. Suppose the population consists of N observations and the population mean is given by μ. The population variance is
σ
µ 2
2
1= −( )
= ∑ x
N
i i
N
(6.8)
Notice that the unit of measure of the variance is the square of the unit of measure of the original data and the mean. It is more convenient to have summary statis- tics (such as the measure of the center and measure of variability) in the same unit of measure as the original data. The measure of variability that is in the same unit of
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measure as the original data and mean is the standard deviation. The standard devi- ation is simply the positive square root of the variance. The sample standard devia- tion is
s
x x
n
i i
n
= −( )
− = ∑ 2
1
1 (6.9)
and the population standard deviation is
σ
µ =
−( ) = ∑ x
N
i i
N 2
1
(6.10)
For example, the sample standard deviation for the blood analysis turnaround times is s = 62.23 min2 = 7.89 minutes.
A.6.c. Shape of the Data
Graphical displays, discussed earlier, include dot plots, box pots, stem-and-leaf- plots, and histograms. These displays can be used to interpret the shape of the sample or population (i.e., the form the data take on). For example, a dot plot (also known as a dot diagram) for the blood analysis turnaround times is shown in Figure 6.6. The display reveals the spread of the data as well as possible outliers.
Data can assume numerous possible shapes. Consider Figure 6.7. Figure 6.7a represents a histogram of the diameters from the drilling operation in Table 6.1. The distribution is symmetric and bell shaped, and there do not appear to be any potential outliers or unusual observations. Figure 6.7b displays lifetime data of a manufactured part. The lifetime data follow a skewed distribution, specifi- cally a right-skewed distribution. Figure 6.7c displays data representing time to show symptoms in rats that have been subjected to a particular treatment. This is a left- skewed distribution. Finally, Figure 6.7d represents a bimodal distribution. This type of distribution has many applications, but sometimes this shape can indicate a mixed distribution of data (data may be coming from two different distributions).
A frequency distribution is a compact summary of data collected. The frequency distribution can be displayed in table form, graphical form, or some functional form. An ungrouped frequency distribution in table form displays the individual observations and the number of times that each value appears in the data set. A frequency distribution of the diameters from the drilling operation is given in Table 6.1.
x 60 63 66 69 72575451
Figure 6.6 Dot plot of blood analysis turnaround times.
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The cumulative frequency distribution contains the observations themselves as well as the frequency of the occurrence of the current and preceding observations. The cumulative frequency distribution of the diameters from the drilling opera- tion is given in the last column in Table 6.1.
Another way to present the diameter data from the previous example would be to group the measurements together as shown in Table 6.2.
0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.0 2.5 5.0 7.5 10.0 x
12.5 15.0 17.5
2 4 6 8 10 12 8 12 16 20 xx
24 28 32
Diameter(a)
(c)
(b)
(d)
Fr eq
ue nc
y
Fr eq
ue nc
y
Figure 6.7 Histograms of variously shaped distributions.
Table 6.1 Frequency and cumulative frequency distributions for the ungrouped diameter data.
Measurement Frequency Cumulative frequency
0.120 1 1
0.121 3 4
0.122 4 8
0.123 7 15
0.124 4 19
0.125 4 23
0.126 3 26
0.127 1 27
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If the data represent numbers of items in nonnumerical groups or categories, a categorical frequency distribution is used. An example of a categorical frequency distribution is displayed in Table 6.3 for several important types of defects in a manufacturing process.
Cumulative frequency distributions should not be used when the groups are categories where order does not matter.
Frequency distributions and cumulative frequency distributions provide a simple way to quickly examine the variability of data around the center. These dis- tributions also aid in the calculation of statistics from the data, such as the sample mean and sample standard deviation. In addition, data from the frequency and cumulative frequency distributions can easily be displayed graphically, such as in a histogram.
B. QuanTiTaTivE ConCEPTS This section lays the foundation for understanding how to draw statistical conclu- sions and how to apply probability terms and concepts.
B.1. Terminology
A population is the entirety of all items or units being studied. A sample is a sub- set of items or measurements selected from the larger population. Since it is often
Table 6.2 Frequency and cumulative frequency distributions for the grouped diameter data.
Group Frequency Cumulative frequency
0.120–0.121 4 4
0.122–0.123 11 15
0.124–0.125 8 23
0.126–0.127 4 27
Table 6.3 Categorical frequency distribution of manufacturing defects.
Defect type Frequency Relative frequency
Chip 3 0.143
Scratch 5 0.238
Ink smear 4 0.190
Fold mark 7 0.333
Tear 2 0.095
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impractical to obtain information on all the items or units in the population, we acquire information on a subset of them in order to draw conclusions about the rest.
A parameter is a quantity that describes characteristics of a population, for example, mean (μ), standard deviation (σ), correlation coefficient (ρ), or fraction nonconforming (p). The value of the population parameter is often unknown and must be estimated. A statistic is a characteristic of a sample and is an estimator of a population parameter. A representative sample is taken from the population and summary statistics calculated, such as the average or mean ( x– ), standard deviation (s), correlation coefficient (r), or fraction nonconforming (p^ ). In this case, x– is an estimator for μ.
Probability is a numerical measure representing the likelihood that a particular outcome will occur. The probability that a particular event occurs is a number between zero and one inclusive. For example, if a lot consisting of 100 parts has 4 nonconforming parts, we would say the probability that a randomly selected part will be nonconforming is 0.04, or 4%.
These concepts are summarized in the following example.
ExaMpLE 6.7
The thickness of a printed circuit board (PCB) is an important characteristic. If the thick- ness does not meet specification, the circuit board is reworked or scrapped. The aver- age thickness of the PCB is assumed to be 0.0630 inches (i.e., μ = 0.0630). The thickness of 25 randomly chosen PCBs is measured and the average thickness is found to be 0.06314 inches (i.e., x– = 0.06314 inches). The probability that the sample average diam- eter is larger than 0.06314 (for a sample size of 25) is 0.0105.
In this particular problem:
• The population is all printed circuit boards manufactured by this company with this process
• The sample is the 25 randomly chosen printed circuit boards
• The parameter is the population average or mean μ and is assumed to be μ = 0.0630 inches
• The statistic is the sample average or sample mean x–, calculated using the sam- ple of 25 printed circuit boards, which was found to be x– = 0.06314 inches
• A probability associated with this sample is 0.0105
B.2. drawing Statistical Conclusions
The purpose of inferential statistics is to infer (arrive at a conclusion by reasoning from evidence) properties of a population through analysis of a sample. This type of study is sometimes referred to as a numeric study. These studies are valid only if the sample is from a stable underlying population. For example, if a control chart is used on a stable process, the data from the chart can be used to conduct a capa- bility study for the material produced while the chart was in use. This capability
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study infers information about the process population based on the sample used for the control chart and would therefore be a numeric study.
Sample data may also be used to study either stable or non- stable processes with the goal of process improvement, which may involve the use of knowledge, experience, creativity, and basic science. Such a study is not numeric, because rather than infer properties of the population, the study seeks to determine the causes that impact the process. Inferential methods are inappropriate because the underlying population is often not stable and the goal is to change it rather than determine its characteristics. W. Edwards Deming called these analytical studies. A control chart, when used to take action on the process to maintain statistical control, is an example of a tool for analytical study.
In section D we describe a number of statistical tests. Each has assumptions or conditions that must be met in order for the test to be valid. It is critical that a test’s assumptions or conditions be satisfied before applying the test. In some cases the discussion accompanying the statistical test may state that the test is robust to minor deviations from the assumptions or conditions. For example, if one of the conditions of a test is that the population be normal, the test may be robust to minor deviations in this condition. This means that even if the population is almost normal, the test could be applied, with caution about the precision of the conclusion. Needless to say, decisions in situations like this require judgment and experience.
B.3. Probability Terms and Concepts
Before discussing probability and probability rules, it is important to define terms that describe the experiment under study. In this context, experiment refers to a random experiment where different outcomes could be obtained even if the experi- ment is repeated under identical conditions.
B.3.a. Sample Spaces and Events
The sample space is the set of all possible outcomes of an experiment or a set of con- ditions (this is also referred to as the universal set in set theory). The sample space is usually denoted by the capital letter S. If the outcomes are finite or countably infinite, then the sample space is considered discrete. If the outcomes are values over an interval of real numbers, then the sample space is considered continuous. (Further discussion of continuous and discrete random variables and distributions is given in section C.) An event is a subset of the sample space and is often denoted by a capital letter such as A, B, C, and so on. If an outcome x is an element or outcome in A, for example, we write this as x ∈ A. To illustrate these concepts, consider an experiment where a single piston ring for an automobile motor is ran- domly selected from a lot and classified as conforming (C) or nonconforming (N). The sample space for this experiment is S = {C, N}. If we are only interested in the event where the piston ring is nonconforming (call this event E), this event would be E = {N}. As a second illustration, suppose the experiment was to determine how long it takes a worker to complete a task (in minutes). Let x represent the time to complete the task. The sample space consists of all positive real numbers. This can be written as S = {x|x > 0}. If the event or sample space has no outcomes in it, then we say it is the empty set, denoted Ø.
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It is often of interest to combine events to form other events in which we are interested. There are three basic set operations used to create new events of interest:
• The union of two events A and B is that event consisting of all outcomes that are contained in A, in B, or in both. The union is denoted as A ∪ B (read “A or B”).
• The intersection of two events A and B is that event consisting of all outcomes that are contained in both A and B. The intersection is denoted as A ∩ B (read “A and B”).
• The complement of any event in a sample space is an event that contains all the outcomes in the sample space that are not in the event itself. The complement of event A is denoted as A′ (read “A complement” or “not A”). Other notation used to represent the complement of an event includes Ã, AC, and sometimes A
– . We will use the prime notation, A′.
It should be noted that if the intersection of any two events results in the empty set (i.e., A ∩ B = Ø), then those two events are called mutually exclusive.
ExaMpLE 6.8
A simple illustration involves the rolling of a single, fair six-sided die. In this random experiment, the sample space is S = {1, 2, 3, 4, 5, 6}. Suppose A is the event where the outcome on a single roll is an even number; so A = {2, 4, 6}. Suppose B is the event where the outcome on a single roll is greater than 3; so B = {4, 5, 6}. In this situation:
• A ∪ B = {2, 4, 5, 6}
• A ∩ B = {4, 6}
• A′ = {1, 3, 5}, B′ = {1, 2, 3}
• A ∩ A′ = Ø (and B ∩ B′ = Ø); the intersection of an event and its complement is always the empty set
• A ∪ A′ = S (and B ∪ B′ = S); the union of any event and its complement always equals the sample space
As stated previously, probability is a numerical measure that represents the likelihood that a particular outcome will occur. The probability that a particular event occurs is a number between zero and one inclusive. The probability of an event, say E, is written as P(E).
If the elements of E are mutually exclusive, then P(E) is equal to the sum of the probabilities of the outcomes that make up that event. To illustrate, suppose an event E contains elements a, b, c, d, and e, that is, E = {a, b, c, d, e}. Then the prob- ability of event E with these five mutually exclusive events would be
P(E) = P(a) + P(b) + P(c) + P(d) + P(e) (6.11)
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Suppose there are N possible mutually exclusive outcomes in an experiment, all equally likely to occur (such as in the rolling of a fair six- sided die). The probability of any one outcome is 1/N. Another way to look at probability is as a relative fre- quency. That is, the probability of an outcome would be the number of times that outcome occurs divided by the total number of possible outcomes.
B.3.b. Probability Properties
Consider a sample space S and two events A and B from that sample space. Then
1. P(S) = 1
2. 0 ≤ P(A) ≤ 1
3. P(A′) = 1 − P(A)
4. If two events A and B are mutually exclusive, then P(A ∩ B) = 0
An important result of rule 4 is sometimes referred to as the special addition rule for mutually exclusive events and is given as
P(A ∪ B) = P(A) + P(B) (6.12)
In other words, when two events are mutually exclusive, the probability that an outcome in event A will occur, an outcome in event B will occur, or an outcome in both A and B will occur can be found by adding the individual probabilities of each event.
ExaMpLE 6.9
Suppose the number of medication errors that occur for a patient at a particular hospital have the following probabilities:
Table 6.4 Probabilities associated with medication errors.
Number of medication errors 0 1 2 3
Probability 0.90 0.07 0.02 0.01
Let A be the event of at most one medication error occurring, that is, A = {0, 1}. Let B be the event where exactly two medication errors occur, that is, B = {2}. We note that these events are mutually exclusive. For this situation:
• P(A) = P(0) + P(1) = 0.90 + 0.07 = 0.97
• P(B) = P(2) = 0.02
• P(A′) = 1 – P(A) = 1 – 0.97 = 0.03
• P(A ∩ B) = 0
• P(A ∪ B) = P(A) + P(B) = 0.97 + 0.02 = 0.99
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The special addition rule applies only to experiments where the two events of interest have no outcomes in common (mutually exclusive). When the events are not mutually exclusive, a more general addition rule applies:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B) (6.13)
ExaMpLE 6.10
Cellular phones are put through several inspections before being shipped to the cus- tomer. Two defect types are of significant importance: critical (C) and major (M) defects. Phones with either critical or major defects are completely reworked. Using recent inspection data, it was determined that 2% of the cell phones have critical defects only, 5% have major defects only, and 1% have both critical and major defects. The manu- facturer wants to know what percentage of all phones would require complete rework.
In this situation, the information given is:
• P(C) = 0.02
• P(M) = 0.05
• P(C and M) = P(C ∩ M) = 0.01
Complete rework is necessary if the phone has critical or major defects or both. This is the event C ∪ M. The percentage of all phones needing rework is then given by P(C ∪ M). Using the addition rule (Equation (6.13)), the percentage of phones needing rework would be
P(C ∪ M) = P(C) + P(M) – P(C ∩ M) = 0.02 + 0.05 – 0.01 = 0.06
Based on this information, roughly 6% of the cell phones will need rework.
B.3.c. Contingency Tables
Suppose each part in a lot is one of four colors (red [R], yellow [Y], green [G], or blue [B]) and one of three sizes (small [S], medium [M], or large [L]). These attri- butes can be displayed in a contingency table like the one in Table 6.5. (Contin- gency tables are also used to determine statistical independence of characteristics. This application is discussed in section D.)
Table 6.5 Contingency table of part color and part size.
Red Yellow Green Blue Totals
Small 16 21 14 19 70
Medium 12 11 19 15 57
Large 18 12 21 14 65
Totals 46 44 54 48 192
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It is often useful to include the row and column totals for calculating quantities of interest, such as probabilities. The total number of parts (N = 192) is written in the bottom right- hand corner of the table. The row and column totals provide a great deal of information about the categories of interest. For example, the column total for red is 46. This indicates that the total number of red parts (regardless of size) in the lot is 46. In addition, the row total for medium parts is 57, which means that the total number of medium parts (regardless of color) is 57. The entries in each cell of the table itself (not including the row and column totals) represent the number of parts that have both characteristics. For example, 16 parts are both small and red.
ExaMpLE 6.11
We want to determine several probabilities using the contingency table given in Table 6.5. Assume that one of the parts is selected at random.
The probability that the part is small would be
P S( ) = = 70
192 0.365
The probability that the part is red would be
P R( ) = = 46
192 0.240
The probability that the part is small and red would be
P S R( )∩ = = 16
192 0.083
Equation (6.13) can be used to calculate the probability that the part is small or red as
P S R P S P R P S R( ) ( ) ( ) ( )
. . .
∪ = + − ∩
= + −
=
0 365 0 240 0 083
0..522
Using the same formulas, the probability that a randomly selected part is yellow would be P(Y) = 0.229. The probability that the part is red or yellow would be
P R Y P R P Y P R Y( ) ( ) ( ) ( )
. .
.
∪ = + − ∩
= + −
=
0 240 0 229 0
0 469
Notice that the events “red” and “yellow” are mutually exclusive (a part cannot be both red and yellow). We could have used the special addition rule to find this probability:
P R Y P R P Y∪( ) = ( ) + ( ) = +
=
0 240 0 229
0 469
. .
.
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B.3.d. Conditional Probability
We begin the discussion of conditional probability with an example.
ExaMpLE 6.12
Continuing with the previous example, suppose the selected part is known to be green. With this knowledge, what is the probability that the part is large?
Solution: It is a given that the part is one of the 54 green parts. Now, the number of the 54 green parts that are large is 21. Therefore, the probability that a part will be large, given that it is green is 21/54 = 0.389.
Example 6.12 involves conditional probability. It is referred to as conditional prob- ability because it is conditioned on the fact that the part is green. In the example, the “probability that the part is large given that it is green” is denoted P(L|G). It is useful to remember that the category to the right of the | sign represents the given condition.
Suppose there are two events A and B. The probability that event A occurs given that event B has already occurred is
P A B
P A B P B
( | ) ( )
( ) =
∩
(6.14)
ExaMpLE 6.13
Using the information given in Table 6.5, find:
a. The probability that a part is small given that it is blue
b. The probability that a small part is blue
c. The probability that a green part is red
Solution (using Equation (6.14)):
a. P S B P S B
P B ( | )
( ) ( )
= ∩
= = = 19 48
.0 396
19 192 b b
48 192 b b
b. The given condition is that the part is small. Given that the part is small, the prob- ability that it is blue is
P B S P B S
P S ( | )
( ) ( )
= ∩
= = = 19 70
.0 271
19 192 b b
70 192 b b
Note that P(B ∩ S) = P(S ∩ B).
c.
P R G P R G
P G ( | )
( ) ( )
= ∩
= = = 0
54 0
54 192 b b
0 192 b b
Note that red and green are mutually exclusive, so P(R ∩ G) = 0.
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If the practitioner has any two of the three probabilities needed to calculate the conditional probability given earlier, the third unknown probability can be found. The conditional probability can be rewritten as
P A B P A B P B( ) ( | ) ( )∩ = (6.15)
This is sometimes referred to as the general multiplication rule. Verifying that this formula is valid will aid in understanding this concept.
ExaMpLE 6.14
Using the contingency table given in Table 6.5, it is known that the probability a part is red and medium is
P R M( )∩ = = 12
192 0.0625
Using the general multiplication rule, we would get the same result:
P R M P R M P M( ) ( | ) ( )∩ = = = = 12 57
57 192
12 192
0.0625b bb b
B.3.e. Independence and the Probability of Independent Events
Events are said to be independent if the occurrence of one event does not depend on the occurrence or lack of occurrence of another (or preceding) event. The prob- ability of two independent events occurring can be found by multiplying the indi- vidual probabilities of each event. If two events A and B are independent of one another, then the probability of both event A and event B occurring is
P(A ∩ B) = P(A)P(B) (6.16)
For more than two independent events, the independence rule can be extended as
P(A ∩ B ∩ C ∩ . . .) = P(A)P(B)P(C) . . . (6.17)
ExaMpLE 6.15
Assume that the probability that a blood specimen contains high levels of lead contami- nation is 0.05. Levels of contamination from one person to the next (thus, one sample to the next) are assumed to be independent. If two such samples are analyzed, then the probability that both will contain high levels of contamination is
P(both contaminated) = P(1st contaminated ∩ 2nd contaminated) = P(1st contaminated)P(2nd contaminated) = (0.05)(0.05) = 0.0025
Recall the definition of conditional probability given in Equation (6.14). If two events A and B are known to be independent, then P(A ∩ B) = P(A)P(B). Therefore,
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if two events are independent, the probability that event A occurs given that event B has already occurred is
P A B P A B
P B P A P B
P B P A( | )
( ) ( )
( ) ( ) ( )
( )= ∩
= =
In other words, knowing that event B has occurred does not affect the probability that event A will occur. In situations where objects or items are selected at random, one after the other, the items are said to be independent if the first item chosen is placed back into the group before the second item is chosen (with replacement).
ExaMpLE 6.16
A box holds 129 parts, of which 6 are defective. A part is randomly drawn from the box and placed in a fixture. A second part is then drawn from the box. This is referred to as draw- ing without replacement. What is the probability that the second part is defective (note that there is no condition on the first part chosen)? Let Di represent the event where the ith part chosen is defective, and let Gi represent the event where the ith part is good.
Solution: We are looking for P(D2). There are two mutually exclusive events that can result in a defective part on the second draw: good on first draw and defective on second or else defective on first and defective on second. Symbolically these two events are (G1 ∩ D2) or else (D1 ∩ D2). The first step is to find the probability for each of these events. By the general multiplication rule (Equation (6.15)):
P G D P G P D G( ) ( ) ( | )1 2 1 2 1 123 129
6 128
0.045∩ = = =b bb b and
P D D P D P D D( ) ( ) ( | )1 2 1 2 1 6
129 5
128 0.002∩ = = =b bb b
Since the two events (G1 ∩ D2) and (D1 ∩ D2) are mutually exclusive, we can use the spe- cial addition rule in order to find the probability that the second part is defective:
P(D2) = 0.045 + 0.002 = 0.047
When drawing two parts at random without replacement, what is the probability that one will be good (G) and one defective (D)?
Solution: Drawing one good and one defective can occur in two mutually exclusive ways (using Equation (6.12)):
P(G ∩ D) = P(G1 ∩ D2) + P(G2 ∩ D1)
From the previous example we know that P(G1 ∩ D2) = 0.045. Use the general multiplica- tion rule to find P(G2 ∩ D1):
P G D P D P G D( ) ( ) ( | )2 1 1 2 1 6
129 123 128
0.045∩ = = =b bb b
Therefore, the probability that one randomly selected part will be good and one will be defective is
P(G ∩ D) = 0.045 + 0.045 = 0.090
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B.3.f. Summary of Key Probability Rules
For events A and B:
Special addition rule: P(A ∪ B) = P(A) + P(B) (Use only if A and B are mutually exclusive)
General addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B) (Always true)
Special multiplication rule: P(A ∩ B) = P(A)P(B) (Use only if A and B are independent)
General multiplication rule: P(A ∩ B) = P(A)P(B|A) (Always true)
Conditional probability: P(B|A) = P(A ∩ B)/P(A)
Mutually exclusive (or disjoint):
1. A and B are mutually exclusive if they cannot occur simultaneously
2. If A and B are mutually exclusive, then P(A ∩ B) = 0
3. If A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B)
Independence:
1. A and B are independent events if the occurrence of one does not change the probability that the other occurs
2. If A and B are independent events, then P(B|A) = P(B) (and P(A|B) = P(A))
3. If A and B are independent events, then P(A ∩ B) = P(A)P(B)
C. ProBaBiLiTy diSTriBuTionS This section focuses on the two kinds of probability distributions: continuous dis- tributions and discrete distributions:
• Continuous distributions are used when the parameter being measured can be expressed on a continuous scale. Examples include the diameter of piston rings, tensile strength, output voltage, and so on.
• Discrete distributions are used when the parameter being measured takes on only certain values, such as integers 0, 1, 2, . . . . Examples include the number of defects or the number of nonconformities.
C.1. Theoretical Probability Functions
Before commencing discussion of the continuous and discrete distributions, important theoretical probability concepts must be introduced. For continu- ous distributions, the probability density function and the cumulative density function will be discussed. These are also referred to as, respectively, probability distribution functions and cumulative distribution functions. For discrete distri- butions, the probability mass function and the cumulative distribution functions will be discussed.
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C.1.a. Probability Density Functions and Cumulative Density Functions
Probability density functions (pdfs) are mathematical expressions that describe the probability distribution of a continuous random variable. The pdf is denoted by f(x). In most cases, the probabilities associated with some random variable can be described by a pdf. Figure 6.8 represents a pdf for a random variable X. The x-axis represents all possible values of the random variable; the y-axis represents the pdf f(x). Suppose we wish to find the probability that our random variable X lies between two real numbers a and b (i.e., P(a < X < b)). Graphically, this probability is the shaded area under the curve f(x) and between the X values of a and b (see Figure 6.8).
For any continuous random variable X, the pdf f(x) is a function with the fol- lowing properties:
f(x) ≥ 0 for all x (6.18)
f x dx( )∫ = 1
−∞
∞
(6.19)
P a X b f x dx
a
b
≤ ≤( ) = ∫ ( )
(6.20)
Equation (6.20) is the area under the curve f(x) and between the values a and b. The first property (Equation (6.18)) guarantees that all probability values are
nonnegative. The second property (Equation (6.19)) can be compared to the con- cept of sample space given in section A. That is, the total area under the curve must equal one (or 100%) and can be verified by integrating f(x) over all real num- bers. Equation (6.20) simply describes how the probability that X will lie between two real numbers a and b can be determined by integrating f(x) over the range [a, b]. Although these calculations may seem complicated, calculus is not neces- sary to find most of the probabilities that we need in quality engineering. Tables with probabilities for specific distributions are available as well as software that
x ba
f(x)
P(a < X < b)
f ( x)
Figure 6.8 A probability density function for a random variable X.
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routinely calculates these probabilities. It should be noted that for continuous dis- tributions, the probability that the random variable equals some specific value is always zero, that is, P(X = a) = 0. As a result
P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a < X < b)
A cumulative distribution function (cdf) is denoted by F(x) and describes the cumula- tive probability for a random variable X:
F x X x f v dv
x
( ) ( ) ( )= ≤ = −∞ ∫P
(6.21)
The cdf can be used to find probabilities of interest for the random variable X. Sup- pose a and b are any real numbers where a < b. Then,
P(X ≤ a) = F(a) (6.22)
P(a < X < b) = P(X ≤ b) – P(X ≤ a) = F(b) – F(a) (6.23)
P(X > a) = 1 – P(X ≤ a) = 1 – F(a) (6.24)
C.1.b. Probability Mass Functions and CDFs
Probability mass functions (pmfs) are expressions that describe the probability out- comes of a discrete random variable X. The pmf is denoted by p(x) and defined for every number the random variable can take on, x, by
p(x) = P(X = x) (6.25)
The pmf for the medication errors in Example 6.9 is shown graphically in Fig- ure 6.9. The probability is plotted on the y-axis and the values for the random vari- able on the x-axis. In Figure 6.9, the probability associated with 0 medication errors is 0.9 and represented by the height of the line in the graph.
Figure 6.9 A line graph of the pmf for random variable X.
1.0
0.8
0.6
0.4
0.2
0.0 0 1 2
Number of medication errors 3
P ro
ba bi
lit y
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For any discrete random variable X, the pmf p(x) is a function with the follow- ing properties:
p(x) ≥ 0 for all x (6.26)
( )p x 1
all x
=∑ (6.27)
The condition specified in Equation (6.26) ensures that all probabilities are non- negative, and the condition specified in Equation (6.27) ensures that the sum of all probabilities is equal to 1.
The cdf for discrete random variables is also denoted F(x) and is defined by
( )p y( )F x y:y≤x
== ( )P X ≤ x ∑ (6.28) For any number x, F(x) is the probability that the observed value of X will be at most x. See Devore (2016) for further details.
C.2. general Form of Expected value and variance
In this section, formulas for expected value and variance for continuous and dis- crete distributions are presented.
The expected value of a continuous random variable X with pdf f(x) is
E( ) ( )X x f x dxX= = ∫µ
−∞
∞
(6.29)
The variance of a continuous random variable X with pdf f(x) is
V( ) ( )X x f x dxX X= = −( )∫2
2
−∞
∞ µσ
(6.30)
The expected value of a discrete random variable X with pmf p(x) is
E( ) ( )X xp x
all x X= = ∑µ
(6.31)
for all outcomes x from the distribution. Remember, f(x) describes the probability that x will occur.
The variance of a discrete random variable X with pmf p(x) is
V( ) ( )X x p xX= = −( )∑σ 2
2
all x
µ
(6.32)
As shown previously, the standard deviation for a random variable X, discrete or continuous, is the positive square root of the variance.
C.3. Common Continuous distributions
In this section we present several common continuous distributions.
C.3.a. Normal Distribution
An important family of continuous distributions is the normal distribution. Exam- ple applications of the normal distribution include modeling height, weight, sam- ple averages, and many other quality characteristics. The normal distribution is a
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symmetric, bell- shaped distribution. The parameters of the normal distribution are the population mean μ and the population variance σ 2. The normal distribution is depicted in Figure 6.10. The center line represents the mean of the distribution.
The area under the curve represents probability (or percentage or propor- tion). The probability can be determined using the standard normal curve table in Appendix E. In a normal distribution, a Z-score for a random variable X is defined as the number of standard deviations between X and the mean of the distribution. Specifically,
Z
X =
− µ σ
(6.33)
To illustrate, suppose X follows a normal distribution with mean μ = 20 and standard deviation σ = 4. The Z-score using Equation (6.33) for an X value of 14 is
Z X
= −
= −
= − µ
σ 14 20
4 1 5.
That is, the value of 14 is 1.5 standard deviations below the population mean of 20.
ExaMpLE 6.17
Recall the number of medication errors provided in Table 6.4. The probability of 0, 1, 2, and 3 errors is 0.9, 0.07, 0.02, and 0.01, respectively. The expected number of medication errors is (using Equation (6.31))
E( ) ( ) ( . ) ( . ) ( . ) ( .X xp xX= = = + + +∑ 0 0 90 1 0 07 2 0 02 3 0 001 0 14) .=µ The variance of the number of medication errors is (using Equation (6.32))
V( ) ( )
( . ) ( . ) ( . ) ( .
X x p x= −( ) = − + −
∑ 2 2 20 0 14 0 90 1 0 14 0 07)) ( . ) ( . ) ( . ) ( . )
.
+ − + −
=
2 0 14 0 02 3 0 14 0 01
0 2204
2 2
µ
Mean
Figure 6.10 Probability density function for the normal distribution.
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By transforming the original random variable into a Z-score, we are able to find probabilities without having to use calculus, as demonstrated in Example 6.18.
C.3.b. Standard Normal Distribution
If a random variable is normally distributed with mean μ = 0 and variance σ 2 = 1, it is called a standard normal random variable and often denoted as Z. Probabili- ties for the standard normal distribution are given in Appendix E. The values in this table are the cumulative probabilities P(Z ≤ z), where capital Z represents a random variable and lowercase z is a real number. The cumulative probabilities in Appendix E can be used to find any probability of interest involving a ran- dom variable that is normally distributed. Some examples illustrating the use of Appendix E follow.
ExaMpLE 6.18
Let Z be a random variable that follows a standard normal distribution. Find the prob- ability that Z will be less than 2.5.
Solution: The probability of interest is P(Z < 2.5). This probability is shown graphically in Figure 6.11. It shows that the probability we are interested in is the area to the left of 2.5.
In Appendix E, the values of Z are written down the left-hand column and across the top of the table. The entries in the body of the table are the cumulative probabilities, P(Z ≤ z). In this example our z value is 2.5. In the table, read down the left-hand column to find the value 2.5, then across until you reach the column heading of 0 (since the value in the second decimal place of our z value is 0). The entry in the body of the table for the row of 2.5 and the column of 0 is 0.9938. Therefore, the probability that the ran- dom variable Z is less than 2.5 is 0.9938.
0 2.5 z
P(Z < 2.5)
Figure 6.11 Probability density function for a standard normal distribution.
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ExaMpLE 6.19
Find the probability that the random variable Z is greater than –2.5.
Solution: The probability of interest is P(Z > –2.5). In Appendix E, the probability given in the body of the table for –2.5 is 0.0062. But the probabilities in this table are the prob- abilities that Z is at most that value. That is, P(Z < –2.5) = 0.0062, and we want P(Z > –2.5). To find this probability, use the result that the total area under the curve (total probability) must equal 1. Therefore, if P(Z < –2.5) = 0.0062, then P(Z > –2.5) = 0.9938. More generally, we can find this probability as follows:
P(Z > –2.5) = 1 – P(Z < –2.5) = 1 – 0.0062 = 0.9938
ExaMpLE 6.20
Find the probability that Z lies between 1.42 and 2.33.
Solution: The probability of interest is P(1.42 < Z < 2.33). The probability is displayed graphically in Figure 6.12.
The area (thus the probability) of interest lies under the curve and between 1.42 and 2.33, as illustrated in Figure 6.12. The probability of interest can be found using the cumulative probabilities from Appendix E and subtraction:
P(1.42 < Z < 2.33) = P(Z < 2.33) – P(Z < 1.42) = 0.9901 – 0.9222 = 0.0679
0 1.42 2.33 z
P(Z < 2.33)
P(Z < 1.42)
P(1.42 < Z < 2.33)
Figure 6.12 Probability density function for a standard normal distribution.
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We can now find the probability for any normal random variable by using a simple transformation.
If X is a random variable that follows a normal distribution with mean μ and variance σ 2, then the random variable Z (from Equation (6.33)) is also normally distributed with mean μ = 0 and variance σ 2 = 1. That is, Z is a random variable that follows a standard normal distribution.
ExaMpLE 6.21
A product-fill operation produces net weights that are normally distributed with mean μ = 8.06 ounces and standard deviation σ = 0.37 ounces. Estimate the percentage of con- tainers that have a net weight less than 7.08 ounces.
Solution: Let X represent the weight of the containers. The probability of interest is P(X < 7.08). Transform X into the random variable Z using the relationship
Z X
= − µ σ
then from Appendix E find the appropriate probability.
P P PX X
Z<( ) = − < − = < −7 08 7 08 8 06 0 37
2 6. . .
. .
µ σ
55 0 0040( ) = .9 9
This indicates that approximately 0.40% of the containers have a net weight less than 7.08 ounces. This can also be stated as the probability that a randomly selected con- tainer will have a net weight less than 7.08 is approximately 0.0040.
C.3.c. Exponential Distribution
The exponential distribution is a continuous probability distribution often used to model problems in reliability. An example application of the exponential dis- tribution includes modeling the time between patient arrivals to an emergency department. In particular, the exponential distribution models the time or distance between successive events (such as failures) when the events follow a Poisson dis- tribution. The Poisson distribution is often a reasonable model of defects in mate- rial or the number of failures in systems. (The Poisson distribution for discrete data is discussed in the next section.) When we wish to determine the average time between failures, we calculate the inverse of the average number of failures or defects. For example, if there is an average of 0.69 failures per hour, then the mean time between failures (MTBF) is 1/0.69 = 1.45 hours. Figure 6.13 displays an exponential distribution with a mean of 1.45.
Suppose X represents the time or distance between successive events of a Pois- son process with mean λ (where λ > 0). The random variable X is said to be an exponential random variable with parameter λ. The pdf for X is
f(x) = λe–λx, for x ≥ 0 (6.34)
The cdf for an exponentially distributed random variable X is given by
F(x) = P(X ≤ x) = 1 – e–λx, for x ≥ 0 (6.35)
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ExaMpLE 6.22
The time between calls to a customer service center is an exponentially distributed random variable with a mean time between calls of two minutes. What is the probability that the next phone call will be received in the next one minute?
Solution: Let X represent the time between phone calls received at the customer service center. X follows an exponential distribution with parameter λ = 0.5 (recall that the mean of an exponential distribution is 1/λ and the mean in this case is two minutes; thus λ = 1/2 = 0.5). The probability of interest is P(X < 1) and can be found using the cdf for the exponential:
P(X < 1) = F(1) = 1 – e–0.5(1) = 1 – 0.6065 = 0.3935
Therefore, the probability that the next phone call will be received within the next minute is 0.3935.
C.3.d. Weibull Distribution
The Weibull distribution is a commonly used distribution in areas such as reliabil- ity. The Weibull is extremely flexible in modeling failure distributions, for exam- ple, the fatigue time of a component or product, that can take on many different shapes. Let X represent a random variable that follows a Weibull distribution. The pdf for the Weibull distribution is
f x
x e
x
( ) = −
− −
−β θ
γ θ
1
, forr x ≥ 0c ^ ^
cc c θ γβ β
(6.36)
where
β is the shape parameter (β > 0)
θ is the scale parameter (θ > 0)
γ is a threshold parameter (γ ≥ 0)
x 0.0 1.5 3.0 4.5 6.0 7.5 9.0
f ( x)
Figure 6.13 Probability density function for an exponentially distributed random variable.
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This pdf is referred to as a three- parameter Weibull distribution. The threshold parameter allows the user to model a distribution that cannot practically begin at zero. It simply shifts the beginning point away from zero. The pdf for the Weibull distribution with β = 1.2 and θ = 20 is displayed in Figure 6.14.
The cdf for a random variable that follows a three- parameter Weibull distribu- tion with parameters β, θ, and γ is
P( ) ( )X x F x e x
≤ = = − −
−
1 ^ ^
βγ θ
(6.37)
The cdf can be used to easily find probabilities associated with the Weibull distribution.
The beauty of the Weibull function is that it takes on many shapes depending on the value of β. For example, when β = 1, the function is exponential, and when β = 3.5, the function is approximately the normal distribution. If the threshold parameter is set equal to zero (i.e., γ = 0), then the three- parameter Weibull reduces to the two- parameter Weibull distribution—another commonly used distribution in reliability. The Weibull function is sometimes used for reliability data when the underlying distribution is unknown. This is discussed in more detail in Chapter 3, section E.
C.3.e. Continuous Uniform Distribution
The continuous uniform distribution is one that has a flat probability distribution between two points a and b. That is, if each value of the random variable has the same probability of occurring, the distribution is called the uniform distribution. The plot of a uniform distribution has a horizontal line as its upper boundary. An example is given in Figure 6.15.
The pdf for the continuous uniform distribution on the interval [a, b] is
f x b a a x b
( ) ,
, = −
≤ ≤ 1
0 otherwise *
(6.38)
x
f ( x)
Figure 6.14 Probability density function for the Weibull distribution.
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The cdf is
P( ) ( )
, ( ) ( )
X x F x
x a x a b a
a x b
b x
≤ = =
< − −
≤ <
≤
0
1
*
(6.39)
The cdf can easily be used to find probabilities associated with the uniform distri- bution, using the relationships shown previously in this section.
ExaMpLE 6.23
The thickness of a manufactured airplane part is uniformly distributed between 2.2 and 2.8 millimeters. We would like to find the probability that the thickness is less than 2.6 millimeters.
Solution: Let X represent the thickness of the airplane part. We want to find P(X < 2.6).
Since we are looking for X less than some value, we have P(X < 2.6) = F(2.6). Since the number that we are interested in (2.6) lies between the endpoints 2.2 and 2.8, we will use F(X ) = (x – a)/(b – a) from Equation (6.39). The probability that the thickness is less than 2.6 millimeters is
P(X < 2.6) = F(2.6) = (2.6 – 2.2)/(2.8 – 2.2) = 0.667
C.3.f. Bivariate Normal Distribution
If there are two variables of interest (such as length and width), each of which is normally distributed, the resulting distribution is called bivariate normal. The bivar- iate normal distribution can be used to describe the distribution of two normally
x 20 25 30 35 40
0.0450
0.0475
0.0500
0.0525
0.0550
f ( x)
Figure 6.15 Continuous uniform probability distribution.
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distributed random variables, say X and Y, that are not necessarily independent. There are five parameters that describe the bivariate normal distribution: μX, μY, σX
2 , σY 2 , and ρ, where
• μX is the mean of the random variable X
• μY is the mean of the random variable Y
• σX 2 is the variance of the random variable X
• σY 2 is the variance of the random variable Y
• ρ is the correlation between X and Y
Computer software is employed for handling problems involving these distributions. The bivariate normal distribution is illustrated in Example 6.24.
ExaMpLE 6.24
The inside and outside diameters of a particular type of tubing are important character- istics to be measured. Let X represent the inside diameter of the tubing and let Y repre- sent the outside diameter of the tubing. The inside and outside diameters are assumed to be normally distributed but not independent. For this problem, μX = 26, μY = 39, σX
2 = 0.16, σY
2 = 0.09, and the correlation between the inside and outside diameters is assumed to be ρ = 0.96.
It is important that both diameters meet the specifications of several customers. Suppose the specifications for one particular customer for X are 25.2 to 26.4 and the specifications for Y are 38.5 to 40.9. The probability that both dimensions are within specifications at the same time is P(25.2 < X < 26.4, 38.5 < Y < 40.9) = 0.7922. This value was calculated using computer software.
C.3.g. Lognormal Distribution
If a variable X follows a normal distribution, then the variable Y = eX follows a lognormal distribution. This distribution has applications in modeling life spans for products, response time, and time- to-failure data, as well as certain economic vari- ables. An example application of the lognormal distribution is modeling systolic blood pressure in adults or modeling fatigue time of a product. Some important properties of the lognormal distribution are the following:
• It assumes only positive values
• It is a right- skewed distribution
• It is the distribution of the random variable whose logarithm follows the normal distribution
Suppose X follows a normal distribution with mean μX and variance σX 2 , and Y =
eX. We then say that Y follows a lognormal distribution with the following mean and variance:
µ µ
Y Y e X X= = +E( ) ( / )1 2
2σ (6.40)
σ σ σ
YY e e X X X= = −+V( ) (2 2
2 2
11)µ (6.41)
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When the data follow a lognormal distribution, a transformation of data can be done to make the data follow a normal distribution so that we can then find probabilities, construct confidence intervals, and conduct tests of hypotheses (all of which depend on the assumption that the log- transformed data follow a normal distribution).
C.3.h. Summary of Continuous Distributions
Table 6.6 summarizes the pdf, mean (expected value), and variance for certain continuous distributions.
C.4. Common discrete distributions
Typical applications for discrete distributions in quality engineering include situa- tions where the variable of interest is either the number of nonconformities or the number of nonconforming units in a sample. The variable represents a count and takes on values of zero or a positive whole number.
C.4.a. Binomial Distribution
The binomial distribution can be applied in situations where the experiment can result in only one of two possible outcomes, for example, good/bad, go/no-go, with/without, conforming/nonconforming, success/failure. In addition, the out- come of one run of the experiment (often referred to as a trial) does not affect the outcomes of subsequent trials; that is, the trials are said to be independent. The outcomes are often referred to as a success or failure. Examples include the num- ber of heads on 50 flips of a fair coin and the number of manufactured parts that are out of specification. In one type of problem that is frequently encountered, the engineer needs to determine the probability of obtaining a certain number of non- conforming units in a sample. Example applications of the binomial distribution include modeling the number of defective units in a lot of n items, the number of
Table 6.6 pdf, mean, and variance for certain continuous distributions.
Distribution pdf (f(x)) Mean Variance
Normal 1
2
2
22
πσ e
x −
−( )µ σ μ σ 2
Exponential λe–λx
1 λ
1 2λ
Weibull (three-parameter)
β θ
γ γ
θ −
− −
− x
e x1
b b ^ ^b b β β
θ γ β
+ +Γ 1
1b bθ θ 2 22 1 1 1Γ Γ+ − +b bb bG Gβ β
Uniform 1 b a−
a b+ 2
( )b a− 2
12
Note: γ(n) = (n – 1)! for integer value n and Γ(z) = xz–1e–xdx∫ 0
∞ for non-integer value z.
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successful tests of n components, and the number of positive outcomes in a clinical trial with 100 participants.
The necessary conditions for a random variable to follow the binomial distri- bution are as follows:
1. There are a fixed number of observations or trials n.
2. The n trials are independent.
3. Each trial results in one of two possible outcomes (success or failure).
4. The probability of a success is denoted by p; the probability of a failure is then 1 − p. The probability of a success is assumed constant trial to trial.
Suppose an experiment consists of n independent trials. Let X represent the num- ber of successes in n trials. Furthermore, let p be the probability of success in one trial. Then, the probability of getting x successes in n trials is described by the pmf
p x X x C p pn x x n x( ) ( ) ( )= = = − −P 1 (6.42)
where
x is the number of successes, with a probability of each success given by p
The number of failures is then n – x, where 1 – p is the probability of a failure
The combination nCx represents the number of ways x successes can occur in n trials, where
n xC
n
x n
x n x n n n n= =
− = −( ) −( )!
!( )! ! .and 1 2 ...(1)c c
(6.43)
ExaMpLE 6.25
Ten manufactured parts are randomly selected from a batch where it is believed that the percent nonconforming is 15%. It is important to determine the probability that exactly 2 out of the 10 manufactured parts will be nonconforming. We define a success to be a nonconforming part. Let X represent the number of nonconforming parts. In this sce- nario, n = 10, x = 2, and p = 0.15. Then, using Equation (6.43),
n xC C= = = − =10 2
10 2
10 2 10 2
45 !
!( )!b b and, using Equation (6.42),
P( ) ( . ) ( . )
( . )(
X C= = −
=
−2 0 15 1 0 15
45 0 0225 0
10 2 2 10 2
.. )
.
85
0 2759
8
=
The probability that a sample of size 10 will have exactly 2 nonconforming parts is approximately 0.2759.
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ExaMpLE 6.26
Consider the previous example. Suppose now we are interested in the probability of finding fewer than 2 nonconforming parts. If X represents the number of nonconform- ing parts, the probability that fewer than 2 nonconforming parts will be found in the sample is P(X < 2).
Solution: The value 2 is not included in this event of interest. So we can rewrite the probability equivalently as
P(X < 2) = P(X ≤ 1) = P(X = 0) + P(X = 1)
In this case, the binomial formula must be applied twice:
P P P P( ) ( ) ( ) ( )
( . ) (
X X X X
C
< = ≤ = = + =
= −
2 1 0 1
0 15 1 010 0 0 .. ) ( . ) ( . )
( )( .
15 0 15 1 0 15
1 1 0 85
10 0 10 1
1 10 1− −+ −
=
C
)) ( . )( . )
. .
.
10 910 0 15 0 85
0 1969 0 3474
0 5443
+
= +
=
The probability that fewer than 2 parts out of 10 will be nonconforming is approximately 0.5443.
C.4.b. Poisson Distribution
When observations take place over a continuum, such as time or space, we do not have a finite series of discrete trials and the Poisson distribution may be used to model these events. Example applications of the Poisson distribution include modeling the number of medication errors for patients in a hospital, the number of nonconformities in a lot of manufactured products, the number of dents on a table, and the number of cases of the flu in a city.
The necessary conditions for a random variable to follow a Poisson distribu- tion are as follows:
1. The counts or occurrences are independent of each other
2. The probability that a count occurs in an interval is the same for all intervals of that size or length
Let λ be a parameter representing the mean number of counts over an interval. Let X represent the number of counts in the interval. Then, the probability that x counts occur in an interval is described by the pmf
p x X x
e x
x x
( ) ( ) !
, , , , ...= = = = −
P for λλ
0 1 2
(6.44)
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ExaMpLE 6.27
A company has a rate of serious accidents of three per year. The probability that at most one serious accident will occur during the next year is as follows. The probability of interest is P(X ≤ 1) = P(X = 0) + P(X = 1).
P P P( ) ( ) ( )
! ! .
X X X
e e
≤ = = + =
= +
= +
− −
1 0 1
3 0
3 1
0 05 0
3 0 3 1
..
.
149
0 199=
Therefore, the probability that at most one serious accident will occur during the next year is approximately 0.2.
C.4.c. Hypergeometric Distribution
When sampling from a finite population where independence is not assumed—for example, drawing cards without replacement from a deck of 52 cards, or select- ing a sample of items from an isolated lot—the probability changes with each observation.
The hypergeometric distribution is used when items are drawn without replace- ment from a population of interest; specifically, the items are not returned to the population before the next items are drawn. It is often used for samples taken from small populations. The items must fall into one of two categories, such as con- forming or nonconforming. Recall that the binomial distribution assumes either an infinite population or sampling with replacement (independent events). There can be a considerable difference when the population is small (results will be similar when the total population is large).
Suppose we have a finite population of size N from which a sample of size n is drawn (without replacement). Furthermore, let A represent the number of noncon- forming units in the population, and let x represent the number of nonconform- ing units in the sample. The probability of obtaining x nonconforming items in a sample of size n for this situation is given by
f x
A
x
N A
n x N
n
( ) =
−
= −
cc cc
c c x n, , , , ...for 0 1 2
(6.45)
where the combinations are as follows:
A x c c is the number of ways of choosing x nonconforming units from A
total possible nonconforming units
N A n x
− −
c c is the number of ways of choosing (n – x) conforming units from a
total of (N – A) conforming units in the population
N n c c is the number of ways of choosing a sample of size n from a population of size N
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ExaMpLE 6.28
The risk of implanting a biomedical device that may be nonconforming is important to quantify. Several assumptions must be made in order to obtain an accurate estimate of the risk, such as whether the devices have the same failure rate, how long the devices are stored before implantation, and so on. In addition, the number of nonconforming medical devices in a population of devices often has to be estimated using prior knowl- edge and/or previous data. For one of these biomedical devices manufactured by a local company, there has been one known failure after implantation within the last month. Based on prior information, it is assumed that out of 200 devices there are 3 that are nonconforming. If 30 medical devices are randomly selected out of the 200 devices, what is the probability that exactly one device will be nonconforming?
Solution: For this situation, N = 200, n = 30, A = 3, and x = 1, and the probability of interest is P(X = 1), which can be calculated with Equation (6.45) as:
P( )X = =
− −
=1
3
1
200 3
30 1
200 30
0..3281
b b
b bb b
The probability of selecting a sample of 30 devices and one is nonconforming is 0.3281.
C.4.d. Other Discrete Distributions
The multinomial distribution is used when an experiment consisting of n trials could result in more than two possible outcomes; the outcome is placed into one of several categories. For example, a randomly selected part could be classified as good, fair, or poor. As another example, a nonconforming part from a manu- facturing process could be due to machine wear, temperature, or a problem with raw material.
The geometric distribution involves independent trials that can result in one of only two possible outcomes, similar to the binomial distribution. However, for the geometric distribution the number of trials is not fixed. The random variable X represents the number of trials until the first success is obtained. An example would be determining the probability that x acceptable parts are produced before the first nonconforming part is generated, or the number of darts necessary until the bull’s-eye is hit.
The geometric distribution is a special case of a more general distribution known as the Pascal distribution (also known as the negative binomial distribu- tion). For the geometric distribution, sampling is terminated once the first suc- cess is obtained. For the Pascal distribution, sampling is terminated only after a fixed number of successes r have been obtained. Obviously, when r = 1 we have the special case of the geometric distribution. An example of the negative binomial distribution is modeling the number of trials necessary until r com- ponents fail.
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C.4.e. Summary of Discrete Distributions
The following rules can be applied for the binomial, Poisson, and hypergeometric distributions:
• Use with binary information (yes/no, conforming/nonconforming) and for nonconformities
• Use the hypergeometric distribution when n > 5%N and when the sample is taken without replacement
• Use the binomial distribution when the sample is taken with replacement or when n < 5%N and the sample is taken without replacement
• Use the Poisson distribution when there can be more than one nonconformity per item or as an approximation for the binomial distribution when n > 100 and np < 10
Figure 6.16 is an illustration of common rules of thumb for approximations. In this figure, H, B, P, and N represent the hypergeometric, binomial, Poisson, and normal distributions, respectively. Table 6.7 summarizes the pmf, mean, and variance for certain discrete distributions.
Table 6.7 pmf, mean, and variance for certain discrete distributions.
Distribution pmf (p(x)) Mean Variance
Binomial n x
x n xC p p( )1 − − np np(1 – p)
Poisson e x
x− λλ !
λ λ
Hypergeometric A x
N A
n x N
n
− −
b b
b bb b nA N
(N – A)(N – n) N(N – 1)
nA N
Figure 6.16 Approximations to probability distributions. Source: D. C. Montgomery, Introduction to Statistical Quality Control, 7th ed. (Hoboken, NJ: John Wiley & Sons, 2013).
p > 0.9
H
B
n N
P
N
The smaller p and larger n the better
Let p′ = 1 – p. The smaller p′ n the better. d d f f
p < 0.1
< 0.1
λ ≥ 15 (The larger the better) 0.1 ≤ p ≤ 0.9
np > 10
and larger
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C.5. Central Limit Theorem
An important statistical principle is the central limit theorem, which states that
The distribution of sample averages will tend toward a normal distribu- tion as the sample size n approaches infinity.
The central limit theorem guarantees at least approximate normality for the distribution of sample averages, even if the population from which the sample is drawn is not normally distributed. A frequent question in the minds of QEs is the validity of x– control charts (discussed in section F) when the population is not normally distributed.
Because the x– control chart involves plotting averages, the central limit the- orem implies that normality is approximately guaranteed. The approximation improves as the sample size n increases. In some cases, the approximation will be applicable for sample sizes as small as 10. In other situations, the required sample size for the approximation to be valid can be quite large (say, n > 100). Finally, if the underlying distribution of the data does not depart significantly from the normal distribution, sample sizes as small as n = 3 can be appropriate. For details on the normality assumption in statistical process control, see Mont- gomery (2013).
Suppose X1, X2, . . . , Xn is a random sample taken from a distribution with mean μ and variance σ 2. If n is sufficiently large, then the sample mean X
– follows approximately a normal distribution with xµ = μ and variance σx
2 = σ 2/n. Notice that the definition does not state that the underlying distribution from
which the sample is drawn must be normally distributed. The true form of the distribution does not have to be normally distributed as long as the sample size is sufficiently large. There have been several recommended cutoffs for large n, including n > 40 (Devore 2016).
However, if the underlying distribution is normal, then the large n require- ment is not necessary. In this case, the sampling distribution of X
– will also follow a normal distribution with mean xµ = μ and variance σx
2 = σ 2/n. An important application of the central limit theorem described here has
been in calculating probabilities associated with the sample mean. In addition, it is important in statistical inference, statistical process control, process capability analysis, and so on, as we will see in the remaining sections.
Recall that if a random variable X follows a normally distributed random variable with mean μ and variance σ2, then the random variable Z, from Equa- tion (6.33), will follow a standard normal distribution with μ = 0 and σ 2 = 1. By the central limit theorem stated above,
Z
X X n
x
x
= −
= −µ µ
σσ / (6.46)
also follows a standard normal distribution with μ = 0 and σ 2 = 1. Using this result, we can find probabilities associated with the sample mean.
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ExaMpLE 6.29
Recall the product-fill operation in Example 6.21, where the product-fill operation pro- duces net weights that are normally distributed with mean μ = 8.06 ounces and standard deviation σ = 0.37 ounces.
a. What is the probability that a randomly selected container will weigh less than 7.08 ounces?
b. What is the probability that a sample of nine randomly selected containers will have an average net weight less than 7.08 ounces?
Solution: Let X represent the weight of the containers.
a. The probability of interest is P(X < 7.08). Transform X into the random variable Z using the relationship
Z X
= − µ σ
then from Appendix E find the appropriate probability.
P P PX X
Z<( ) = − < − = < −7 08 7 08 8 06 0 37
2 6. . .
. .
µ σ
55 0 0040( ) = .c c
Therefore, the probability that a randomly selected container will have a net weight less than 7.08 is approximately 0.0040. (Note that this is the same answer we found previously.)
b. The probability of interest here is P(X – < 7.08). Transform X
– into the random
variable Z using the relationship
Z X
n =
− µ σ /
:
P P P( . ) /
. . . /
X X
n Z< =
− <
− = <7 08
7 08 8 06 0 37 9
µ σ
−−( ) ≅7 95 0.c c
Therefore, it would be very unlikely that a random sample of nine such containers would have an average net weight less than 7.08 ounces.
Note: In Example 6.29, the value z = –7.95 is not given in the standard nor- mal table in Appendix E. The table values extend only from –3.59 to 3.59. If the z value is not in the table, this does not mean that it is not a possible value. When the value of z is not in the table, the probability of interest will be practi- cally zero or one, depending on the area of interest under the curve and the sign on the z-value. For example, P(Z < –7.95) ≅ 0, P(Z > –7.95) ≅ 1, P(Z < 7.95) ≅ 1, P(Z > 7.95) ≅ 0.
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C.6. Sampling distributions
A sampling distribution is the probability distribution of a sample statistic. The following sampling distributions are used in the inferential statistics sections (D–H) of this chapter. Each of the following sampling distributions can be defined in terms of normally distributed random variables. Their theoretical bases are introduced here.
C.6.a. Chi- Square (χ2) Distribution
Suppose Z1, Z2, Z3, . . . , Zk are independent standard normal random variables. Then the random variable
χ 2
1 2
2 2
3 2 2= + + + +Z Z Z Zk... (6.47)
follows a chi-square distribution with k degrees of freedom. The pdf for the chi- square distribution is
f x k
x e x k
k x( ) , /
( / ) /= ≥− − 1
2 2
0, k = 1, 2, ... 2
2 1 2
Γ for
c c
(6.48)
The expected value and variance for the chi- square distribution are μ = k and σ 2 = 2k, respectively. Several chi- square distributions are displayed in Figure 6.17.
f ( x)
x 0 10
k = 10 k = 6
k = 2
20
Figure 6.17 Several chi-square distributions.
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C.6.b. Student’s t Distribution
Let Z be a standard normal random variable, and let W be a χ2 random variable with k degrees of freedom, where Z and W are statistically independent. Then the random variable T, defined as
T Z
W k
=
(6.49)
has a pdf given by
f x
k
k k x
k
k ( )
( )/ =
+
+ +
Γ
Γ
1 2
2
1
1 2 1 2
π
,, ,,, k = 1, 2, ...for − ∞ < < ∞x c c
c e ec q r
(6.50)
which follows a t distribution with k degrees of freedom. For k > 1, the mean and variance for the t distribution are μ = 0 and σ 2 > 1. For k = 1, the t distribution is a Cauchy distribution, which has no mean or variance. Figure 6.18 displays t dis- tributions for various degrees of freedom. The t distribution is very similar to the standard normal distribution since both are symmetric, are bell- shaped, and have μ = 0. However, the tails of the t distribution are heavier than the standard nor- mal; in other words, there is more probability in the tails (extreme values) of the t distribution than in the standard normal distribution. Notice that as the degrees of freedom go to infinity, the form of the Student’s t distribution becomes the stan- dard normal distribution.
x
k = 10
k = 8
k = 2
0 2 4 6–6 –4 –2
Figure 6.18 Probability density functions for three t distributions.
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C.6.c. F Distribution
Suppose Y and W are independent chi- square random variables with k1 and k2 degrees of freedom, respectively. Then the random variable
F
Y k W k
= / /
1
2 (6.51)
is said to follow an F distribution with k1 and k2 degrees of freedom.
C.6.d. Sampling Distribution of the Sample Mean
Another important statistical principle refers to the distribution of sample means. The sample mean is a statistic calculated from a sample of data. Since the sample mean can take on different values for different samples taken from the same popu- lation, the sample mean is a random variable. As such, the sample mean has its own distribution. The distribution of the sample mean is referred to as the sam- pling distribution of the sample mean. An important statistical principle states:
If samples of size n are randomly drawn from a population with mean μ and standard deviation σ, then the distribution of sample means has the following properties:
• Its mean, denoted µx, is equal to the population mean: µx = μ
• Its variance, denoted σx 2 , is equal to the population variance divided by the
sample size n:
σ
σ x n 2
2
=
(6.52)
• Its standard deviation is equal to the positive square root of the variance:
n σ
σ x =
(6.53)
The standard deviation of the sampling distribution of x– is referred to as the stan- dard error.
ExaMpLE 6.30
Consider the time it takes to repair a robotic arm in a manufacturing facility. The mean repair time is μ =1.27 hours and standard deviation σ = 0.17 hours. Assume that the under- lying distribution is normally distributed. Compute the mean and standard deviation of the mean repair times for a sample of 10 robotic arm repairs.
Solution: The goal is to determine the sampling distribution for the mean repair time for robotic arms. The mean is µx = μ = 1.27 hours. The standard error is σx = σ/ n = 0.17/ 10 = 0.054.
The mean repair time for sample of size 10 has a normal distribution with mean 1.27 hours and standard deviation 0.054 hours.
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In theory, the expected value of any single observation xi taken from a random sample is equal to the mean of the population μ from which the observation has come. The notation is given as E(xi) = μ. In essence, this states that an unknown value xi from some population of interest would be expected to be equal to the true population mean.
ExaMpLE 6.31
A part is selected at random from a population. The true diameter of this population of parts is believed to be 12 mm (i.e., μ = 12). Therefore, without actually measuring the part selected at random, we would expect the diameter to be 12 mm (i.e., E(xi) = 12).
Now, the expected value of the sample mean is also μ, the mean of the population. The notation for expected value of the sample mean is E(x–) such that
E(x–) = μ
for a sample of size n.
Example 6.32 summarizes the use of expected value and standard error of the mean.
ExaMpLE 6.32
A sample of 100 parts is selected at random from a population. The true diameter of the population of parts is believed to be 12 mm (μ = 12) with a standard deviation of 0.05 mm (i.e., σ = 0.05). Therefore, we would in theory expect the sample of 100 parts to have an average diameter of 12 mm or
E(x–) = μ = 12
The standard error of the sample mean is found as
σ σ
x n
= = = 0 05 100
0 005 .
.
The value 0.005 represents the variability associated with the mean of the 100 parts cho- sen at random.
C.7. Probability Plots
Graphical displays of data are important tools that can help determine important properties of the data. Using graphical displays, the overall shape, location of the center, and measure of variability can be approximately estimated. These displays can also be used to possibly determine the type of distribution that the data may follow. One graphical display that can be used for determining the type of distri- bution the data may follow is the probability plot.
The probability plot displays the actual data on the x-axis, plotted against percentiles based on the hypothesized or assumed distribution of interest on the
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y-axis. For example, the normal probability plot displays the actual data against percentiles from a normal distribution. If the data fall—at least approximately— along a straight line, then they are said to be approximately normally distributed. Probability plots can be constructed for many distributions, including the normal, lognormal, Weibull, and exponential. Two of the most commonly used are the nor- mal and Weibull probability plots.
The assumption of at least approximate normality is often necessary to satis- factorily apply many statistical tests. The normal probability plot can be used to determine whether a given set of data come from a population that is normally distributed. In general, the data are plotted on a probability plot where the vertical axis has been scaled according to a normal distribution. It is unnecessary to create these plots manually. Most statistical software packages will generate these plots for any set of data. If the data fall at least approximately along a straight line, then the distribution of interest (in this case the normal distribution) is assumed to be a reasonable form for the data.
ExaMpLE 6.33
Sumithra and Bhattacharya (2008) present a study on the toasting of corn flakes. Appro- priately toasted flakes possess the desired moisture content, texture, and color. In their study, the authors investigated the effect of three independent variables—moisture content, toasting temperature, and toasting time—on several responses of interest. One response was the force needed to puncture the toasted flake. The puncture force data (measured in Newton) for this experiment are:
5.34, 6.62, 2.90, 2.07, 5.87, 4.02, 3.45, 2.24, 3.80, 3.80,
2.27, 6.62, 3.95, 4.12, 2.95, 2.80, 2.81, 2.80, 2.90, 2.95
A normal probability plot of the puncture force data is given in Figure 6.19. The data do not appear to fall along a straight line, so the normal distribution may not be the best to model puncture force.
0 1 2 3
Puncture force
4 5 6 7 0
5
10
20 30 40 50 60 70 80
90
95
99
P e
rc e n
t
Figure 6.19 Normal probability plot of puncture force for toasted corn flakes.
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The Weibull probability plot is used to determine whether a particular set of data follows a Weibull distribution. The Weibull distribution is often used in reli- ability problems. Similar to the normal probability plot, the actual data are plotted against a percentile that is based on the Weibull distribution. The Weibull prob- ability plot is more difficult to construct by hand than the normal probability plot and so will not be outlined here. Many statistical packages have the capability to construct Weibull probability plots.
ExaMpLE 6.34
The following data represent the life of a particular part used in the semiconductor manufacturing industry. Fifteen parts are selected at random and their life (in hours) recorded when the parts are in use. The data are:
479.23, 43.17, 3219.41, 558.46, 56.00, 705.37, 12.02, 280.42,
3867.95, 6672.37, 8494.07, 1220.94, 66.92, 2078.13, 6431.02
It is important to determine the distribution that the data may follow. The Weibull distri- bution could be investigated. The Weibull probability plot for this set of data is shown in Figure 6.20.
The data mostly fall along a straight line. Therefore, the data appear to follow a Weibull distribution. The Weibull distribution appears to be valid for this set of data.
Life (in hours) 1 10 100 1000 10,000
1
99
10
20 30 40 50 60 70 80 90
5 3 2
P er
ce nt
Figure 6.20 Weibull probability plot for life of a part.
Interpretation of probability plots can be subjective. One person may interpret the data as normally distributed, for example, while someone else examining the same plot could say that they are not normally distributed. The closer the data fall along a straight line, the more evidence there is that the distribution of interest is reasonable for that particular set of data. If the plot exhibits curvature or an “S”
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shape, then other distributions should possibly be investigated. Goodness- of-fit tests are often more reliable approaches to determining the appropriateness of a particular distribution (see Devore 2016).
d. STaTiSTiCaL dECiSion MaKing
d.1. Point Estimates and Confidence intervals
Suppose an estimate is needed for the average coating thickness of a population of 1000 circuit boards received from a supplier. Rather than measure all 1000 boards, randomly select a sample of 40 for measurement. Suppose that the average coating thickness of these 40 boards is 0.003 and the standard deviation of the 40 coating measurements is 0.005. The estimate for the average coating thickness on the entire lot of 1000 is then around 0.003. This value is called the point estimate. In this case, the sample mean is a point estimator of the population mean (recall the definitions of population mean, sample mean, parameter, and statistic given in section B).
To find information about a population of interest, it is important to obtain information about the parameters that describe the population. Recall that param- eters include the population mean μ and population variance σ 2, for example. It is important to be able to estimate the parameters using information acquired from a sample taken from the population. A statistic, such as the sample mean, is used as a point estimator for a parameter, in this case the population mean (keep in mind that “statistic” refers to a value obtained from a sample and “parameter” is a char- acteristic of a population).
The point estimator is said to be unbiased if its expected value is equal to the parameter that it is estimating. Consider the sample and population means. If a sample consists of n observations, X1, X2, . . . , Xn, taken from a normal population, then the sample mean X
– is known to be an unbiased estimator of the population mean μ, that is, E(X
– ) = μ. It can also be shown that the sample variance s2 for the same situation (n observations taken from a normal distribution) is an unbiased estimator for the population variance σ 2, that is, E(s2) = σ 2. However, the sample standard deviation s is not an unbiased estimator for σ, yet the bias is often negli- gible for all but very small sample sizes.
Sometimes there is more than one possible unbiased point estimator for a parameter. One point estimator for a parameter is said to be more efficient than another if the variance of the point estimator is smaller than the variance of its competitor. As a simple example, consider a random sample of observations, X1, X2, . . . , Xn, taken from a population with mean μ and variance σ
2. The sample mean X – is one point estimator for the population mean μ. However, any one observation Xi is also a possible point estimator for μ. Recall from section C that the variance of the sampling distribution of X
– is σ 2/n (i.e., V(X – ) = σx
2 = σ 2/n) and the variance of a single observation from a population with variance σ2 is simply σ 2 (i.e., V(Xi) = σ
2). Since V(X
– ) < V(Xi), X – is said to be a more efficient point estimator than Xi for μ.
The standard error of a statistic was briefly introduced in section C. In general, the standard error (s.e.) of a point estimator provides a measure of precision of the estimate and is simply the square root of the variance of the point estimator. For example, we know that the variance of the sampling distribution of the point esti- mator X
– is σ 2/n. The standard error is then σ/ n . You sometimes see this written as s.e. (X
– ) = σ/ n .
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When we calculate a statistic of interest, such as the sample mean, from real pop- ulations of interest, we know that this estimate is not exactly equal to the true popu- lation mean. We could select 15 different samples of the same size n from the same population and calculate 15 sample means, and most if not all of them would be different. So which of the 15 would be the best estimate? Well, any one of the sample means would be an appropriate estimate of the population mean. As the sample size n increases, our estimates become more precise. If we were to use the entire popula- tion, our estimate would be perfect, but that is rarely found in practice. In nearly all cases, the sample means will be different even though they are drawn from the same population. The variability in the estimates needs to be quantified somehow.
d.2. Confidence intervals
In this section we present confidence intervals for the following:
• A single population mean μ
• A single population variance σ 2
• A single population standard deviation σ
• A single population proportion p
In the example given in section D.1, is the population mean exactly 0.003? Almost surely not, due to sampling error. To capture information about a parameter, we need to quantify the variability in this estimate and then report it in a meaningful way. One type of estimation that can be used is called interval estimation. One of the most useful interval estimation approaches is constructing confidence inter- vals on the parameter of interest.
In general, a confidence interval on a population parameter depends on the following:
• A point estimate for the parameter of interest
• An estimate of the standard error of the point estimate
• A stated level of confidence, denoted by 1 – α
• In some cases, an idea of the approximate distribution of the underlying population
For example, suppose we wish to construct a 95% confidence interval on a popula- tion mean μ (complete details and discussion of confidence intervals are presented later in this section). We would take a random sample from the population, calcu- late the necessary statistics (e.g., sample mean and sample variance—if population variance is not known), and construct an interval on μ using the sample informa- tion (and some other information). Suppose we found the 95% confidence interval on μ to be 16 ≤ μ ≤ 22. This is much more informative than just reporting that the sample mean was found to be x– = 19.
The general form of a 100(1 − α)% two- sided confidence interval for any popu- lation parameter (which we denote as θ) is
L ≤ θ ≤ U
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where
L = lower endpoint of the confidence interval
U = upper endpoint of the confidence interval
θ = parameter of interest, such as μ or σ
α = significance level
There are two general interpretations of the results from a confidence interval: the probabilistic interpretation and the practical interpretation.
The probabilistic interpretation of the confidence level is the proportion of all confidence intervals constructed on that parameter (under repeated sampling and identical conditions) that would contain the true parameter. For example, a 95% confidence interval on μ would be interpreted as the percentage of all confidence intervals constructed (under repeated sampling and identical conditions) that would contain the population mean μ. Consider a normal population with known mean μ = 25 from which 20 samples of size n = 100 are selected. Twenty 95% confi- dence intervals on μ are constructed, with the following results:
(24.4713, 25.6473) (24.0548, 25.2308)
(24.5202, 25.6962) (23.9702, 25.1461)
(24.3994, 25.5754) (24.8004, 25.9763)
(24.1861, 25.3620) (24.4020, 25.5780)
(24.7750, 25.9510) (23.8412, 25.0172)
(24.0617, 25.2377) (24.4872, 25.6632)
(24.2578, 25.4337) (24.4695, 25.6455)
(24.0038, 25.1798) (25.0248, 26.2008)
(24.1231, 25.2991) (24.4982, 25.6742)
(24.2900, 25.4659) (24.1859, 25.3619)
What percentage of these confidence intervals contain the true population mean μ = 25? In this scenario, 19 out of 20 (or 95%) confidence intervals constructed contain the true population mean μ = 25. That also means that 5% of all intervals constructed do not contain the true population mean.
Since the true parameter value, such as the population mean μ, is usually unknown in practice, the probabilistic interpretation may not be very useful. Additionally, we often have only one sample of data, not 20. The practical interpre- tation of a confidence interval constructed in this situation would be a statement of degree of belief that the confidence interval contains the true μ. For example, we are 95% confident that the 95% confidence interval will contain the true popula- tion mean μ.
It should be noted that the practical interpretation should never be miscon- strued as saying that the “probability that the confidence interval contains the true value μ is 0.95.” Remember, the true value of the population mean μ exists but is unknown. Therefore, when we construct a confidence interval on μ, either the interval contains the true value of μ (probability of 1) or it does not (probability of 0).
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Next, confidence intervals involving single samples are covered. Confidence intervals involving two samples are presented in section D.3 as part of the discus- sion on hypothesis testing.
D.2.a. Confidence Intervals for a Single Population Mean μ
In this section, the presentation of confidence intervals on a population mean is divided into two cases:
1. Confidence intervals on μ when the population variance σ 2 is assumed known
2. Confidence intervals on μ when the population variance σ 2 is assumed unknown
Case 1: Population Variance σ 2 Is Assumed Known
Let X1, X2, . . . , Xn represent a random sample taken from a normal distribution with known variance σ 2 but unknown population mean μ. From section C, we know that the sampling distribution of X
– is normally distributed with mean μ and variance σ 2/n. And
Z X X
n x
x
= −
= −µ µ
σσ /
follows a standard normal distribution. It can be shown that a general 100(1 – α)% confidence interval on μ is given by
x z
n x z
n − ≤ ≤ +α α
σ µ σ/ /2 2
(6.54)
where
x– = sample mean.
n = sample size. σ n
= standard error of the mean.
zα/2 = multiple of the standard error of the mean, which determines the width of the interval. It is a direct result of the level of confidence 1 − α and is found using the standard normal distribution. The subscript α/2 is the area to the right of the number zα/2 under the standard normal curve.
ExaMpLE 6.35
A manufacturing process has been running in control for some length of time. The qual- ity characteristic of interest is the diameter of the manufactured part (measured in mm). It is believed there may have been a shift in the process mean due to the change of a raw material. A sample of 25 items is randomly selected from the process and measured, and the sample average is found to be 103 mm. The in-control process average has been 102 mm, which is nominal. The standard deviation during the time the process was believed to be in control was 3 mm. The customer wants to construct a 95% confidence interval on the true process mean to determine whether the process has shifted away from nominal.
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Solution: The necessary information is:
• x– = 103.
• σ = 3.
• n = 25.
• 1 − α = 0.95, and solving for α, we get α = 0.05. This value is necessary to find the multiple zα /2 from the standard normal table.
From the standard normal table in Appendix E with α = 0.05, zα /2 = z0.05/2 = z0.025 = 1.96. The resulting 95% confidence interval (using Equation (6.54)) is then
103 1 96 3 25
103 1 96 3 25
103 1 176 103 1
− ≤ ≤ +
− ≤ ≤ +
. .
. .
µ
µ 1176 101 824 104 176. .≤ ≤µ
With a 95% level of confidence, we can say 101.824 mm ≤ μ ≤ 104.176 mm. Since the nominal value of 102 mm is contained in this interval, there is insufficient evidence to conclude at the 95% level of confidence that the process mean has shifted.
ExaMpLE 6.36
Reconsider the previous example. What conclusions could be reached if the level of confidence were changed to 0.90?
Solution: With all other information held constant, the only change is in the multiple Zα /2. In this case 1 – α = 0.90, so α = 0.10 and zα /2 = z0.10/2 = z0.05 = 1.645. The 90% confidence interval (using Equation (6.54)) is then
103 1 645 3 25
103 1 645 3 25
103 0 987 103
− ≤ ≤ +
− ≤ ≤ +
. .
.
µ
µ 00 987 102 013 103 987
.
. .≤ ≤µ
With a 90% level of confidence, we can say 102.013 mm ≤ μ ≤ 103.987 mm. Since the nomi- nal value of 102 mm lies outside this interval, there is sufficient evidence to conclude that the process mean has shifted. In fact, we would conclude based on the data that the pro- cess mean has shifted upward. Notice that the nominal value of 102 mm lies just outside the lower bound of the confidence interval. Whether this shift would result in corrective action should be determined by personnel familiar with the manufacturing process.
Examine the 95% and 90% confidence intervals for Examples 6.35 and 6.36. Decreasing the confidence level from 95% to 90% decreased the width of the con- fidence interval. Decreasing (increasing) the level of confidence—while all other quantities remain constant—will always result in a decrease (increase) in the width of the confidence interval.
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There are many situations where the practitioner would like to know how large a sample is needed in order to estimate a population mean with some level of precision. This information can be obtained using the 100(1 – α)% confidence interval and making some reasonable assumptions.
The length of the confidence interval (from Equation (6.54)) provides a mea- sure of precision of estimation. The margin of error in the confidence interval for- mula is
z
n α
σ /2
(6.55)
This quantity provides information about the accuracy of the confidence interval. That is, when x– is used as the estimate for μ, the error
E x= − µ (6.56)
will be at most
z n
α σ
/2
So, if
E z n
= α σ
/2
is an upper bound on the amount of error the practitioner is willing to live with, an appropriate sample size can be determined. That is, for a given level of confi- dence, an assumed value of σ, and a margin of error E one is willing to live with, a minimum sample size can be found using the relationship
E z
n = α
σ /2
(6.57)
and solving for n:
n
z
E ≥
σ α/2 2
d d
(6.58)
If the value of n is not an integer, round up to the nearest integer.
ExaMpLE 6.37
Suppose the turnaround time for basic blood analysis for the emergency room at a local hospital is of interest. A goal is to be able to estimate the true average turnaround time μ. Specifically, it is of interest to obtain an estimate that is within three minutes of the true average turnaround time with 95% confidence. Based on prior information it is assumed that σ = 8 min. How many turnaround times should be obtained in order to estimate the true average turnaround time and meet the requirements stated? That is, what is n?
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Solution: First, note that the problem states that the estimate is “within three minutes of the true average.” This implies that we will allow the estimate to be at most three minutes less than the true average or at most three minutes greater than the true average. In this case, the margin of error we are willing to live with is E = 3 min. The level of confidence is 95%, so 1 – α = 0.95, α = 0.05, and zα/2 = z0.025 = 1.96. With σ = 8 min, the minimum sample size needed (using Equation (6.58)) is
n z E
≥ = = = σ α/ ( . )2
2 2 28 1 96
3 (5.227) 27..32b b bb
Therefore, to meet the requirements of 95% confidence, a margin of error of no more than three minutes, and σ = 8 min, the minimum number of times to obtain would be n = 28.
Case 2: Population Variance σ 2 Is Assumed Unknown
Suppose the population of interest is normally distributed with unknown mean μ and unknown variance σ 2. Let X1, X2, . . . , Xn be a random sample taken from the population, and let the sample mean and variance be denoted by x– and s2, respec- tively. When σ is unknown, it can be estimated by s. Furthermore, if the sample size n is small, the standard normal distribution is no longer an appropriate distri- bution for the sample mean. In its place we will use Student’s t distribution (recall the t distribution from section C).
Let X1, X2, . . . , Xn be a random sample taken from a normal population, and let the sample mean and variance be denoted by x– and s2, respectively. The random variable
T Z
W k
=
follows a t distribution with k = n – 1 degrees of freedom as shown in Equa- tion (6.49). It should be noted that the t distribution is defined by its degrees of freedom, which are directly related to the sample size n. As the degrees of freedom go to infinity, the t distribution approaches the standard normal distribution. The degrees of freedom are of particular interest. The degrees of freedom for T is n – 1. This is a direct result of using s2 as an estimate for σ 2. Recall that the sample vari- ance is based on the deviations x1 – x
– , x2 – x – , . . . , xn – x
– , and that
x xi
i
n
−( ) = = ∑
1
0
(6.59)
Since Equation (6.59) holds, this implies that if we specify n – 1 of these deviations, the nth deviation is automatically determined (and cannot vary freely). That is, only n – 1 of these deviations can vary freely. The degrees of freedom for the t dis- tribution represent the number of deviations that can vary freely.
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It can be shown that the 100(1 – α)% confidence interval on the population mean μ is given by
x t
s
n x t
s
n k k− ≤ ≤ +α αµ/ , / ,2 2
(6.60)
where tα/2,k is the multiple of the standard error of the mean (s/ n ) and deter- mines the width of the interval. It is a direct result of the level of confidence 1 − α and is found using Student’s t distribution. The subscript α/2 is the area to the right of the number tα/2,k that we need, and k = n – 1 is the degrees of freedom for a particular problem. Probabilities and values for the t distribution are given in Appendix O.
ExaMpLE 6.38
A study on the effect the type of joystick has on powered wheelchair driving perfor- mance was conducted. In the study, one type of joystick used was a position-sensing joystick. Suppose eight subjects are asked to use a position-sensing joystick, and the time to complete a predetermined task is recorded (in minutes). The results are:
30.7, 31.2, 26.1, 29.4, 34.6, 26.8, 33.1, 25.5
Assume that time to complete the task is well approximated by a normal distribution. The investigators would like to construct a 95% confidence interval on the true time to complete the task using the position-sensing joystick.
Solution: First, we need to calculate the values x– and s:
x x
n
s x x
n
i i
n
i i
n
= =
= −
− =
=
=
∑
∑
1
2
1
29 675
1 3 3363
.
( ) .
For a level of confidence of 95%, 1 – α = 0.95, and solving for α, we get α = 0.05. The degrees of freedom for this problem is k = n – 1 = 7. These values are necessary to find the multiple tα/2,k from the t distribution. From the t distribution table in Appendix O, with α = 0.05, tα/2,k = t0.025,7 = 2.306. The resulting 95% two-sided confidence interval (using Equation (6.60)) is then
29 675 2 365 3 3363
8 29 675 2 365
3 3363 8
29 67
. . .
. . .
.
− ≤ ≤ +
=
µ
55 2 790 29 675 2 790
26 89 32 47
− ≤ ≤ +
= ≤ ≤
. . .
. .
µ µ
With a 95% level of confidence we believe the true time to perform the task using the position-sensing joystick lies between 26.89 minutes and 32.47 minutes.
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In Example 6.38 the sample size was small, n = 8. In cases where the sample size is small (Devore [2016] suggests n < 40), the population is known to be nor- mally distributed, and the population variance is unknown, the t distribution is an appropriate distribution. In cases where the sample size is sufficiently large, the population is known to be normally distributed, and the population variance is unknown, the standard normal distribution or the t distribution could be used. Again, the t distribution and standard normal distribution are approximately of the same form, as the sample size n goes to infinity.
Now in addition to the population mean and population variance being unknown, assume that the underlying distribution is not necessarily normally dis- tributed. In this situation, if the sample size is large, then the central limit theorem applies and the 100(1 – α)% confidence interval
x z
s
n x z
s
n − ≤ ≤ +σ σµ/ /2 2
(6.61)
can be used. However, if the sample size is small and the underlying distribution is decidedly non- normally distributed, then nonparametric methods should be used to construct the confidence interval (see Devore 2016).
D.2.b. Confidence Intervals on a Population Variance and Standard Deviation
As stated previously, the sample variance s2 can be used as the point estimate for the population variance σ 2. The sample standard deviation s can be used as the point estimate for σ. Suppose instead of relying on a point estimate of σ 2, we can construct a confidence interval. Assume a normally distributed population from which a random sample X1, X2, . . . , Xn is selected.
The 100(1 − α)% two- sided confidence interval on the population variance σ 2 is given by
( ) ( )
/ , / ,
n s n s
k k
− < <
−
−
1 12
2 2
2 2
1 2 2χ χ
σ α α
(6.62)
where χ2α/2,k and χ 2 1–α/2,k are values obtained from the chi- square distribution with
α/2 and (1 − α/2) being the areas under the chi- square curve and to the right of the value of interest to be put into the confidence interval. Again, k = n – 1 is the degrees of freedom. Table values of the chi- square distribution are given in Appen- dix J. (Recall the chi- square distribution from section C.6.)
ExaMpLE 6.39
A heart rate stress test was administered to 10 men. The heart rates (in beats per minute [bpm]) were recorded. The average heart rate was found to be x– = 100 bpm with a stan- dard deviation of s = 16.2 bpm. A confidence interval was constructed on the mean heart rate, and it is now desired to construct a 95% confidence interval on the variance in heart rate.
Continued
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Solution: The necessary information is:
s2 = (16.2 bpm)2 = 262.44 bpm2
k = n – 1 = 9
1 − α = 0.95, therefore α = 0.05 and the corresponding chi-square values obtained from a chi-square table are
χ
χ χ χ
χ χα
α
/ , . / , . ,
/ ,
.2 2
0 05 2 9 2
0 025 9 2
1 2
19 023k
k
= = =
− 22
1 0 05 2 9 2
0 975 9 2 2 700= = =− . / , . , .
The resulting 95% two-sided confidence interval (using Equation (6.62)) on the popula- tion variance is
( ) . .
( ) . .
10 1 262 44 19 023
10 1 262 44 2 700
124
2− < < −
=
σ
.. .16 874 802< <σ
With a 95% level of confidence, we believe that the true population variance in heart rate lies within 124.16 bpm2 and 874.80 bpm2.
An approximate confidence interval on the population standard deviation σ can easily be found by taking the square of the bounds on the population variance. An approximate 100(1 – α)% confidence interval on σ is given by
( ) ( )
/ , / ,
n s n s
k k
− < <
−
−
1 12
2 2
2
1 2 2
σ χχ αα
(6.63)
ExaMpLE 6.40
Reconsider the previous example. A 95% confidence interval (using Equation (6.63)) on the standard deviation of heart rate is
124 16 874 80
11 14 29 58
2. .
. .
< <
< <
σ σ
With a 95% level of confidence, we believe the true population standard deviation will lie between 11.14 bpm and 29.58 bpm.
D.2.c. Confidence Intervals on a Population Proportion p
The population proportion p often represents the fraction nonconforming or the fraction of defective items in a population of items. Since it is often impossible or impractical to find the exact value of p, a point estimate is used to estimate the true
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proportion. A sample proportion denoted p̂ will be used as a point estimate for p. The sample proportion is calculated using
p̂
x n
=
(6.64)
where x is the number of successes out of n trials. If n is sufficiently large, then it can be shown that the random variable
Z p p
p p n
= −
−
ˆ
( )1
(6.65)
follows a standard normal distribution. This relationship allows for the develop- ment of a confidence interval on the true population proportion of interest.
A 100(1 – α)% confidence interval on a population proportion p is given by
ˆ
ˆ( ˆ ) ˆ
ˆ( ˆ ) / /p z
p p n
p p z p p
n −
− ≤ ≤ +
− α α2 2
1 1
(6.66)
ExaMpLE 6.41
Billing statements for 1000 patients discharged from a particular hospital were randomly selected for errors. Out of the 1000 billing statements, 102 were found to contain errors. Using this information, it is desired to construct a 99% confidence interval on the true proportion of billing statements with errors.
Solution: In this problem, a “success” is a billing statement with errors. Therefore, X = 102 and n = 1000. The sample proportion (using Equation (6.64)) is then
ˆ .p = = 102
1000 0 102
and provides our best estimate for the true proportion of billing statements with errors. The corresponding 99% confidence interval (using Equation (6.66)) is
ˆ ˆ ( ˆ )
ˆ ˆ ( ˆ )
. .
/ /p z p p
n p p z
p p n
− −
≤ ≤ + −
−
α α2 2 1 1
0 102 2 576 0.. ( . )
. . . ( . )102 1 0 102
1000 0 102 2 576
0 102 1 0 102 1
− ≤ ≤ +
− p
0000
0 077 0 127. .≤ ≤p
With a 99% level of confidence, we believe the true proportion of billing statements that contain errors would be between 7.7% and 12.7%.
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Note: The procedure outlined here for the population proportion assumes that the normal distribution is a good approximation to the binomial. That is, the param- eter of interest is a proportion that is well modeled by the binomial distribution, but if the sample size is large enough, the normal distribution is a good approxi- mation to the binomial.
Recommendations have been that np ≥ 5 and np(1 – p) ≥ 5. However, if the sample size is small, the confidence interval given previously may not be a good approximation, and the binomial distribution rather than the standard normal distribution should be used to find the multiple of the standard error in the con- fidence interval. Agresti and Coull (1998) present an alternative form of a confi- dence interval for small n (in addition, see Devore [2016]).
D.2.d. Statistical Tolerance Intervals
Consider a population of manufactured steel rods. Suppose the diameters of these steel rods follow a normal distribution with mean μ = 25 mm and standard devia- tion σ = 4 mm. Then the interval (μ – 1.96σ, μ + 1.96σ) = (17.16, 32.84) would capture the diameters of 95% of the steel rods. This is a result of the fact that 95% of the area under the normal curve lies between –1.96 and 1.96.
In many situations, μ and σ are unknown and are estimated using x– and s. The interval ( x– – 1.96s, x– + 1.96s) may not actually contain 95% of the values in the population (since there is variability in the estimates in this case, there is no guar- antee that the tolerance interval will contain 95% of the values). The solution to the problem is to replace 1.96 with some value that will make the resulting interval contain 95% of the values with a high level of confidence. This interval is referred to as a tolerance interval.
Suppose we wish to capture at least τ% of the values in a normal distribution with a 100(1 – α)% level of confidence. The appropriate two- sided tolerance inter- val is
x – – Ks, x– + Ks (6.67)
where K is a tolerance interval factor found in Appendix C. Only selected values of K are given in the table, in particular for 99% of the population for 90%, 95%, and 99% levels of confidence.
ExaMpLE 6.42
Suppose the time to complete a task is of interest. A sample of n = 10 times is collected and it is found that x– = 31.50 and s = 2.764. Time to complete the task is assumed to be normally distributed. Suppose we want to find a tolerance interval for time that includes 99% of the values in the population with 95% confidence.
From Appendix C with n = 10 and confidence level of 0.95 we find K = 4.433. The resulting tolerance interval is
(x– – Ks, x– + Ks) = (31.50 – 4.433(2.764), 31.50 + 4.433(2.764)) = (19.25, 43.75)
We can be 95% confident that at least 99% of all times to complete the task are between 19.25 minutes and 43.75 minutes.
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d.3. Hypothesis Testing
The hypothesis test, another tool used in inferential statistics, is closely related to confidence intervals, a relationship that is illustrated in this section. Textbooks tend to treat hypothesis tests as somewhat more formal procedures. Many list seven or eight steps to be followed for each type of test. Although not all books agree on the steps themselves, they share some generic steps:
1. State the null hypothesis (H0) and alternative hypothesis (Ha)
2. Choose the level of significance α
3. Determine the rejection region for the statistic of interest
4. Calculate the test statistic
5. Decide whether the null hypothesis should be rejected
6. State the conclusion in terms of the original problem
General descriptions of each step are provided.
1. Write the assumption that is claimed to be true in a null hypothesis. The null hypothesis is denoted by H0. This is the statement that we assume to be true and that we are trying to find evidence against. For example, from past experience the average time to complete a task is five minutes, and we would like to test this claim. The null hypothesis would be H0: μ = 5.
State the alternative hypothesis. The notation for the alternative hypothesis is usually Ha. If the null hypothesis is rejected, then an alternative must be provided. The alternative hypothesis contains the statement that we would eventually like to support (we can support the alternative only if the null hypothesis is rejected). For example, suppose we wish to show that the average time to complete a task is actually longer than five minutes. The alternative hypothesis would be Ha: μ > 5.
2. Choose the level of significance for the test. The significance level (denoted α) is the probability of making the mistake of rejecting the null hypothesis when it is in fact true. This type of mistake is called a type I error (as opposed to a type II error, which is the mistake of not rejecting the null hypothesis when it is in fact false). The probability of committing a type I error should be small (i.e., α ≤ 0.10) and should be set before performing the test. The significance level α is the same α used in the level of confidence in constructing confidence intervals (recall that 1 – α is the level of confidence, so α is the level of significance). Choosing the level of significance is sometimes an economic decision. It is established based on how expensive a particular mistake may be (this expense is not necessarily monetary and could include personal injury or loss of life).
3. Determine the rejection region. The rejection region consists of all those values of the test statistic for which the null hypothesis would be rejected. The rejection region is determined by the stated level of significance and the alternative hypothesis. Critical values are those values that determine the rejection region (they are cutoff values for the test statistic—discussed in the next step).
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4. Calculate the test statistic. The test statistic is simply a statistic (such as the sample mean or sample variance) that has been transformed in order to compare this value with some standard (critical values). An example of a test statistic is a sample mean X
– transformed into a Z-score. The Z-score is used to determine whether the null hypothesis should be rejected.
5. Compare the test statistic with the critical value. If the test statistic falls within the rejection region, then the null hypothesis would be rejected. Large test statistic values (in absolute value) will provide evidence against the null hypothesis. It is important to note that rejecting the null hypothesis is a strong claim. It takes significant evidence to reject the claim associated with H0. As a result of using a small level of significance, rejecting the null hypothesis is a strong claim. Failing to reject the null hypothesis is a weak claim. That is, by failing to reject the null hypothesis, we cannot say that the claim is true, just that we did not have sufficient evidence to reject the claim.
6. Once it has been determined whether the null hypothesis can be rejected, the results are written in terms of the problem statement.
D.3.a. Hypothesis Tests for a Single Population
In this section we present hypothesis tests for the following:
• A single population mean μ
• A single population variance σ 2
• A single population proportion p
D.3.a.i. Hypothesis Tests for a Single Population Mean μ
Let μ0 be a real value that is hypothesized or assumed to be the true value of the population mean μ. There are three types of hypothesis tests:
H0: μ ≤ μ0 Ha: μ > μ0 (right-tailed test)
H0: μ ≥ μ0 Ha: μ < μ0 (left-tailed test)
H0: μ = μ0 Ha: μ ≠ μ0 (two-tailed test)
As with confidence intervals, there are two cases under study:
1. Hypothesis tests on μ when the population variance σ 2 is assumed known
2. Hypothesis tests on μ when the population variance σ 2 is unknown
Case 1. Hypothesis Tests on μ When the Population Variance σ2 Is Assumed Known Assume that the underlying population of interest is normally distributed with known population variance σ 2. Let X1, X2, . . . , Xn be a random sample from the population, and let X
– represent the sample mean. An appropriate test statistic for this situation is
z
x
n 0
0= − µ
σ/ (6.68)
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The rejection region, shown in Table 6.8, depends on the alternative hypothesis and level of significance α.
Table 6.8 Rejection regions for a single sample mean, variance known.
Alternative hypothesis Reject H0 if
Ha: μ > μ0 z0 > zα
Ha: μ < μ0 z0 < –zα
Ha: μ ≠ μ0 z0 < –zα/2 or z0 > zα/2
ExaMpLE 6.43
A vendor claims that the average weight of a shipment of parts is 1.84 kg. The customer randomly chooses 64 parts and finds that the sample has an average weight of 1.88 kg. Suppose that the standard deviation of the population is known to be 0.3 kg. Using a level of significance of 0.05, we want to test the hypothesis that the true average weight of the shipment is 1.84 kg. We assume that the weights are normally distributed.
Solution: Since the population standard deviation is known, the standard normal distribution is used. In addition, since the problem statement gives no indication as to whether the true mean weight is less than or greater than 1.84 kg, a two-sided hypothesis test is appropriate. The steps are as follows:
1. The null and alternative hypotheses are
H0: μ = 1.84 Ha: μ ≠ 1.84.
2. The level of significance is α = 0.05.
3. There are two areas for the rejection region since we have a two-tailed test. The critical values are found from a standard normal table with α = 0.05:
zα/2 = z0.05/2 = z0.025 = 1.96 and –zα/2 = –1.96
Therefore, reject the null hypothesis if the test statistic is greater than 1.96 or less than −1.96.
4. The test statistic (using Equation (6.68)) is calculated to be:
z x
n 0
0 1 88 1 84 0 30 64
1 07= −
= −
= µ
σ/ . . . /
.
(Note: This test statistic says the following: the sample mean is roughly 1.07 standard deviations away from the population mean. Is this a large difference? The next step tells us how large is large.)
5. Comparing 1.07 with the critical values in step 3, we see that −1.96 < 1.07 < 1.96. Since the test statistic, 1.07, does not fall into the rejection region, we cannot reject the null hypothesis at the 0.05 level of significance. We fail to reject the null hypothesis (weak claim).
6. There is insufficient evidence to conclude that the true mean weight of the shipment is different from 1.84.
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Case 2. Hypothesis Tests on μ When the Population Variance σ2 Is Unknown Assume that the underlying population of interest is normally distributed with unknown population variance σ 2. Let X1, X2, . . . , Xn be a random sample from the population, and let X
– represent the sample mean. If the sample size is small, an appropriate test statistic for this situation is
t
s n 0 =
/
x 0− µ
(6.69)
which follows a t distribution with k = n – 1. The rejection region, shown in Table 6.9, depends on the alternative hypothesis and level of significance α.
It is important to note here that the fact that the null hypothesis is not rejected does not mean it is true. It only indicates that with this sample of data we could not find evidence against the null hypothesis. If a second sample of data is taken, it is possible that the null hypothesis could be rejected.
A summary of the situations outlined for testing the population mean is shown in Table 6.10 (assume the underlying population is normally distributed).
ExaMpLE 6.44
A cutoff saw has been producing parts with a mean length of 4.125 mm. A new blade is installed and we want to know whether the mean has decreased. We select a random sample of 20 parts, measure the length of each part, and find the mean length to be 4.120 mm and the sample standard deviation to be 0.008 mm. Assume that the population is normally distributed. Using a significance level of 0.10, determine whether the mean length has decreased. Since the population standard deviation is unknown, the t-test will be used.
Solution: 1. The null and alternative hypotheses are
H0: μ = 4.125 Ha: μ < 4.125.
2. The level of significance is α = 0.10.
3. There is only one rejection region since this is a left-tailed test (again, this is based on the alternative hypothesis). The critical value is found from the t distribution table in Appendix O with α = 0.10: tα,k = t0.10,19 = 1.328. Therefore, reject the null hypothesis if the test statistic is less than –1.328 (since the alternative hypothesis is “less than”).
4. The test statistic (using Equation (6.69)) is calculated to be:
t x s n
0 0 4 120 4 125
0 008 20 2 80=
− =
− = −
µ /
. . . /
.
5. Comparing –2.80 to the critical value in step 4, we see that –2.80 < –1.328. Since the test statistic falls into the rejection region, we can reject the null hypothesis at the 0.10 level of significance (strong claim).
6. There is sufficient evidence to indicate that the average length of the part has decreased.
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D.3.a.ii. Hypothesis Tests on a Single Population Variance σ 2
The hypothesis testing procedure can be applied in the case of testing a standard or hypothesized value of the population variance. For example, we may be inter- ested in testing the claim that a particular population variance is eight. The null hypothesis is H0: σ
2 = 8. Assume that the underlying distribution is normal. Let σ0 2
be a real value that is hypothesized or assumed to be the true value of the popula- tion variance σ 2. There are three types of hypothesis tests on a population variance:
H0: σ 2 ≤ σ0
2 Ha: σ 2 > σ0
2 (right-tailed test)
H0: σ 2 ≥ σ0
2 Ha: σ 2 < σ0
2 (left-tailed test)
H0: σ 2 = σ0
2 Ha: σ 2 ≠ σ0
2 (two-tailed test)
The chi- square distribution introduced in section C is an appropriate distribution for testing a population variance. The test statistic is given by
χ
σ0 2
2
0 2
1 =
−( )n s
(6.70)
Table 6.9 Rejection regions for a single sample mean, variance unknown.
Alternative hypothesis Reject H0 if
Ha: μ > μ0 t0 > tα,k
Ha: μ < μ0 t0 < –tα,k
Ha: μ ≠ μ0 t0 < –tα/2,k or t0 > tα/2,k
Table 6.10 Summary of situations outlined for testing the population mean.
Assumption Distribution Test statistic
σ 2 known, population normally distributed
Standard normal distribution z
x
n 0
0= − µ
/σ
σ 2 unknown and n small (n ≤ 30), population normally distributed
t distribution t
x
s n 0
0= − µ
/
σ 2 unknown and n large (often n > 40), population not necessarily normal
Standard normal distribution or t distribution* orz
x
s n 0
0= − µ
/
t x
s n 0
0= − µ
/
* Recall that as n goes to infinity (n → ∞), the form of the t distribution becomes indistinguishable from the standard normal distribution.
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The test statistic follows a chi- square distribution with k = n – 1 degrees of free- dom, where
• s2 is the sample variance for a sample taken from the population of interest
• σ0 2 is the hypothesized value of the variance
• n is the sample size chosen from the population of interest
The hypothesis testing steps outlined earlier also apply to this situation, and the rejection regions are shown in Table 6.11.
The critical values for the chi- square distribution can be found in Appendix J.
Table 6.11 Rejection regions for a single sample hypothesis test on the variance.
Alternative hypothesis Reject H0 if
Ha: σ 2 > σ 0
2 χ0 2 > χ α
2 ,k
Ha: σ 2 < σ 0
2 χ0 2 < χ 1
2 –α,k
Ha: σ 2 ≠ σ 0
2 χ0 2 < χ 1
2 –α/2, k or χ 0
2 > χ α 2 /2,k
ExaMpLE 6.45
A process has been running for some time with a variance of 6.25 for a critical dimen- sion. In an effort to improve throughput, a methods engineer increases the drive motor speed. A sample of 13 items is randomly selected from the new process. The variance of the critical dimension in this sample is found to be 6.82. Is there sufficient evidence to conclude that the true process variance has increased? Use α = 0.05. Assume that the critical dimension follows a normal distribution.
Solution:
1. The null and alternative hypotheses are
H0: σ 2 = 6.25 H1: σ
2 > 6.25
2. The level of significance is α = 0.05
3. Since this is a right-tailed test, we do reject the null hypothesis if the test statistic is greater than χ 2α ,k = χ
2 0.05,12 = 21.026
4. The test statistic (using Equation (6.70)) is calculated to be
χ σ0
2 2
0 2
1 13 1 6 82 6 25
13 1= −
= −
= ( ) ( ) .
. .
n s
5. Since 13.1 < 21.026, we fail to reject the null hypothesis (weak claim)
6. There is insufficient evidence to conclude that the variance for this critical dimen- sion has increased at the 0.05 level of significance
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D.3.a.iii. Hypothesis Tests on a Single Population Proportion p
The hypothesis testing procedure can be applied in the case of testing a standard or hypothesized value of a population proportion. For example, we may be inter- ested in testing the claim that a particular population proportion is 0.50. The null hypothesis is then H0: p = 0.50.
Let p0 be a real value that is hypothesized or assumed to be the true value of the population proportion. There are three types of hypothesis tests on a popula- tion proportion:
H0: p ≤ p0 Ha: p > p0 (right-tailed test)
H0: p ≥ p0 Ha: p < p0 (left-tailed test)
H0: p = p0 Ha: p ≠ p0 (two-tailed test)
The null hypothesis is often written simply as H0: p = p0 for each of the three cases. Either notation is acceptable.
For sufficiently large sample sizes, the normal approximation to the binomial distribution is valid and the test statistic
z p p
p p n
0 0
0 01 =
− −
ˆ
( )
(6.71)
follows a standard normal distribution where p̂ is the sample proportion (see Equation (6.64)) and p0 is the hypothesized value of the population proportion.
If the sample sizes are relatively small, then the appropriate hypothesis test to use is based directly on the binomial distribution. See Devore (2016) for more details on small- sample tests.
The rejection region, shown in Table 6.12, depends on the alternative hypoth- esis and level of significance α.
Table 6.12 Rejection regions for a single sample hypothesis test on the proportion.
Alternative hypothesis Reject H0 if
Ha: p > p0 z0 > zα
Ha: p < p0 z0 < –zα
Ha: p ≠ p0 z0 < –zα/2 or z0 > zα/2
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ExaMpLE 6.46
Billing statements for discharged patients from a particular hospital sometimes contain errors. It is believed that the percentage of billing statements that contain errors is 15%. Out of 1000 billing statements randomly selected from the population, 102 were found to contain errors. Based on this information, can we conclude that the proportion of billing statements that contain errors is actually less than 15%? Use a 10% level of significance.
Solution: For this problem:
• p = the true proportion of billing statements with errors
• x = number of statements with errors, so x = 102
• n = sample size, so n = 1000
• The sample proportion is then ˆ .p x n
= = = 102
1000 0 102
1. The null and alternative hypotheses are
H0: p = 0.15 Ha: p < 0.15
2. The level of significance is α = 0.10
3. Since this is a left-tailed test, reject the null hypothesis if the test statistic is less than –zα = –z0.10 = –1.28
4. The test statistic (using Equation (6.71)) is
z p p p p
n
0 0
0 01 0 102 0 15 0 15 1 0 15
100
= −
− =
− −
ˆ ( )
. . . ( . )
00
4 25= − .
5. Since – 4.25 < –1.28, we reject the null hypothesis in favor of the alternative at the 0.10 level of significance (strong claim)
6. There is sufficient evidence to conclude that the true percentage of statement errors is less than 15%
D.3.b. Hypothesis Tests and Confidence Intervals for Two Independent Populations
Hypothesis tests and confidence intervals for two independent population means, variances, and proportions are presented in this section.
D.3.b.i. Hypothesis Tests and Confidence Intervals for Two Independent Population Means μ1 and μ2
There are many important cases that involve comparing two populations of inter- est, such as two processes, two vendors, and so on. Of interest here is testing the difference between two population means μ1 – μ2. One test in particular is that of no difference between the two populations μ1 – μ2 = 0.
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Either the standard normal distribution or Student’s t distribution can be used. If the population variances are known, use the standard normal distribution. If the population variances are unknown and the sample sizes are relatively small, use Student’s t distribution.
The basic assumptions are as follows:
• X1, X2, . . . , Xn1 is a random sample from a population with mean μ1 and variance σ 1
2
• Y1, Y2, . . . , Yn2 is a random sample from a population with mean μ2 and variance σ 2
2
• The two samples are independent of each other
A good point estimator for μ1 – μ2 would be the difference between the two sample means, X
– – Y – . The expected value and variance for this point estimator are
E X Y−( ) = −µ µ1 2 (6.72) and
V X Y
n nX Y −( ) = = +−σ
σ σ2 1 2
1
2 2
2 (6.73)
If we assume that the populations are normally distributed, then X – – Y
– will also be normally distributed. Therefore, the random variable
Z
n n
=
+ σ σ
X Y−( ) − −( )µ µ1 2 1 2
1
2 2
2
(6.74)
follows a standard normal distribution. As a result, hypothesis tests and confi- dence intervals can be constructed on the parameter μ1 – μ2.
Case 1. Hypothesis Test on μ1 – μ2, with Population Variances Known Let ∆0 be a real value that represents the difference between μ1 and μ2 that is of interest to be tested. Specifically, the null hypothesis would be H0: μ1 – μ2 = ∆0. For the test of no difference, ∆0 = 0. The hypothesis tests, test statistic, and rejection regions (Table 6.13) are as follows:
Null hypothesis: H0: μ1 – μ2 = ∆0
Test statistic: z x y
n n
0 0
1 2
1
2 2
2
= −( ) −
+ σ σ
Δ (6.75)
The null hypothesis does not always have to be μ1 – μ2 = ∆0; it could be ≤ or ≥ also. We will use the null hypothesis of H0: μ1 – μ2 = ∆0.
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Example 6.47 illustrates the use of hypothesis testing for the differences between two population means. The steps for the hypothesis tests are identical to the six steps outlined earlier.
ExaMpLE 6.47
Two different formulations of gasoline are being tested to study their road octane num- bers. Formulation 1 has a variance octane number of σ 21 = 1.45 while the variance for for- mulation 2 is σ 22 = 1.5. Ten samples (n1 = 10) are selected from formulation 1 and fifteen samples (n2 = 15) are selected from formulation 2. For sample 1, the average octane num- ber was found to be x– = 89, and for sample 2 the average octane number was found to be y– = 91. Is there significant evidence to indicate that a difference exists between the two formulations? Use a 0.05 level of significance.
Solution: The parameter of interest is the difference in average octane number, μ1 – μ2. Since there is no indication that μ1 > μ2 or μ1 < μ2, a two-sided test is used.
1. H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0
2. α = 0.05
3. Since this is a two-tailed test and α = 0.05, we will reject the null hypothesis if the test statistic is less than –zα/2 or greater than zα/2, where zα/2 = z0.025 = 1.96
4. The test statistic (using Equation (6.75)) is
z x y
n n
0 0
1 2
1
2 2
2
89 91 0 1 45 10
1 5 15
4= −( ) −
+ =
−( ) −
+ = −
∆
σ σ . . ..04
5. Since – 4.04 < –1.96, we reject the null hypothesis (strong claim)
6. We conclude there is a significant difference in average octane number for the two formulations at the 0.05 level of significance
A 100(1 – α)% confidence interval on the parameter μ1 – μ2 is given by
x y z
n n n −( ) x y z−( )− + ≤ − ≤ − +α α
σ σ σ σ µ µ/ /2
1 2
1
2 2
2 1 2 2
1 2
1
++ 2 2
2n (6.76)
Table 6.13 Rejection region for a hypothesis test on the means of two independent samples, variance known.
Alternative hypothesis Reject H0 if
Ha: μ1 – μ2 > Δ 0 z0 > zα
Ha: μ1 – μ2 < Δ 0 z0 < –zα
Ha: μ1 – μ2 ≠ Δ 0 z0 < –zα/2 or z0 > zα/2
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ExaMpLE 6.48
Reconsider the formulation problem given in the previous example. The 95% confi- dence interval on μ1 – μ2 is
x y z n n
z n
−( ) x y−( )− + ≤ − ≤ +α σ σ α σ σµ µ
µ µ
µ µ
/ /2 1 2
1
2 2
2 1 2 2
1 2
1
++
−( ) − + ≤ − ≤ −
2 2
2
1 289 91 1 96 1 45 10
1 5 15
89 91
n
. . . (( ) + +
− ≤ − ≤ −
1 96 1 45 10
1 5 15
2 97 1 031 2
. . .
. .
We are 95% confident that the true difference between the average octane numbers lies between –2.97 and –1.03. Note that the hypothesized value ∆0 = 0 given in the null hypothesis of the previous problem is not contained in this interval. Therefore, we again say there is a significant difference in the two population means (i.e., we reject the null hypothesis).
Case 2. Hypothesis Test on μ1 – μ2, with Population Variances Unknown If the sample sizes are relatively large, regardless of the underlying distributions of the populations of interest, we can use the standard normal distribution, as in the case of known variances. The sample variances are used as estimates of the population variances.
If the sample sizes are relatively small and the underlying distributions are normally distributed, then the t distribution can be used to conduct hypothesis tests and construct confidence intervals. There are two methods for this situation: (1) the population variances are unknown but assumed roughly equal, and (2) the population variances are unknown and not necessarily equal. The three basic assumptions given earlier still hold for the following methods. Let 1
2s represent the variance for sample 1 and 2
2s represent the variance for sample 2.
Method 1. σ 2 = 2 σ 2σ 1 =
2 (The population variances are unknown but assumed roughly equal.) Since 1
2s and 2 2s estimate the same common variance σ 2, yet the
sample variances may not be equal, we can combine the sample variances to obtain a single point estimate for σ 2. This is commonly called the pooled variance:
s
n s n s n np
2 1 1 2
2 2 2
1 2
1 1 2
= − + −
+ − ( ) ( )
(6.77)
The pooled standard deviation can be found by taking the square root of the pooled variance. The random variable T below
T X Y
σ n n
0
1 2
1 1 =
−( )−
+
Δ
(6.78)
follows a t distribution with k = n1 + n2 – 2 degrees of freedom.
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The hypothesis tests, test statistic, and rejection region (Table 6.14) are as follows:
Null hypothesis: H0: μ1 – μ2 = ∆0
Test statistic: t x y
s n np
0 0
1 2
1 1 =
−( )− +
∆ (6.79)
where k = n1 + n2 – 2 is the total degrees of freedom.
Table 6.14 Rejection regions for a hypothesis test on the means of two independent samples, variance equal, but unknown.
Alternative hypothesis Reject H0 if
Ha: μ1 – μ2 > Δ 0 t0 > tα,k
Ha: μ1 – μ2 < Δ 0 t0 < –tα,k
Ha: μ1 – μ2 ≠ Δ 0 t0 < –tα/2,k or t0 > tα/2,k
ExaMpLE 6.49
Two vendors of a valve diaphragm present significantly different cost quotations. The wall thickness is the critical quality characteristic. Use the following data to determine whether the average thickness of the products from vendor 1 is greater than that from vendor 2. Assume that the populations are normally distributed and that the samples are independent. Furthermore, the population variances are unknown but assumed to be equal. The test is to be conducted at the 0.05 significance level. The wall thickness mea- surements for both vendors are shown in Table 6.15.
Table 6.15 Wall thickness measurements for two vendors.
Vendor 1: 86 82 91 88 89 85 88 90 84 87 88 83 84 89
Vendor 2: 79 78 82 85 77 86 84 78 80 82 79 76
Solution: The necessary summary statistics are:
Vendor 1: x– = 86.7 s1 = 2.76 n1 = 14
Vendor 2: y– = 80.5 s2 = 3.26 n2 = 12
Since the population variances are unknown but assumed equal, the pooled variance and pooled standard deviation should be calculated (using Equation (6.77)):
s n s n s
n np 2 1 1
2 2 2
2
1 2
21 1 2
14 1 2 76 1 =
− + − + −
= − +( ) ( ) ( )( . ) ( 22 1 3 26
14 12 2 9
9 3
2− + −
=
= =
)( . )
sp
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The parameter of interest is the difference in average wall thickness, μ1 – μ2.
1. H0: μ1 – μ2 = 0 Ha: μ1 – μ2 > 0
2. α = 0.05
3. Since this is a right-tailed test and α = 0.05, we will reject the null hypothesis if the test statistic is greater than tα,k where from Appendix O we find tα,k = t0.05,24 = 1.711
4. The test statistic (using Equation (6.78)) is:
t x y
s n np
0 0
1 2
1 1 86 7 80 5 0
3 1
14 1
12
= −( )−
+
= −( )−
+
= . .
55 25. Δ
5. Since 5.25 > 1.711, we reject the null hypothesis (strong claim)
6. We conclude that the average thickness of the products from vendor 1 is greater than that from vendor 2 at the 0.05 level of significance
A 100(1 – α)% two- sided confidence interval on the parameter μ1 – μ2 is given by
t s
n n t sk p k− ( ) + ≤ − ≤ +/ , / ,2
1 2 1 2 2
1 1 pp n n
( ) +1 1 1 2
x y−( ) x y−( )α αµ µ
(6.80)
It is important to note that the test statistic based on the t distribution is robust to the common variance assumption. In addition, it is not necessarily a good idea to do formal testing on the equality of two variances (see Box [1954]).
Method 2. σ 2 2σ 1 ≠
2 (The population variances are unknown and not necessarily equal.) The main differences between methods 1 and 2 are the calculation of the test statistic and the calculation of the degrees of freedom. Since the population variances are not necessarily equal, 1
2s and 2 2s do not estimate a common popula-
tion variance; 1 2s is an estimate for σ 1
2, and 2 2s is an estimate of σ 2
2, and they can be used directly in the test statistic
t s n
s n
0
1 2
1
2 2
2
=
+
x y 0−( )− Δ
(6.81)
This test statistic is based on random variable T, which follows a t distribution with degrees of freedom:
ν = +
( ) −
+ ( )
s n
s n
s n
n
s n
1 2
1
2 2
2
2
1 2
1
2
1
2 2
2
2
1
/ /
n2 1−
e e
(6.82)
which is rounded down if it is not an integer value. Devore (2016) recommends using method 2 unless there is strong evidence that the variances of the two popu- lations are equal.
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A 100(1 – α)% two- sided confidence interval on the parameter μ1 – μ2 is given by
t
s n
s n
t s
− + ≤ − ≤ ( ) +/ , / ,2 1 2
1
1 2
2 1 2 2
11 2
1
1 2
2n s n
+αµ µx y−( ) α v vx y−
(6.83)
D.3.b.ii. Hypothesis Tests and Confidence Intervals for Two Population Variances σ 1 2, σ 2
2
A test on the variances of two populations can be used to determine whether the variance of one is greater than the other. For example, the null hypothesis could be H0: σ 2
2σ 1 = 2 , where σ 1
2 is the variance of population 1 and σ 2 2 is the variance of
population 2. This test uses the F distribution, presented in section C. Let Y1, Y2, . . . , Yn1 be a random sample from a normal distribution with vari-
ance σ 1 2, and let W1, W2, . . . , Wn2 be a random sample from a normal distribution
with variance σ 2 2. Furthermore, assume that the samples are independent of each
other and 1 2s and 2
2s are the sample variances for Yi’s and Wj’s, respectively. The random variable
F
s s
= 1 2
1 2
2 2
2 2
/ / σ σ
(6.84)
follows an F distribution with k1 = n1 – 1 and k2 = n2 – 1 degrees of freedom. The hypothesis tests, test statistic, and rejection regions (Table 6.16) are as
follows:
Null hypothesis: H0: σ 2 2σ 1 =
2
Test statistic: F0 = 2 2
1/s s 2 (6.85)
The critical values for the F distribution can be found in Appendixes F, G, and H. The notation Fα,k1,k2 is the F value with area of α to its right, k1 represents the numera- tor degrees of freedom, and k2 represents the denominator degrees of freedom. A probability from this table represents the area under the curve and to the right of the F value of interest. For example, for a 0.05 level of significance with k1 = 15 and k2 = 5 degrees of freedom, the F value would be F0.05,15,5 = 4.62. The F table provides values for specific values of α (level of significance). To find critical values when the area to the right is 1 – α, such as F1–α,k1, k2, we have to use the following relationship:
F
Fk k k k 1 1 2
2 1
1 − =, ,
, , α
α (6.86)
Table 6.16 Rejection regions for a hypothesis test on two independent variances.
Alternative hypothesis Reject H0 if
Ha: σ 1 2 > σ 2
2 F0 > Fα,k1, k 2
Ha: σ 1 2 < σ 2
2 F0 < F1– α,k1, k 2
Ha: σ 1 2 ≠ σ 2
2 F0 < F1– α/2,k1, k 2 or F0 > Fα/2,k1, k 2
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To illustrate, suppose we want to find a critical value for a left- tailed test where k1 = 10, k2 = 8, and α = 0.05. Then F1–α,k1, k2 = F0.95,10,8 and
F F0 95 10 8 0 05 8 10
1 1 3 07
0 326. , , . , , .
.= = =
ExaMpLE 6.50
Two chemical companies can supply a particular material. The concentration of an element in this material is important. The mean concentration for both suppliers is approximately the same, but we suspect that the variability in concentration may differ between the two companies. The standard deviation of concentration in a random sample of n1 = 10 batches produced by company 1 is s1 = 3.8 g/l, while for company 2, a random sample of n2 = 16 batches yields s2 = 4.2 g/l. Using a level of significance of 0.10, we would like to determine if there is sufficient evidence to conclude that the two popu- lation variances differ.
Solution: Let:
• σ 21 be the population variance of the element concentration in the material from company 1
• σ 22 be the population variance of the element concentration in the material from company 2
1. The null and alternative hypotheses are
H0: σ 2 1 = σ
2 2 H1: σ
2 1 ≠ σ
2 2.
2. The level of significance is α = 0.10.
3. There are two rejection regions since we have a two-tailed test. Furthermore, the degrees of freedom are k1 = n1 – 1 = 9 and k2 = n2 – 1 = 15. The critical values are
Fα/2,k1,k2 = F0.05,9,15 = 2.59
and
F F Fk k
0 950 9 15 2 0 05 15 9
1 1 1 3 01
0 332 2 1
. , , / , , . , , .
.= = = = α
Therefore, reject the null hypothesis if the test statistic is less than 0.332 or greater than 2.59.
4. The test statistic (using Equation (6.85)) is
F s s0
1 2
2 2
2
2
3 8
4 2 0 819= =
( ) ( )
= . .
.
5. Since 0.332 < 0.819 < 2.59, we fail to reject the null hypothesis at the 0.10 level of significance (weak claim).
6. There is insufficient evidence to conclude that the variances of the element concentration in this material from the two suppliers are not equal.
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A 100(1 – α)% two- sided confidence interval on the parameter σ 2 2σ 1/
2 is given by
s s
F s s
Fk k k k 1 2
2 2 1 2
1 2
2 2 21 2 1 2−
≤ ≤/ , , / , ,σ 2 2
σ 1 2
α α
(6.87)
The confidence interval on the ratio of the two variances can be used to determine whether there is a significant difference between the two variances. In hypothesis testing we assume the null hypothesis is true, that is, H0: σ 2
2σ 1 = 2 . We can rewrite
this equality as
1= σ 2
2
σ 1 2
When constructing a two- sided confidence interval on the ratio of the two vari- ances, we would reject the null hypothesis if the value 1 is not contained within that interval. For example, the 95% two- sided confidence interval on the ratio of variances of the previous example is
0 217 2 555. .≤ ≤ σ 2
2
σ 1 2
Therefore, we are highly confident that the true ratio of the variances lies between 0.217 and 2.555. Since this interval contains the value 1, we would conclude based on our samples that there is no statistically significant difference between the two variances.
D.3.b.iii. Hypothesis Tests and Confidence Intervals on Two Population Proportions p1, p2
There are many situations where we want to determine whether two populations differ with respect to some proportion of successes or failures. For example, we may wish to determine whether two machines from the same process are producing the same proportion of nonconforming items. The hypothesis could be H0: p1 – p2 = 0, where p1 is the proportion of successes from population 1 and p2 is the proportion of successes from population 2. The estimates of p1 and p2 are, respectively,
ˆ , ˆp x n
p x n1
1
1 2
2
2
= =
where
n1 is the size of the sample chosen from population 1
n2 is the size of the sample chosen from population 2
x1 is the number of successes out of a sample of size n1
x2 is the number of successes out of a sample of size n2
Note that these proportions are the same calculations from Equation (6.64), but with respective sample sizes and successes.
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The parameter of interest is the difference in the two population proportions p1 – p2. The point estimator for this parameter is p̂1 – p̂2. The expected value and variance for the point estimator are
E ˆ ˆp p p p1 2 1 2−( ) = − (6.88)
and
V ˆ ˆ
( ) ( ) ˆ ˆp p
p p n
p p np p1 2
2 1 1
1
2 2
2 1 2
1 1 −( ) = = − + −−σ
(6.89)
The random variable
z p p p p
p p n
p p n
1 2 1 2
1 1
1
2 2
2
1 1 =
− − − −
+ −
( ˆ ˆ ) ( )
( ) ( )
(6.90)
follows an approximate standard normal distribution when the approximations as described in Figure 6.16 hold for both samples. Under the assumption that the null hypothesis (H0: p1 – p2 = 0) is true, then the random variable Z, from Equa- tion (6.90), can be written as
z p p
p p n n
1 2
1 2
0
1 1 1
= − −
− +
( ˆ ˆ )
ˆ( ˆ )d d
where p̂ is the proportion resulting from the combination of the two samples (esti- mate of the overall proportion when we are testing p1 = p2). The formula is
p̂
x x n n
= + +
1 2
1 2 (6.91)
The hypothesis tests, test statistic, and rejection regions (Table 6.17) are as follows:
Null hypothesis: H0: p1 – p2 = 0
Test statistic: z p p
p p n n
0 1 2
1 2
0
1 1 1
= − −
− +
( ˆ ˆ )
ˆ( ˆ )d d (6.92)
Table 6.17 Rejection regions for a hypothesis test on two independent proportions.
Alternative hypothesis Reject H0 if
Ha: p1 – p2 > 0 z0 > zα
Ha: p1 – p2 < 0 z0 < –zα
Ha: p1 – p2 ≠ 0 z0 < –zα/2 or z0 > zα/2
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ExaMpLE 6.51
Two machines produce the same parts. A random sample of 1500 parts from machine 1 has 36 that are nonconforming, and a random sample of 1680 parts from machine 2 has 39 that are nonconforming. Is there evidence to suggest that machine 1 has a higher non- conforming rate than machine 2? Test at the 0.01 level of significance.
Solution: For this problem:
• p1 represents the proportion of nonconforming units produced by machine 1.
• p2 represents the proportion of nonconforming units produced by machine 2.
• p̂1 is the sample proportion estimating p1; it is
ˆ .p x n1
1
1
36 1500
0 024= = =
• p̂2 is the sample proportion estimating p2; it is found to be
ˆ .p x n2
2
2
39 1680
0 0232= = =
• p̂ is the proportion resulting from the combination of the two samples (estimate of the overall proportion when we are testing p1 – p2 = 0). It is found (using Equation (6.91)) to be
ˆ .p x x n n
= + +
= + +
= =1 2 1 2
36 39 1500 1680
75 3180
0 0236
The parameter of interest is the difference in the two population proportions p1 – p2.
1. The null and alternative hypotheses are
H0: p1 – p2 = 0 Ha: p1 – p2 > 0
2. The level of significance is 0.01
3. Since this is a right-tailed test, we will reject the null hypothesis if the test statistic is greater than zα = z0.01 = 2.33
4. The test statistic (using Equation (6.92)) is
z p p
p p n n
0 1 2
1 2
1 1 1
0 024 0 =
−
− +
= −( ˆ ˆ )
ˆ( ˆ )
( . .00232
0 0236 1 0 0236 1
1500 1
1680
0 1 )
. ( . )
.
− + = 448
c c a a
5. Since 0.148 < 2.33, we fail to reject the null hypothesis (weak claim)
6. There is insufficient evidence to conclude that machine 1 has a higher nonconforming rate than machine 2 at the 0.01 level of significance
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A 100(1 – α)% confidence interval on the parameter p1 – p2 is
( ˆ ˆ ) ˆ( ˆ ) (/p p z p p n n
p p1 2 2 1 2
1 21 1 1
− − − + ≤ − ≤ ˆ̂ ˆ ) ˆ( ˆ )/p p z p p n n1 2 2 1 2 1
1 1 − + − +d d d dαα
(6.93)
D.3.c. Hypothesis Tests and Confidence Intervals for Paired Data
Paired-comparison hypothesis testing involves a two- sample t-test for two sam- ples that are believed to be dependent, that is, when an observation from one sam- ple can be logically paired with an observation from the other sample. The pairing of two observations is based on some characteristic they have in common. By pair- ing the data when necessary, we can reduce the effect of the common characteristic that may influence the results.
To illustrate, 40 people are placed on the same diet program. Each person is weighed on day 1 (before weight), put through the program, then weighed again at the end of the program (after weight). The difference between the person’s before weight and after weight is recorded. This is a paired comparison—pairing each individual person’s before weight with their after weight (it would not make sense to compare the before weight of person X with the after weight of person Y, due to individual differences in people).
The data from two samples should be paired when there is a logical relation- ship between the two observations. Table 6.18 presents heart rates (in beats per minute) for individuals who used two types of exercise equipment, A and B.
There is no indication of whether the heart rates recorded involved five people using both types of equipment or five people using type A and five people using type B. Therefore, there is no indication that the data should be paired from equip- ment A to equipment B. Consider the set of data in Table 6.19, again representing the heart rates of people who used the exercise equipment. Now we have more information that indicates the heart rates have a characteristic in common. Since basal heart rates vary a great deal from person to person, we would pair the heart rates by person in order to minimize the effect of individual differences. Therefore, if a significant difference is found, there is a better chance that the difference is due to the type of equipment and not the person using it. If there is any indication that the data should be paired in a particular problem, then pairing should be done (i.e., if in doubt, pair the data).
Table 6.18 Heart rate data for two types of exercise equipment.
A 161 172 166 189 180
B 155 191 187 174 171
Table 6.19 Heart rate data for paired observations.
Person 1 2 3 4 5
A 161 172 166 189 180
B 155 191 187 174 171
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Suppose we have n independently selected pairs given by (X1, Y1), (X2, Y2), . . . , (Xn, Yn). Furthermore, let E(Xi) = μ1 and E(Yi) = μ2. The paired- comparison test is a test conducted on the differences between the two groups. Define the differences as Di = Xi – Yi for all n pairs. Assume the differences Di are normally distributed with the following parameters:
• Mean difference μD, where μD = μ1 – μ2 • Variance of the differences σ 2D
For a sample of n independently selected pairs (Xi, Yi), let di (i = 1, 2, . . . , n) rep- resent the actual differences from the sample. The sample mean d
– and sample
standard deviation sd for the differences are, respectively,
d
d
n
i i
n
= = ∑
1
(6.94)
and
s
d d
nd i
i
n 2
= −( )
− = ∑
1
1 (6.95)
It is important to note that once the differences are calculated, the paired t-test is equivalent to the t-test for a single population mean presented earlier. The test statistic that will be used for the paired t-test is
t
d
s nd 0 =
/ (6.96)
which follows a t distribution with n – 1 degrees of freedom. Follow the six- step procedure for performing a hypothesis test originally introduced in section D.3 using Equation (6.96) in step 4.
ExaMpLE 6.52
Consider the study on two types of exercise equipment given earlier. The following set of data represents heart rates (in beats per minute) for individuals who used the two types of exercise equipment, A and B. The last row represents the differences in heart rate for each person, di = Ai – Bi for i = 1, 2, . . . , 5.
Table 6.20 Paired heart rate data with combined differences.
Person 1 2 3 4 5
A 161 172 166 189 180
B 155 191 187 174 171
di = Ai – Bi 6 –19 –21 15 9
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Is there a significant difference in heart rate due to the type of exercise equipment used? Use a 0.05 level of significance. We will assume that the differences follow a normal distribution.
Solution:
• Let μ1 represent the mean heart rate after using equipment A.
• Let μ2 represent the mean heart rate after using equipment B.
• The logical pairing involves the differences in heart rates for both types of equipment by person. There are n = 5 heart rates in each sample.
• Let μD represent the true mean difference between the two populations μD = μ1 – μ2.
The necessary summary statistics (using Equations (6.94) and (6.95), respectively) are
d d
n
i i
n
= = + − + − + +
= −= ∑
1 6 19 21 15 9 5
2 0 ( ) ( )
.
and
s d d
nd i
i
n
= −( )
−
= − −( )( ) + − − −( )( )
= ∑
2
1
2
1
6 2 0 19 2 0. . 22 2 2 2
21 2 0 15 2 0 9 2 0
5
+ − − −( )( ) + − −( )( ) + − −( )( ) −
. . .
11 16 76= .
The steps are:
1. H0: μD = 0 Ha: μD ≠ 0
2. α = 0.05
3. Since this is a two-tailed test, reject H0 if the test statistic is less than –tα/2,n–1 or greater than tα/2,n–1 where tα/2,n–1 = t0.025,4 = 2.776
4. Calculate the test statistic (using Equation (6.96)):
t d
s nD 0
2 0 16 76 5
0 27= = −
= − /
. . /
.
5. Since –2.776 < – 0.27 < 2.776, we fail to reject the null hypothesis (weak claim)
6. There is insufficient evidence to conclude that the type of exercise equipment sig- nificantly affects heart rate at the 0.05 level of significance
A 100(1 – α)% two- sided confidence interval on the parameter μD is given by
d t
s
n d t
s
n n–1
d n–1
dµD− ≤ ≤ +/ , / ,2 d d d d2αα
(6.97)
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d.4. The p-value approach to Hypothesis Testing
Up to this point, hypothesis testing has been presented using the critical value (or fixed significance level) approach and rejection regions. Critical values are determined based on a stated level of significance (α), among other quantities. The critical value approach is somewhat lacking for two reasons: (1) it does not completely quantify the degree to which a null hypothesis is rejected or not rejected, and (2) it imposes a specific significance level on the practitioner or others making the decision.
For example, suppose a right- tailed test on a population mean with α = 0.05 was carried out. It was reported that the critical value found from the t table was t = 2.306 and the null hypothesis was rejected—but you are not told the value of the test statistic itself. If you are given no further information, can you determine the value of the test statistic? Obviously the value of the test statistic was greater than 2.306 (since the null hypothesis was rejected), but by how much? There is no indication of whether the test statistic was 2.310 or 10.310 given just this informa- tion. In addition, the decision was based on a specific level of significance. If a 0.05 level of significance was used, but after conducting the test it was determined that a 0.01 level of significance should have been used, new critical values must be determined. In many engineering problems, an acceptable level of significance may be known, but not in every situation.
A second approach that offers some measure of the degree to which the test statistic is significant involves the use of p-values. Given the null hypothesis is true, a p-value is the smallest level of significance at which the null hypothesis would be rejected. The p-value is the probability of obtaining a more extreme value than the observed test statistic. The p-value is also referred to as the observed sig- nificance level. A small p-value is evidence against the null hypothesis in favor of the alternative.
For example, suppose we are testing H0: μ = 50 against Ha: μ > 50 and the test statistic is found to be z0 = 2.47. The p-value for this test would be P(Z > 2.47) = 0.0068. That is, the probability that we should have obtained a test statistic of 2.47 or more extreme (in this case, larger than 2.47 since we have a right- tailed test) if H0: μ = 50 is true is 0.0068. This is a highly unlikely event. It is highly unlikely that we should have obtained a test statistic of 2.47 if the mean really is 50. But we did get a test statistic of 2.47, so what went wrong? Remember that the claim μ = 50 is a hypothesis that can be proved incorrect (based on collected data). Therefore, the null hypothesis is probably false.
For a predetermined level of significance, the decisions would be as follows:
• If p-value < α, reject H0 • If p-value ≥ α, fail to reject H0
Using the p-value approach allows the practitioner flexibility in making decisions. If the significance level is changed for a particular test, no new calculations need to be done in order to make a decision. The p-value will not change for a test even if the level of significance does.
The p-value is easy to interpret, and most computer software packages will report a p-value for a hypothesis test. Reconsider the wall thickness example for two competing vendors given in Example 6.49.
The parameter of interest is the difference in average wall thickness, μ1 – μ2, and the hypotheses are H0: μ1 – μ2 = 0 and Ha: μ1 – μ2 > 0. The level of significance
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was α = 0.05, and we concluded that since our test statistic (5.25) was greater than the critical value (1.711), we would reject the null hypothesis.
The problem was worked again, this time using a statistical software package, with the output shown in Figure 6.21.
The p-value is given in bold and reported to be 0.000 (this value is most likely some extremely small number very close to zero; so for all practical purposes, the p-value is zero). Since the p-value < α (i.e., 0.000 < 0.05), we reject the null hypoth- esis. What is telling about the p-value is that we could easily change the level of significance and be able to make a decision without having to do any further calculations. The p-value will remain 0.000 regardless of the level of significance.
The six- step hypothesis testing procedure given earlier would also apply when using p-values. These steps are as follows:
1. State H0 and Ha.
2. State α (a common default value is 0.05 if not explicitly stated).
3. Calculate the test statistic.
4. Calculate the p-value.
5. Reject or fail to reject H0. Reject H0 if p-value < α; otherwise fail to reject H0.
6. State your conclusions in terms of the problem statement.
The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis. It is important to note that the p-value approach and the critical value approach will lead to the same conclusion for the same problem. This is true as long as all quantities and assumptions are identical when using both approaches. Finally, as with all of the procedures given in this section, it is recommended that the calculations be done using a reliable statistical software package.
D.4.a. Significance Level, Power, Type I and Type II Errors
Since every hypothesis test involves analyzing samples to infer properties of a population, there is some chance that the conclusion may be incorrect, even if the analysis is flawless. These sampling errors are not errors in the usual sense, because they cannot be corrected (without using 100% sampling with no measure- ment errors). The two possible types of errors that can occur in hypothesis testing are the type I error and type II error, briefly introduced earlier in this section.
A type I error occurs when a true null hypothesis is rejected. The probabil- ity of committing a type I error is denoted α—the significance level in hypothesis
Two-sample T for Vendor1 versus Vendor2
Difference = mu (Vendor1) – mu (Vendor2)
T-Test of difference = 0 (vs >): T-Value = 5.27 p-Value = 0.000 DF = 24
Figure 6.21 Statistical software output of a t-test.
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testing. The critical values for a hypothesis testing procedure are based on a pre- determined level of significance. That is, the maximum allowable probability of rejecting a true null hypothesis (α) is fixed for a particular problem. A type II error occurs when a false null hypothesis is not rejected. The probability of committing a type II error is denoted β. A summary of the possible decisions and errors is given in Figure 6.22.
Again,
α = P(type I error) = P(rejecting H0 when in fact H0 is true)
β = P(type II error) = P(failing to reject H0 when in fact H0 is false)
In general, it is desirable to have small α and β, but there is often a trade- off. For a fixed sample size in hypothesis testing, decreasing α will result in an increase in β. Because of the manner in which the null and alternative hypotheses are specified in hypothesis testing, it is generally true that committing a type I error is more serious than committing a type II error. Therefore, controlling the probability of committing a type I error (setting α) is often a higher priority than controlling the probability of committing a type II error. In fact, it is difficult to specify an exact value of β since that would require knowing the true value of the parameter being tested.
The power of a hypothesis test is defined as the probability of correctly reject- ing a false null hypothesis. The power is given by 1 – β, since β is the probability of failing to reject the null hypothesis when it is false. The power provides some measure of the test’s ability to detect differences. This ability to detect differences is often referred to as the sensitivity of the hypothesis test.
D.4.b. Statistical versus Practical Significance
In some situations, it may be possible to detect a statistically significant difference between two populations when there is no practical difference. In hypothesis test- ing, the goal is to make a decision about a claim or hypothesis. The decision as to whether the null hypothesis is rejected in favor of the alternative hypothesis is based on a sample taken from the population of interest. If the null hypothesis is rejected, we say there is statistically significant evidence against the null hypothe- sis in favor of the alternative hypothesis. But statistical significance does not imply practical significance. Rejecting the null hypothesis in favor of the alternative hypothesis by a very small margin may be the result of a relatively large sample size. Large sample sizes will almost always lead to rejection of the null hypothesis.
Consider an automobile manufacturer’s claim that one particular make of car averages 31 miles per gallon (mpg) on the highway. A consumer group tests 75 cars of the same make under identical conditions and finds the average to be 30.6 mpg. We could conduct a hypothesis test of H0: μ = 31 versus Ha: μ < 31. The sample average is x– = 30.6. If we are able to reject H0 in favor of Ha, we say there is
Figure 6.22 Possible decisions and errors in hypothesis testing.
Result of Hypothesis Test
Reject H0 Fail to reject H0
H0 is actually
True Type I error Correct decision
False Correct decision Type II error
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statistically significant evidence to indicate that the true average is less than 31 mpg. But, is 30.6 really different from 31 in the practical sense? In this situation, a statisti- cal significance was found, but not necessarily a practical significance, in the differ- ence between the hypothesized value (31 mpg) and the estimated value (30.6 mpg).
Rejecting the null hypothesis in favor of the alternative hypothesis by a very small margin may be the result of a relatively large sample size. Again, large sam- ple sizes can often result in statistically significant results, even though the differ- ence may not be of practical significance. In summary, a rejected null hypothesis implies statistical significance but not necessarily practical significance. Some practitioners prefer using confidence intervals for making decisions since they allow one to see if a practical difference exists.
d.5. analysis of variance (anova)
In the previous section, we discussed one- and two-sample hypothesis tests. In the case where we are interested in comparing a continuous random variable across more than two samples, the ANOVA procedure can be used. We discuss one-way and two-way ANOVA in this section.
D.5.a. One- Way ANOVA
If we are interested in comparing more than two samples, the previous tests are not valid. Suppose we want to compare a populations. Sometimes the popula- tions are referred to as treatments or levels of a factor. Let μ1, μ2, . . . , μa represent the means for the populations. The goal is to determine whether the treatments applied significantly affect the outcome or response of interest.
ExaMpLE 6.53
Alternative energy sources to traditional fossil fuels are in high demand. Several vari- ables are believed to influence the conversion of waste vegetable oil into biodiesel fuel. One variable of interest is the amount of catalyst (%) at three levels—0.6, 1.0, and 1.4—used in the conversion process. The response of interest is conversion rate (wt%) (larger values indicate that more of the waste vegetable oil was successfully converted into useable biodiesel fuel). The experiments are conducted in random order and with the following results shown in Table 6.21.
Table 6.21 Conversion rates for three levels of catalyst.
Catalyst (%)
0.6 1.0 1.4
71.34 84.62 78.33
76.11 78.21 76.89
73.16 82.39 71.42
76.02 76.55 76.60
Continued
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In this study:
• Some questions of interest are: Does the amount of catalyst have a significant effect on the conversion rate? If so, which catalyst amount will result in a high conversion rate?
• Catalyst is the factor of interest or independent variable.
• The factor of interest (catalyst) has three levels (0.6%, 1.0%, and 1.4%). You will also see levels referred to as treatments or groups. In this example, we would say there are three treatments applied or three groups being studied.
• Conversion rate is the response of interest (or dependent variable).
• There are four replicates (n = 4) for each level of catalyst.
Three important assumptions for the use of a one- way ANOVA are as follows:
1. The observations follow a normal distribution (i.e., the populations are normally distributed)
2. The observations are independent
3. The treatments have constant variance (homogeneity of variances)
In summary, the treatment distributions (populations) for all a treatments should be normally distributed, each with the same variance σ 2. An additional assumption that is not always included deals with the number of replicates. It is not necessary that each level of the factor or treatment have the same number of replicates. But large differences in the number of observations from group to group can affect the validity of the analysis. We assume that the number of replicates n will be equal or near equal for all treatments. For more details, see Devore (2016), Montgomery and Runger (2013), and Vining and Kowalski (2011).
It should be noted that the ANOVA procedure outlined next is fairly robust to slight departures from these assumptions. The assumptions should always be verified. This is discussed later in this section.
Suppose we have one factor of interest with a levels of that factor (we could also say a treatments are being compared). Let yij represent the jth response in the ith treatment, where i = 1, 2, . . . , a and j = 1, 2, . . . , n. For example, the third observation for the second treatment (catalyst amount = 1.0%) given earlier would be denoted y23 = 82.39. There are a total of a × n observations in the experiment. A general table of results could be set up as in Table 6.22, where
yi. = the sum of the n observations in the ith treatment (the dot subscript indicates summation over the subscript it replaces)
y.. = the sum of all an observations
y– i. = average of the n observations in the ith treatment
y– .. = average of all an observations
A model provides a way to describe the observations (yij) as a function of the mean for each treatment level (i). The means model is written as
yij = μi + εij (6.98)
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where μi is the mean of the ith treatment level and εij is the random error. The hypotheses of interest regarding the means model are
H0: μ1 = μ2 = . . . = μa Ha: at least two of the means differ (that is, μi ≠ μj , i ≠ j)
The quantities needed to determine whether a significant difference exists among the treatments (or levels of the factor) are the sum of squares and the degrees of freedom.
The total sum of squares, denoted SST, is an important quantity that provides a measure of the total overall variability in the response:
SS y yT ij
j
n
i
a
= −( ) == ∑∑ ..
2
11 (6.99)
where yij is a single observation and y –
.. is the average of all an responses. The total sum of squares can be partitioned into two sources of variability: variability due to the treatments applied (SSTreatments) and variability due to error or unknown sources (SSE). In other words,
SST = SSTreatments + SSE (6.100)
The sum of squares due to treatments (SSTreatments) is a portion of the total sum of squares (SST). It is a quantity that measures the proportion of total variability that can be explained by or that is due to the different treatments applied:
SSTreatments = −( )
= ∑n y yi i
a
. ..
2
1 (6.101)
The error sum of squares (SSE) is that portion of the total sum of squares (SST) that represents the inherent variability. This is variability that is not due to the treat- ments applied or levels used. In conducting an experiment involving a single fac- tor, we assume that all variables (other than the treatments) that could possibly influence the response are held constant; any variability that cannot be attributed to the treatment is said to be error. Under these conditions the “leftover” vari- ability is considered inherent. For example, if temperature is known to possibly
Table 6.22 A typical table of data for an experiment with one factor.
Treatment
1 2 . . . a Totals
y11 y12
y1n
y21 y22
y2n
. . .
. . .
. . .
. . .
ya1 ya2
yan
y.1 y.2
y.n
Totals y1. y2. . . . ya. y..
Averages y–1. y –
2. . . . y –
a. y –..
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influence the conversion rate in our previous example, but temperature is not of interest at this point, we would hold temperature (this is the temperature of a water bath in the reaction experiment) at a constant and only vary the factor that is of interest (such as a catalyst). The error sum of squares can be found by subtrac- tion from Equation (6.100):
SSE = SST − SSTreatments
It is important to note that if we are trying to show that the treatments cause a signifi- cant effect on the response, then we want the variability due to treatments to be large as compared with the inherent variability. It is desirable to have the variability due to error or unknown sources be as small as possible (i.e., we want SSE to be small).
A ratio involving the SSTreatments and SSE is used to reject or fail to reject the hypothesis of interest that all treatment means are equal:
H0: μ1 = μ2 = . . . = μa
This ratio involving the SSTreatments and SSE (whose exact calculations will be shown next) should be significantly large in order to reject the null hypothesis. A large value of the ratio indicates that most of the total variability is attributed to the treatments applied and is not just variability due to error. If the error variability was comparable to (or larger than) the variability due to treatments, this would indicate that there is very little difference in treatments applied and that most of the variability is uncontrollable.
Degrees of freedom are associated with each source of variability. The degrees of freedom are necessary values in the computation of a test statistic and are as follows:
• The total degrees of freedom are an – 1
• For a treatments, the degrees of freedom are a − 1
• The degrees of freedom for error are a(n – 1)
• The total degrees of freedom can be partitioned into degrees of freedom for treatments and degrees of freedom for error:
an – 1 = a − 1 + a(n − 1)
The degrees of freedom are used in the ratio involving the sums of squares discussed previously. The sum of squares divided by the appropriate degrees of freedom pro- vides a measure of variability adjusted for sample size and number of treatments in the study. These resulting measures are referred to as mean squares (MS). Under H0, they are estimates of the error variance. The mean square for treatments is
MS
SS Treatments
Treatments= −a 1
(6.102)
The error mean square is
MS
SS E
E
a n =
−( )1 (6.103)
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Now, we can directly compare the two mean squares. If MSTreatments > MSE, then there may be evidence that the treatments have a significant effect on the response. The ratio of the mean squares can be used as a test statistic to make a decision about rejecting or failing to reject the null hypothesis H0. The ratio is our test sta- tistic, denoted by F0:
F
E 0 =
MS MS Treatments
(6.104)
which can be shown to follow an F distribution with k1 = a – 1 and k2 = a(n – 1) degrees of freedom. If F0 is large, then we may have evidence that the different treatments significantly affect the response of interest.
Recall the section on testing two population variances. The F distribution was appropriate for modeling the ratio of two variances. Therefore, we will compare the test statistic F0 with an appropriate critical value found from the F distribution (Appendixes F, G, and H). The appropriate critical value is given by Fα,a–1,a(n–1), where α is the probability of a type I error (discussed in previous sections), a − 1 is the degrees of freedom for the numerator of F0, and a(n − 1) is the degrees of free- dom for the denominator of F0. Therefore, if F0 > Fα,a–1,a(n–1), we will reject the null hypothesis and conclude that the treatments or levels are significantly different at the α level of significance. We only reject the null hypothesis for large F0 since a large ratio indicates that the variability due to the treatments is larger than the variability due to error (i.e., MSTreatments >> MSE).
The p-values discussed in section D.4 can also be calculated for the F test given here. The same interpretation would apply: if the p-value < α, then reject the null hypothesis; otherwise, fail to reject the null hypothesis. The p-values are easy to interpret and are provided by most statistical software packages when conducting an ANOVA.
The degrees of freedom (df), sum of squares (SS), mean squares (MS), F value, and p-value are often summarized in an ANOVA table. Table 6.23 is a one- way ANOVA table.
The calculations and the resulting ANOVA table can be easily obtained using modern computer software or the formulas given in this section. Software pack- ages that have statistical capabilities will automatically report some form of this ANOVA table.
We now return to our conversion rate of biodiesel fuel example presented at the beginning of this section. In this example, we will set up only the null and alternative hypotheses and present the results of an ANOVA table obtained using a commercially available and reliable statistical package.
Table 6.23 One-way ANOVA table.
Source of variability df SS MS F p-value
Treatments a – 1 SS Treatments MSTreatments F0 P(F > F0)
Error a(n – 1) SSE MSE
Total an – 1 SST
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ExaMpLE 6.54
The data are shown in Table 6.21. The treatment of interest is the amount of catalyst (%) and the response is conversion rate. Is there significant evidence to conclude that the amount of catalyst significantly affects the conversion rate?
Solution: The null and alternative hypotheses are
H0: μ1 = μ2 = μ3
Ha: at least two μi are different, for i ≠ j
Before a formal analysis is conducted on the data, it is often informative to display the results graphically. The box plot is one such appropriate graphical display. Box plots for catalyst amount are shown in Figure 6.23.
Based on the box plots, it appears that the catalyst amount of 1.0% results in higher conversion rates than either 0.6% or 1.4%. There does not appear to be a difference in conversion rate between catalyst amounts of 0.6% and 1.4%. Since interpretation of the graphical displays can be subjective, a more formal analysis such as an ANOVA would be more reliable. The ANOVA table is shown in Table 6.24.
86
84
82
80
78
76
74
72
70
0.6 1.0 Amount of catalyst (%)
1.4
C o
n v e rs
io n
r a te
( %
)
Figure 6.23 Box plots for catalyst amount.
Table 6.24 ANOVA table for conversion rate data.
Source of variability df SS MS F p-value
Catalyst 2 84.92 42.46 4.50 0.044
Error 9 85.01 9.45
Total 11 169.93
The p-value is reported as 0.044. Since the p-value is small, we can reject the null hypoth- esis and conclude that there is a difference in mean conversion rate for at least one pair of catalysts.
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At this point, the assumptions given earlier should be verified. Some simple tools can be used to verify that the normality and constant variance assumptions are valid. To assess normality, a normal probability plot of the observations can be constructed. A simple method for assessing constant variance is to examine the standard deviations of each treatment. A quick and dirty rule of thumb is that the constant variance assumption is plausible as long as the largest treatment stan- dard deviation is not much more than two times the smallest treatment standard deviation. (See Devore [2016] for more details on these and other methods.) More formal methods for assessing constant variance involve analyzing residuals. These methods will be discussed in later sections.
In order to adequately assess independence, we must have the order in which the data were collected. Without the order, it is difficult to determine the validity of the independence assumption. In addition, if the experiment was conducted randomly (carried out randomly) it is often assumed that this randomization will minimize any dependency among the observations.
D.5.b. Two- Way ANOVA
The two-way ANOVA hypothesis test can be used when there are two factors of interest in the experiment. In our biodiesel fuel example there was one fac- tor (catalyst amount) with three levels. When more than one factor is under investigation in an experiment, a factorial experiment should be used. A factorial experiment is one where all possible combinations of the factor levels are inves- tigated. To illustrate, suppose we have two factors A and B with levels a and b, respectively. In a full- factorial experiment, there would be ab total combina- tions, which are the treatments of interest. We are interested in the differences among the levels of factor A, the differences among the levels of factor B, and whether an interaction exists between the two factors. An interaction between factors can be demonstrated graphically. Figure 6.24 represents the interaction plots of two factors A and B, where A has two levels (a = 2) and B has three levels (b = 3). Figure 6.24a indicates that there is no significant interaction between fac- tors A and B. Notice that as we move across the levels of factor B, the levels of factor A maintain identical patterns. Figure 6.24b, on the other hand, indicates a significant interaction between factors A and B. Notice that by changing from level 1 to level 2 of factor A, the response is quite different. By changing the
Figure 6.24 Interaction plots of factors A and B.
32
30
28
26
24
22 1 2
B 3
M ea
n re
sp on
se
27.5
25.0
22.5
20.0
17.5
15.0
1 2 B(a) (b)
3
M ea
n re
sp on
se
2 1 A
2 1 A
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level of factor A, and keeping factor B at level 2, for example, the response has changed.
When analyzing two- factor experiments, the effects to be analyzed are the main effects of factor A, the main effects of factor B, and the interaction between them.
ExaMpLE 6.55
Reconsider the biodiesel fuel example given earlier; now a second factor is of interest. Along with the catalyst, the temperature of the water bath for the process is also of inter- est. There are two temperatures, 30°C and 60°C. Two replicates of each combination of catalyst and temperature are recorded, with the results shown in Table 6.25.
Table 6.25 Conversion rates for experiment with two factors.
Catalyst (%)
0.6 1.0 1.4
Temperature 30 75.22, 76.81 83.10, 79.55 69.24, 71.64
60 77.01, 75.39 75.33, 72.67 72.00, 74.57
In this study:
• Some questions of interest are: Does the amount of catalyst have a significant effect on conversion rate? Does temperature have a significant effect on conver- sion rate? Is there a significant interaction between temperature and the amount of catalyst?
• Catalyst and temperature are the factors of interest or independent variables.
• The factor “catalyst” has three levels (0.6%, 1.0%, and 1.4%). The factor “temper- ature” has two levels (30°C and 60°C).
• Conversion rate is the response of interest (or dependent variable).
• There are two replicates (n = 2) for each combination of catalyst and temperature.
Suppose we have two factors of interest, A and B, with a and b levels, respec- tively. Furthermore, suppose there are n replicates for each combination ab. The total sum of squares (SST) can be calculated for the two- way table and it can be partitioned into four sources of variability:
SST = SSA + SSB + SSAB + SSE (6.105)
where SSA is the sum of squares for factor A, SSB is the sum of squares for factor B, SSAB is the sum of squares for the interaction AB, and SSE is the error sum of squares. The mean squares can be calculated and an ANOVA table created (see Table 6.26).
The numerator and denominator degrees of freedom needed to find the appro- priate critical value or calculate the p-value will vary depending on which factor is being tested. Consider factor A: the correct degrees of freedom needed to find the appropriate critical value are (a – 1), ab(n – 1) (i.e., the degrees of freedom for the
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numerator of F0 and the degrees of freedom for the denominator of F0). As with the one- way ANOVA, the F-statistic measures the ratio between the effect and the experimental error. If the variation due to the effect is a sufficiently large multiple of the error, the effect is considered statistically significant. Using p-values, this means that we would reject the null hypothesis if the p-value < α.
ExaMpLE 6.56
Reconsider the conversion rate example with two factors of interest, catalyst and tem- perature. The resulting ANOVA table is shown in Table 6.27.
Using p-values, we would conclude that the amount of catalyst has a significant effect on the conversion rate (p-value = 0.009). But there is insufficient evidence to con- clude that temperature has a significant effect on the conversion rate (p-value = 0.209). Finally, it appears that there is a significant interaction between catalyst and temperature (p-value = 0.016).
The interaction plot is shown in Figure 6.25. There is a clear indication that a signifi- cant interaction exists between temperature and catalyst. If the goal is to maximize conver- sion rate, it appears that 1.0% catalyst and a temperature of 30°C would be a good choice.
Table 6.27 ANOVA table for conversion rate example with two factors of interest.
Source of variability df SS MS F p-value
Catalyst 2 72.104 36.0520 11.63 0.009
Temperature 1 6.149 6.1490 1.98 0.209
Catalyst × temperature interaction
2 55.635 27.8174 8.97 0.016
Error 6 18.598 3.0996
Total 11 152.486
Table 6.26 Two-way ANOVA table.
Source of variability df SS MS F p-value
Factor A a – 1 SSA MSA F A E
0 = MS MS
P(F > F0)
Factor B b – 1 SSB MSB F B E
0 = MS MS
P(F > F0)
AB interaction (a – 1)(b – 1) SSAB MSAB F AB E
0 = MS MS
P(F > F0)
Error ab(n – 1) SSE MSE
Total abn – 1 SST
Continued
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76
74
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0.6 1.0 Catalyst
1.4
M e a n
r e s p
o n
s e
60
30
Temperature
Figure 6.25 Interaction plot for temperature and catalyst.
The details of the general notation and formulas are not presented here, but the reader is encouraged to see Devore (2016); Montgomery and Runger (2013); Montgomery, Runger, and Hubele (2010); or Vining and Kowalski (2011).
As a final note on the two- way ANOVA, it is important to realize that if each combination of the factors consists of only one observation (n = 1), you cannot estimate the two- factor interaction. When n = 1, there will not be enough degrees of freedom left over for error, the important quantity MSE cannot be estimated, and MSE is needed to calculate the test statistic. If it is acceptable to only estimate the main factors (A and B), then the interaction is removed (not tested) and the SSAB and degrees of freedom for the interaction are moved into error. If the interaction is possibly important, then it is recommended that at least two replicates for each combination of factors be collected.
d.6. Hypothesis Tests for discrete data
In this section we discuss goodness- of-fit tests and contingency tables.
D.6.a. Goodness- of-Fit Tests
Chi-square and other goodness- of-fit tests help determine whether a discrete sam- ple has been drawn from a known population. The probability distribution may be of a specific form, such as the Poisson, binomial, geometric, and so on, or it may be simply a table of outcomes and their assumed probabilities. For example, sup- pose that all rejected products have exactly one of four types of nonconformities (that render them nonconforming), and historically they have been distributed as in Table 6.28. Data on rejected parts for a randomly selected week in the current year are shown in Table 6.29.
The question we need to answer is this: Is the distribution of nonconformity types different from the historical distribution? The test that answers this question
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is referred to as the χ2 goodness-of-fit test. To get a feel for this test, construct a table that displays the number of nonconforming units that would be expected in each category if the sample exactly followed the historical percentages (the historical percentages are used as estimates of the probabilities that the nonconformities are present on the product). The expected number will be referred to as the expected frequency. The expected frequency is found by multiplying the total number of items in the sample n by the probability for a particular category, as shown in Table 6.30.
The question to be answered is whether the difference between the expected frequencies and the observed frequencies will be sufficiently large. If the difference is large, there may be significant evidence to conclude that the distribution the cur- rent sample came from is not the same as the historical distribution; we may even have enough evidence to conclude that the historical (assumed) distribution is no longer valid. The test statistic that can be used to determine whether the assumed distribution is still valid is
χ0 2
2
1 =
−( ) = ∑
O E E
i i
ii
k
(6.106)
Table 6.28 Historical percentages of nonconformities for rejected products.
Nonconformity Percentage of
nonconforming products
Paint run 16%
Paint blister 28%
Decal crooked 42%
Door cracked 14%
Total 100%
Table 6.29 Number of nonconformities for a random week.
Nonconformity Number of
nonconforming products
Paint run 27
Paint blister 60
Decal crooked 100
Door cracked 21
Total 208
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It can be shown that the test statistic follows a chi- square distribution with k – 1 degrees of freedom (where k = number of categories). Let pi represent the pro- portion of the population that falls into the ith category, for i = 1, 2, . . . , k. Let pi,0 represent the hypothesized value of pi. The null and alternative hypotheses would be
H0: p1 = p1,0; p2 = p2,0; . . . ; pk = pk,0
Ha: pi ≠ pi,0 for at least one i = 1, 2, . . . , k
The procedure for conducting a goodness- of-fit test is as follows:
1. Determine the null and alternative hypotheses.
2. State the level of significance α.
3. Determine the rejection region. For this test the critical value is χ2α,k–1.
4. Calculate the test statistic, as in Equation (6.106).
5. If χ20 ≥ χ 2 α,k–1, reject H0; otherwise fail to reject H0.
6. State the conclusions in terms of the problem statement.
ExaMpLE 6.57
We will complete the goodness-of-fit test for the problem involving nonconforming products, based on the data shown in Table 6.30, using a 5% level of significance.
1. H0: p1 = 0.16; p2 = 0.28; p3 = 0.42; p4 = 0.14
Ha: pi ≠ pi,0 for at least one i = 1, 2, . . . , 4
2. α = 0.05
3. For this test the critical value is χ 2α,k–1 = χ 2 0.05,3 = 7.815
Table 6.30 Observed and expected frequencies for nonconformity data.
Nonconformity Observed
frequency (Oi) Probability ( pi)
Expected frequency (Ei)
[Ei = npi]
Paint run 27 0.16 33.28
Paint blister 60 0.28 58.24
Decal crooked 100 0.42 87.36
Door cracked 21 0.14 29.12
Total n = 208 1
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4. Calculate the test statistic (using Equation (6.106)):
χ0 2
2
1
2 27 33 28
33 28 60 58
= −( )
= −( )
+ −
= ∑
O E E
i i
ii
k . .
.224 58 24
100 87 36 87 36
21 29 12 29
2 2 2( ) +
−( ) +
−( ) .
. .
. ..
.
12
5 33=
5. Since 5.33 < 7.815, we fail to reject H0 (weak claim)
6. We conclude that the data do not indicate that there has been a change in the pro- portion of observed nonconforming products when compared with historical proportions, at the 0.05 level of significance
The chi- square goodness- of-fit test is valid as long as the expected frequencies are not too small. Some recommendations for a minimum value have included 3, 4, and 5. Other recommendations have been that some of the expected frequencies can be as small as 1 or 2 as long as most of the expected frequencies exceed 5. For more details on other applications of the chi- square test and further recommendations, see Devore (2016), Montgomery and Runger (2013), and Vining and Kowalski (2011).
D.6.b. Contingency Tables
In this section, a test concerning count data will be presented. Suppose a sample of n items has been collected and each item can be classified into two different categories at the same time. Data that can be classified according to two different criteria (or factors) can be displayed in a two-way contingency table. In cases such as this, it is often of interest to determine whether the two categories are statistically independent of one another. For example, consider the population of high school graduates. We may want to determine whether the hourly wage for an entry- level job is independent of graduating from high school.
Suppose there are r levels of factor 1 and c levels of factor 2. Each criterion can have several different levels. An r × c contingency table could be written as in Table 6.31, where
• The r rows represent the levels of the first factor
• The c columns represent the levels of the second factor
• Oij represents the number of observations that fall into category i of factor 1 and category j of factor 2, where i = 1, 2, . . . , r and j = 1, 2, . . . c
Table 6.31 A generic contingency table.
Columns
1 2 . . . c
Rows
1 O11 O12 . . . O1c
2 O21 O22 . . . O2c
: : : . . . :
r Or1 Or2 . . . Orc
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ExaMpLE 6.58
A company operates two machines on three different shifts. The company wants to determine if machine breakdowns that occur during operation are independent of the shift on which the machine is used. The data are recorded in a 2 × 3 contingency table, shown in Table 6.32.
Table 6.32 A contingency table for machine breakdown.
Shift
Machine 1 2 3
1 30 40 20
2 30 40 10
In this study:
• r = 2, the number of machines.
• c = 3, the number of shifts.
• Oij represents the number of breakdowns that occur on the ith machine when used on the jth shift. For example O12 = 40. Therefore, 40 breakdowns have been recorded on machine 1 when it was used on the second shift.
We are interested in determining whether the two factors are independent of one another.
The expected frequencies are calculated based on the assumption that the two factors of interest are independent of one another. Denote the expected frequencies as Eij (for i = 1, 2, . . . , r and j = 1, 2, . . . , c). Calculate the expected frequencies for each entry in the contingency table using Eij = naibj, where
• n = total number of observations, that is, n = O11 + O12+ . . . + Orc • ai is the probability that a randomly selected observation will fall into
the ith category of factor 1 (the row factor) and is found using the following formula:
a
O
ni
ij j
c
= = ∑
1
(6.107)
(Summing over the columns for the ith row)
• bj is the probability that a randomly selected observation will fall into the jth category of factor 2 (the column factor) and is found using the following formula:
b
O
nj ij
i
r
= = ∑
1
(6.108)
(Summing over the rows for the jth column)
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If the two factors are independent, then we would expect the observed frequency and the expected frequency for each cell to be similar. This assumption can be tested using the test statistic
χo
ij ij
ijj
c
i
r O E
E 2
2
11
= −
== ∑∑
( )
(6.109)
which follows a chi- square distribution with (r – 1)(c – 1) degrees of freedom. The test statistic can be compared with a critical value from the chi- square distribution with (r – 1)(c – 1) degrees of freedom and a significance level α. The critical value is denoted χ 2α,(r–1)(c–1). If the test statistic is greater than the critical value, we say there is enough evidence to conclude that the two factors are not independent. The test is valid as long as the expected frequency of each cell is at least five.
ExaMpLE 6.59
A company operates two machines on three different shifts. The company wants to deter- mine if machine breakdowns are independent of shift. The data are shown in Table 6.32.
Solution: Calculate the expected frequency for each cell, where n = 30 + 40 + 20 + 30 + 40 + 10 = 170. The row probabilities (probabilities associated with the machines) are found using Equation (6.107)):
Machine 1: a1 = (30 + 40 + 20)/170 = 90/170 = 0.529
Machine 2: a2 = (30 + 40 + 10)/170 = 80/170 = 0.471
The column probabilities (probabilities associated with the shifts) are found using Equa- tion (6.108)):
Machine 1: b1 = (30 + 30)/170 = 60/170 = 0.353
Machine 2: b2 = (40 + 40)/170 = 80/170 = 0.471
Machine 3: b3 = (20 + 10)/170 = 30/170 = 0.176
Expected frequency computations for two of the six cells are
E11 = na1b1 = 170(90/170)(60/170) = 31.765
E23 = na2b3 = 170(80/170)(30/170) = 14.118
The expected frequencies for all six cells are provided in Table 6.33.
Table 6.33 Expected frequencies for machine breakdown.
Shift
Machine 1 2 3
1 31.765 42.353 15.882
2 28.235 37.647 14.118
Continued
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The test statistic (using Equation (6.109)) is
χ0 2
2
230 31 765 31 765
40 42 3
= −
= −
+ −
∑∑ ( )
( . ( .) .
O E
E ij ij
ij
553 42 353
2) .
. . . ++
2.755
210 14.118 14.118
( )−
=
With α = 0.05 and degrees of freedom (r – 1)(c – 1) = 2, the critical value is χ2α,(r–1)(c–1) = χ 2 0.05,2 =
5.99. Since 2.755 < 5.99, we fail to reject the hypothesis of independence. Breakdown of a particular machine appears to be independent of which shift is using the machine.
E. rELaTionSHiPS BETWEEn variaBLES This section covers four kinds of relationships between variables: simple linear correlation, linear regression, multiple linear regression, and time series analysis.
E.1. Simple Linear Correlation
Correlation measures the strength of the linear relationship between two variables. A linear relationship exists between two variables if one variable increases as the other increases or decreases. Graphically, this will be seen if the values of the two variables plot along a straight line. Recall the discussion on the bivariate normal distribution in section C. One of the parameters that define the bivariate normal pdf is the population correlation coefficient ρ. The population correlation coefficient is often unknown and can be estimated using sample data. Suppose x and y are jointly normally distributed random variables. The sample correlation coefficient, denoted r, can be used as the estimate of the population correlation coefficient ρ. The sample correlation coefficient is
r S
S S xy
xx yy
=
(6.110)
where
S x x y y x y
x
xy i i i
n
i i i
n i i
n
= −( ) −( ) = − = =
=∑ ∑ ∑
1 1
1 = ∑ y
n
i i
n
1 c cc c
(6.111)
S x x x
x
nxx ii
n
i
i i
n
i
n
= −( ) = − =
=
= ∑
∑ ∑2
1
2 1
2
1
c c
(6.112)
S y y y
y
nyy ii
n
i
i i
n
i
n
= −( ) = − =
=
= ∑
∑ ∑2
1
2 1
2
1
c c
(6.113)
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The sample correlation coefficient r measures the strength of the linear relation- ship and has the following properties:
• 1 ≤ r ≤ 1; the closer the value is to −1 or 1, the stronger the linear relationship
– If r is negative, this indicates that as one variable is increasing, the other is decreasing
– If r is positive, this indicates that both variables are increasing or both variables are decreasing
• If r = 0, then there is no linear relationship between the two variables
• r has no units attached to it, such as pounds, inches, feet, and so on
Figure 6.26 illustrates two variables x and y. Figure 6.26a displays two random variables that are positively correlated. In this case, the value of r is positive and near 1. Figure 6.26b displays two random variables that are negatively correlated. In this case, the value of r is negative and near –1. Figure 6.26c displays two vari- ables that are not linearly related at all. In this situation, there does appear to be some relationship between x and y, but it is not linear. Therefore, the correlation coefficient r would be zero.
Figure 6.26 Various scatter plots for two variables x and y.
90
85
80
75
70
65
60 110 120 130 140 150 160 170 180
X (a)
190
Y
90
85
80
75
70
65
60 110 120 130 140 150 160 170 180
X (b)
190
Y
90
85
80
75
70
65
60 0 10 20 30 40 50
X (c)
60
Y
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There is a direct relationship between the correlation coefficient and the slope of a simple linear model. The sign on the correlation coefficient is the same as the sign on the slope. This will be discussed further in the next section.
There is some subjectiveness in the interpretation of the correlation coefficient estimate. If −0.6 ≤ r ≤ 0.6, for example, there is not always agreement as to whether the association between the two variables is significantly correlated. A t-test on the population correlation coefficient can be conducted to determine the statistical significance of the correlation. For complete details on testing the significance of the population correlation coefficient, see Devore (2016); Montgomery and Runger (2013); Montgomery, Runger, and Hubele (2010); or Kutner et al. (2004).
ExaMpLE 6.60
When data have been collected relating two variables, it is often useful to find an equa- tion that models the relationship. Then the value of the dependent variable can be predicted for a given value of the independent variable. For example, suppose a chemi- cal engineer is investigating the relationship between the operating temperature of a process and product yield. In this case, it might be useful to control the operating tem- perature (independent variable) in order to control or predict yield (dependent vari- able). For this example, eight readings are taken, and shown in Table 6.34. In an actual application more data would be desirable.
Table 6.34 Temperature and yield data.
Temperature, °C (x) 115 125 135 145 155 165 175 185
Yield, % ( y) 62 64 69 77 78 81 82 88
The first step in the investigation is to plot the data, as in Figure 6.27, to determine if it seems reasonable to approximate the relationship with a straight line.
90
85
80
75
70
65
60
110 120 130 140 150 160 170 180 Temperature
190
Y ie
ld
Figure 6.27 Scatter plot of temperature and yield.
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For these data, the sums of squares needed to determine the sample correlation coef- ficient can be found using Equations (6.110), (6.111), and (6.112), respectively. They are
S x y x y
nxy i ii
n i i
n
i i
n
= − = =
= =∑ ∑ ∑
1
1 1 1545 b bb b
S x x
nxx i i
i
n
i
n
= − == =
∑ ∑ 2 1
2
1
4200 b b
S y y
nyy i i
i
n
i
n
= − == =
∑ ∑ 2 1
2
1
592.875 b b
resulting in the below calculation (from Equation (6.110))
r S
S S xy
xx yy
= = = 1545
4200 592.875 0 979.
The sample correlation coefficient of 0.979 indicates that there is a very strong positive linear relationship between temperature and yield.
E.2. Linear regression
Linear regression models are important statistical tools developed to relate two or more variables of interest. This relationship often takes the form of a linear equa- tion or linear model. In this section, simple linear regression is presented. Simple linear regression is the situation where there are exactly two variables:
• One independent variable (often denoted by x)
• One dependent variable (often denoted by y)
Although a perfect straight line cannot be drawn through these points in Exam- ple 6.60, the trend looks linear. The next step is to find an equation that best fits the data. Before creating regression lines for particular problems, some assumptions and basic definitions must be presented.
E.2.a. Notation and Definitions
Reconsider the scatter plot in Figure 6.27. The scatter plot indicates that the two variables may be linearly related. This is indicated by the fact that the observations fall approximately along a straight line. A simple linear regression model is one that characterizes a linear relationship between the response of interest y and an inde- pendent (explanatory or regressor) variable x:
y = β0 + β1x + ε (6.114)
where β0 and β1 are called regression coefficients and ε represents a random error term (recall that the data will rarely lie exactly along a straight line, so when a straight line is fit to the data, there will be some error).
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Let x1, x2, . . . , xn represent real values of the independent variable (also called an explanatory variable) and y1, y2, . . . , yn represent real values of the dependent variable (also called the response). The sample consists of n pairs of data (x1, y1), (x2, y2), . . . , (xn, yn). The general model given in Equation (6.114) can be written in terms of the individual observations:
yi = β0 + β1xi + εi, for i = 1, 2, . . . , n (6.115)
For the simple linear regression model:
• The coefficients β0 and β1 are parameters that define the mathematical relationship between the independent and dependent variables. β0 is the intercept and β1 is the slope
• The intercept is the value of y when x = 0. This is the height at which the regression line crosses the y-axis
• The slope represents the change in the response for every one unit change in the independent variable x
Given a series of values for an independent variable x and the corresponding dependent variable y, we can calculate point estimates for β0 and β1. The point estimates are denoted b0 and b1. Once the point estimates are obtained, a fitted regression line can be given by
ŷ b b xi i= +0 1
(6.116)
where ŷ i is the predicted value of the response for a given value of the indepen- dent variable.
E.2.b. Estimating the Parameters β0 and β1 and Making Predictions
The statistics b0 and b1 need to be calculated in such a way that the resulting fitted line will provide predicted values that will be close to the actual value for each value of x. One method for calculating b0 and b1 is based on minimizing the error between the actual value of y and the predicted value of y for each pair of data, that is, ei = yi – ŷ i. This is an appropriate method since a goal of simple linear regres- sion is to make predictions after fitting a model between the independent variable and the response. A method frequently employed is the least squares method. The formulas for finding b1 and b0 are as follows:
b
S
S xy
xx 1 =
(6.117)
b y b x0 1= − (6.118)
where Sxy and Sxx are defined in Equations (6.111) and (6.112), respectively. Equa- tions (6.116) and (6.117) are often referred to as the least squares estimates. These formulas result in estimates that will give us a best- fitting line. By “best fitting,”
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we mean that the line with these estimates for the coefficients will result in the smallest error sum of squares.
Let ei represent the error associated with the ith observation ei = yi – ŷ i for i = 1, 2, . . . , n (a realized value of the error is referred to as a residual). We want a fitted line that will minimize the n errors as much as possible. More specifically, we want to find the fitted regression line that will minimize the quantity
ei
i
n 2
1= ∑SSE =
(6.119)
(the error sum of squares). The formulas for b0 and b1 given previously, in Equa- tions (6.117) and (6.118), respectively, result in a fitted line that will make this quantity as small as possible. In fact, there are no other estimates of β0 and β1 that will result in a smaller error sum of squares.
ExaMpLE 6.61
Reconsider the yield data from Example 6.60. Temperature is the independent variable x and yield is the response y. We want to fit a regression line relating temperature to yield.
The necessary calculations for b1 and b0 are:
n
x xi i
i i
=
= + + + = = = = ∑ ∑
8
115 125 185 1200 115 1
8 2
1
8
... 22 2 2125 185 184 200+ + + =... ,
y x yi i
i i i
n
= = ∑ ∑= + + + = = +
1
8
1
62 64 88 601 115 62 1... ( ) 225 64 185 88 91 695
150 75 125
( ) + + ( ) =
= =
... ,
.x y
Calculate b1 and b0 (using Equations (6.116) and (6.117)):
S x y x y
nxy i ii
n i i
n
i i
n
= − = =
= =∑ ∑ ∑
1
1 1 911 695 1200 601
8 1545
2
1
1
, − ( )( )
=
= − =
=∑S x x
xx i i
n i i
n
∑∑ = −
( ) =
2
2
184 200 1200
8 4200
n ,
b bb b
b b
b S
S
b y b x
xy
xx 1
0 1
1545 4200
0 368
75 125 0 368 1
= = =
= − = −
.
. . 50 19 9( ) = .
The final fitted regression line is then
ŷ = 19.9 + 0.368
Continued
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The fitted regression line is plotted along with the original data in Figure 6.28.
{e4
90
85
80
75
70
65
60 110 120 130 140 150 160 170 180
Temperature 190
Yi el
d
Figure 6.28 Scatter plot and fitted regression line for the yield data.
The fitted regression line represents the predicted value ŷ for each value of x. Graphically, the residuals mentioned earlier are the vertical differences between the actual value of y and the predicted value of y for each value of x. See Fig- ure 6.28. The vertical line represents the residual for x4 = 145 (e4 = y4 – ŷ 4 = 77 – 73.26 = 3.74).
The fitted regression model is often used to make predictions of new or future observations for the response. Let x0 be a value of the independent variable. The point estimator of the new value, y0, is given using Equation (6.116) by
ŷ 0 = b0 + b1x0
For example, suppose we wish to predict the yield for a temperature of 150°C. The predicted value of the yield would be:
ŷ 0 = 19.9 + 0.368x0
= 19.9 + 0.368(150)
= 75.14%
Because the value is a single point estimate calculated from sample data, there is variability or error in this prediction. It is sometimes of interest to construct an interval estimate for a future observation. A prediction interval provides a mea- sure of the estimate and error of prediction. For complete details on prediction intervals, see Devore (2016); Montgomery and Runger (2013); Montgomery, Run- ger, and Hubele (2010); Kutner et al. (2004); or Vining and Kowalski (2011).
The equation for the regression line is based on observed data. Therefore, pre- dictions of new observations with X values outside the range of original data (xmin, xmax) should be used with caution. This is called extrapolation.
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E.2.c. Hypothesis Testing in Simple Linear Regression
Fitting a simple linear regression model involves a sample of data. As a result, there will naturally be some error in the estimates of the coefficients β0 and β1. Recall that when we are estimating a population mean with a sample mean there is some variability in this estimate. The same is true for the point estimates of the coefficients b0 and b1. The expected value and variance of the point estimate of β0 are
E b( )0 0= β (6.120)
V b
n x Sxx
( )0 2
21 = +σ 7 7
(6.121)
The expected value and variance for the point estimate of β1 are
E b( )1 1= β (6.122)
V b
Sxx ( )1
2
= σ
(6.123)
where Sxx was defined in Equation (6.111) and σ 2 is the error variance (also referred
to as the process variability). An estimate of σ 2 is
σ̂ 2
2 = =
− SS n
E MSE
(6.124)
where n – 2 is the error degrees of freedom and SSE represents the error sum of squares defined previously:
SS y y eE i i
i
n
i i
n
= − = = = ∑ ∑( ˆ )2
1
2
1 (6.125)
The standard errors for b0 and b1 are
s.e.( ) ( )b V b
n x Sxx
0 0 2
21 = = +σ 7 7
(6.126)
s.e.( ) (b V b1 = 11
2
) = σ Sxx
(6.127)
An important test in simple linear regression is a test on the coefficient β1. In particular, a test on the significance of regression would involve testing β1 = 0. If β1 = 0, then Equation (6.114) would be
y = β0 + β1x + ε
= β0 + ε
which indicates no significant linear relationship between x and y.
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A t-test can be conducted on the slope β1. The hypotheses of interest are
H0: β1 = 0
Ha: β1 ≠ 0
The test statistic is
t b
Sxx
0 1
2
0 =
−
σ̂
(6.128)
A large value of t0 would lead to rejection of the null hypothesis. An appropriate critical value is tα/2,n–2, found from the t table in Appendix O. If a p-value is calcu- lated, the null hypothesis would be rejected if the p-value is small (p-value < α), and would indicate a significant linear relationship between x and y.
A t-test can also be conducted on the intercept β0. The hypotheses of inter- est are
H0: β0 = 0
Ha: β0 ≠ 0
The test statistic is
t b
n x Sxx
0 0
2 2
0
1 =
−
+σ̂ 7 7
(6.129)
A large value of t0 would lead to rejection of the null hypothesis. An appropriate critical value is tα/2,n–2, found from the t table in Appendix O. If a p-value is calcu- lated, the null hypothesis would be rejected if the p-value is small (p-value < α).
The calculations do not need to be done by hand. A statistical software pack- age can be used to carry out all the necessary calculations.
ExaMpLE 6.62
Reconsider the yield and temperature data in Examples 6.60 and 6.61. The hypotheses of interest are
H0: β1 = 0 versus Ha: β1 ≠ 0 and
H0: β0 = 0 versus Ha: β0 ≠ 0
The output from a particular statistical package for this problem is:
Predictor Coef SE Coef T P Constant 19.946 4.735 4.21 0.006 x 0.36786 0.03120 11.79 0.000
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The output can be interpreted as follows:
• The row “Constant” is the hypothesis test on the intercept: H0: β0 = 0 versus Ha: β0 ≠ 0.
• The row “x” is the hypothesis test on the slope: H0: β1 = 0 versus Ha: β1 ≠ 0.
• The column labeled “Coef” contains the point estimates for each parameter, that is, b0 = 19.946 and b1 = 0.36786. Note that these values were calculated in Exam- ple 6.61, but may differ slightly due to rounding.
• The column labeled “SE Coef” provides the standard error of each estimate, that is, s.e.(b0) = 4.735 and s.e.(b1) = 0.0312.
• The column labeled “T” represents the test statistic for the intercept and slope; t0 = 4.21 (test on the intercept) and t0 = 11.79 (test on the slope).
• The last column, labeled “P,” contains the p-value for each parameter.
The p-value for testing β1 = 0 is 0.000. Since this value is small, we can reject the hypoth- esis that β1 = 0 and conclude that the slope is not zero. That is, there appears to be a statistically significant linear relationship between temperature and yield.
The test on the intercept β0 = 0 also indicates that the intercept is significant. If we fail to reject the null hypothesis β0 = 0, it is often left to the practitioner to determine if it makes practical sense to leave the intercept in the model.
The 100(1 – α)% two- sided confidence interval on the intercept is given by
b t b b t bn n0 2 2 0 0 0 2 2 0− ≤ ≤ +− −α αβ/ , / ,( ) ( )s.e. s.e. (6.130)
The 100(1 – α)% two- sided confidence interval on the slope is given by
b t b b t bn n1 2 2 1 1 1 2 2 1− ≤ ≤ +− −α αβ/ , / ,( ) ( )s.e. s.e. (6.131)
ExaMpLE 6.63
For the yield and temperature data in Example 6.60, we wish to construct 95% confidence intervals on the slope and intercept. In this case, the value tα/2,n–2 = t0.025,6 = 2.447 and the resulting confidence intervals (using Equations (6.129) and (6.130), respectively) are
b t b b t bn n0 2 2 0 0 0 2 2 0
1
− ≤ ≤ +− −β/ , / ,( ) ( )s.e. s.e. 99 946 2 447 4 735 19 946 2 447 4 735
8 3 0. . ( . ) . . ( . )
.
− ≤ ≤ +β 6 3 1 530≤ ≤β .
α α
and b t b b t bn n1 2 2 1 0 1 2 2 1
0
− ≤ ≤ +− −β/ , / ,( ) ( )s.e. s.e. .. . ( . ) . . ( . )
.
368 2 447 0 0312 0 368 2 447 0 0312
0 2 1− ≤ ≤ −β
992 0 4441≤ ≤β .
α α
The confidence intervals do not contain zero, so we have evidence to indicate that the slope and intercept are both nonzero. Again, since the 95% confidence interval on β1 does not contain zero, we have evidence to indicate that there is a significant linear rela- tionship between temperature and yield.
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E.2.d. Test for Significance of Regression Using ANOVA
ANOVA can also be used to test for significance of regression. The null and alter- native hypotheses of interest are H0: β1 = 0 and Ha: β1 ≠ 0, respectively. Note that only the test on the slope will determine significance of regression. The total sum of squares (SST) is a measure of the total variability. SST can be partitioned into two sources of variability: (1) the regression line that we have fit and (2) error. This is similar to the total sum of squares discussed in section D. The error sum of squares (also referred to as the residual sum of squares) defined in this section is a measure of the unexplained variability in the responses y. The variability due to the regression model that we have fit is the regression sum of squares (SSR). The partition is
SST = SSR + SSE (6.132)
where
S y y y y
nyy ii
n
i
i i
n
i
n
= −( ) = − =
=
= ∑
∑ ∑2
1
2 1
2
1
c c
(6.133)
and
SS y y eE i i i
n
i i
n
= − = = = ∑ ∑( ˆ )2
1
2
1
It is desirable to have the regression sum of squares be large in comparison with the error sum of squares. A large value of SSR would indicate that most of the vari- ability in the response can be explained by the regression model that has been fit. As we did in section D, we have to take into account the sample size and adjust the sum of squares using the appropriate degrees of freedom. An ANOVA table can be constructed (see Table 6.35).
If critical values are used, we would reject the null hypothesis if F0 > Fα,1,n–2. The critical value is found from the F table in Appendix G.
Table 6.35 ANOVA table for testing significance of regression.
Source df SS MS F p-value
Regression 1 SSR MSR = SSR/1 F0 = MSR/MSE P(F > F0)
Error n – 2 SSE MSE = SSE/(n – 2)
Total n – 1 SST
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ExaMpLE 6.64
The ANOVA table associated with the test for significance of regression based on the model fit in Example 6.62 is given below. The null and alternative hypotheses are
H0: β1 = 0
Ha: β1 ≠ 0
Table 6.36 ANOVA table for temperature and yield regression model.
Source df SS MS F p-value
Regression 1 568.34 568.34 138.98 0.000
Error 6 24.54 4.09
Total 7 592.87
Since the p-value is approximately zero, we would reject the null hypothesis and con- clude again that there is a statistically significant linear relationship between yield and temperature. If critical values are used, the critical value is F0.05,1, 6 = 5.99 (assuming a 0.05 level of significance). Since 138.98 > 5.99, we again reject the null hypothesis.
The ANOVA is useful not only for testing the significance of regression but also for providing an estimate of σ̂ 2. Specifically, σ̂ 2 = MSE. In this problem, σ̂ 2 = MSE = 4.09.
E.2.e. Coefficient of Determination
The coefficient of determination R2 gives a measure of the adequacy of the current regression model for a particular set of data. It is the proportion of the total vari- ability in the response that can be explained by the regression line. The coeffi- cient of determination for the simple linear regression model can be computed by taking the square of the correlation coefficient r. (We often use the notation R2 for this value even though correlation coefficient is denoted by lowercase r.) Since –1 ≤ r ≤ 1, then 0 ≤ R2 ≤ 1. In general, the coefficient of determination can be calculated for any linear regression model by
R R
T
E
T
2 1= = − SS SS
SS SS
(6.134)
ExaMpLE 6.65
For the yield and temperature problem, the coefficient of determination is
R R T
2 568 34 592 87
0 959= = = SS SS
.
. .
We would conclude that approximately 95.9% of the total variability in the response (yield) can be explained by the regression model involving temperature.
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E.2.f. Assumptions in Regression Analysis
There are a number of assumptions in linear regression. For the most part, the least squares approach and the results or conclusions drawn from the data are fairly robust to these assumptions; that is, it takes a substantial deviation from the norm to affect the results. The assumptions for the least squares approach to regression analysis are as follows:
1. The errors ei are independent
2. The errors ei are normally distributed with mean zero
3. The errors ei have constant variance σ 2
The assumptions can be checked using residual analysis. Residuals plotted against the fitted values ŷ or against the independent variable x can provide some infor- mation about the validity of the constant variance assumption. A normal prob- ability plot of residuals can be used to assess the assumption of normality. The independence assumption can be verified by examining a plot of the residuals against the time sequence—if the time sequence is known. With the exception of the normal probability plot of the residuals, all residual plots should exhibit no obvious patterns in the residuals. If the residuals fall along a straight line in the normal probability plot, then the normality assumption is assumed to be valid. Caution should be used when interpreting residual plots for small sets of data. For small sample sizes, patterns on residual plots can often occur by chance. Keep in mind that the least squares method is fairly robust to slight departures from these assumptions. Residual analysis is further discussed in section H.5 of this chapter.
E.3. Multiple Linear regression
Multiple least- squares linear regression is an extension of simple linear regression. The response variable y is a function of k independent variables, x1, x2, . . . , xk. The equation for the multiple linear regression is
y = β0 + β1x1 + β2x2 + . . . + βkxk + ε (6.135)
where
β0 represents the intercept
β1, β2, . . . , βk are the coefficients of the k independent variables
ε represents random error
A fitted regression model would be given as
ŷ b b x b x= + +0 1 1 2 2 b x. . .+ + k k (6.136)
where b0 is an estimate for β0, b1 is an estimate for β1, and b2 is an estimate for β2, and so on. Each bi is an estimate for βi.
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ExaMpLE 6.66
An article by Menesatti et al. (1999) in the Journal of Agricultural Engineering Research describes the modeling of damage susceptibility of peaches to several independent variables. Two independent variables that are believed to impact peach damage are the height at which the peach is dropped (mm) and peach density (g/cm3). Data that are typi- cal of this type of experiment are shown in Table 6.37:
Table 6.37 Peach damage data.
y x1 x2
7.22 371.05 0.99
4.24 315.02 1.11
8.50 550.10 0.97
9.32 400.00 1.02
5.87 336.00 0.96
7.12 361.10 0.95
8.04 499.24 1.01
6.62 403.58 1.00
10.06 482.33 1.04
8.96 451.65 0.98
The multiple regression line for this set of data is (found using a statistical software package)
ˆ . . .y x x= + −5 380 0 016 4 4811 2
where b0 = 5.380, b1 = 0.016, and b2 = – 4.481. Hypotheses tests were conducted on the parameters of interest β0, β1, and β2, and it
was determined that peach density (x2) is insignificant (p-value > 0.10). Regression analy- sis was carried out again but this time relating only height dropped (x1) and damage (y). (Note that the intercept is also found insignificant, but the researchers determined that it is appropriate to leave it in.) The final fitted model is
ˆ . .y x= +0 662 0 017 1
The fitted model may be useful for predicting damage, but further analysis is necessary. The coefficient of determination was calculated for the final fitted model and found to be 52.9%. That is, approximately 53% of the total variability in damage can be explained by the fitted regression line involving only drop height. Although peach density is not a significant variable for damage, there may be other independent variables that should be investigated. Residual analysis should also be done for this problem.
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For complete details on multiple regression, see Devore (2016); Montgomery and Runger (2013); Montgomery, Runger, and Hubele (2010); Kutner et al. (2004), or Vining and Kowalski (2011).
As a final note on regression analysis, finding a fitted regression line statis- tically significant does not imply a causal relationship between the independent and dependent variables. Two variables could be linearly related and have a very strong association, but this does not infer that one variable caused the change in the other variable. Causation could be concluded only if a designed experiment were conducted for a particular problem. Design of experiments is discussed in section H of this chapter.
E.4. Time Series analysis
Time series analysis in mathematical statistics involves mathematical techniques for determining cycles and trends in data over time. Two specific tools are dis- cussed in this section:
• Moving average smoothing
• Trend analysis
Moving average smoothing and trend analysis are two methods of analyzing data. There are other methods, but they are beyond the scope of this discussion.
Run charts display a plot of data obtained on sequential samples taken from a process. This plot has an x-axis of sequence or time. The y-axis is that of the measurement taken on the sample. For example, the rate of non- mission-capable equipment or systems due to maintenance issues (NMCM) is important to moni- tor. Figure 6.29 displays the monthly NMCM rates over a two- year period.
Note that in this case we can see that there might be some sort of cycle present in the NMCM rate. The process may not have a random pattern of NMCM rates
Figure 6.29 Run chart for NMCM rate.
35
30
25
20
15
2 4 6 8 10 12 1614 18 2220 Month
24
R a te
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around the average rate. We have to know the scale or time period of the x-axis very well in order to correctly interpret the run chart.
The control chart, discussed in section F, is a special type of run chart. The control chart has a center or central line (the mean) drawn in to facilitate our eyes seeing the data move back and forth across this line. When the measurement for one sample tends to be dependent on the measurement for the previous sample, these data are called autocorrelated. As will be shown in section F, one assumption necessary for control charts to be valid is that the observations are independently distributed. When autocorrelation is present in the data, standard control charts do not work well in monitoring the process. Typical modeling techniques such as linear regression are also no longer valid because autocorrelated data are not inde- pendent, one of the key assumptions of linear regression models. Suppose there are n measurements taken in some time sequence. A measure of sample autocor- relation is given by
r x x x x
x x kk
t t k t
n k
n
t t
= −( ) −( )
−( ) =
+ =
−
=
∑
∑ 1
2
1
, for 0 1, , ..., K
(6.137)
where
k represents the number of time periods between measurements
x– is the average of all measurements
xt is the measurement taken at time t
xt+k is the measurement taken at time t + k
K is the total number of time periods
For example, if it is believed that measurements taken one after another are autocorrelated, then k = 1. In many problems, we may need to compute rk for several values of k, where an autocorrelation plot can be used to identify the value of k. See Bisgaard and Kulahci (2011) for more information on autocorrela- tion plots.
Detecting autocorrelation can be accomplished using several methods (e.g., graphically, as described above). Two analytic methods include moving average smoothing and trend analysis. Moving average smoothing involves smoothing the data over a short interval of time. In particular, consecutive observations in a series are averaged over a chosen window of time in order to remove as much noise as possible from the system (see Equation (6.163)). Moving average is dis- cussed in section F.5.d as part of the discussion on control charts. Trend analysis fits a general trend model to time series data and provides forecasts. Some models commonly fit include the linear, quadratic, exponential, and S-curve. Both meth- ods work well when no seasonal component is present in the data.
Time series data often exhibit seasonal or cyclic patterns. This type of behav- ior is often present in financial data. Consider sales of a department store, for example. While there may be a trend in sales (sales are increasing over time), there is also likely to be a seasonal pattern to these data. Sales in January of
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the current year are correlated not only with the previous month’s sales but also with sales in January of the previous years. More advanced modeling tech- niques can take advantage of this cyclic behavior. Autoregressive integrated moving average (ARIMA) models, for example, are commonly used to better model and predict seasonal data. For complete details of time series analysis, see Montgomery, Jennings, and Kulahci (2015) or Bisgaard and Kulahci (2011). For discussion of autocorrelation with respect to statistical process control, see Montgomery (2013).
F. STaTiSTiCaL ProCESS ConTroL (SPC) This section covers eight aspects of statistical process control: objectives and benefits, common and special causes of variation, selection of variable, ratio- nal subgrouping, control charts, control chart analysis, pre- control charts, and short- run SPC.
F.1. objectives and Benefits
Statistical process control (SPC) is quantitative problem solving consisting of diag- nostic techniques to assist in locating problem sources and prescriptive tech- niques to help solve problems. Many of these techniques are based on statistical principles.
A process is any repeatable sequence of events or operations leading to either a tangible or an intangible outcome. The use of SPC will show that a process is either in statistical control (i.e., the process variation appears to be random) or out of statistical control (i.e., the process exhibits nonrandom variation). SPC also makes it possible to determine whether the process is improving.
SPC is a tool for communicating information to engineering, product opera- tions, and quality control personnel. The principal elements of a successful SPC framework are analysis (to understand the process), methods (to measure the pro- cess), and leadership (to change the process).
A number of benefits can be attributed to SPC. Continuous improvement and maintenance of quality and productivity can be achieved, and process complex- ity can be reduced. By identifying and reducing process complexity, errors will be reduced and productivity improved through the substitution of sampling for 100% inspection. SPC also provides a common internal language for management, supervision, quality assurance/control, and product operations to discuss prob- lems, solutions, decisions, and actions.
F.2. Common and Special Causes of variation
Every process has variation. The sources of process variation can be divided into two categories: special and common. Common cause variation is that which is inher- ent in the process and generally is not controllable by process operators. Examples
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of common causes include variation in raw materials and variation in ambient temperature and humidity. In the case of service processes, common causes typi- cally include such things as variation in input data, variations in customer load, and variation in computer operations. Some authors refer to common cause varia- tion as natural variation.
Special causes of variation include unusual events that, when detected, can usually be removed or adjusted. Examples include tool wear, gross changes in raw materials, and broken equipment. Special causes are sometimes called assignable causes.
A principal problem in process management is the separation of special and common causes. If the process operator tries to adjust a process in response to com- mon cause variation, the result is usually more variation rather than less. This is sometimes called overadjustment or overcontrol. If a process operator fails to respond to the presence of a special cause of variation, this cause is likely to produce addi- tional process variation. This is referred to as underadjustment or undercontrol.
The principal purpose of control charts is to help the process operator recog- nize the presence of special causes so that appropriate action can be taken. Control charts are discussed in detail in the sections that follow.
F.3. Selection of variable
When a control chart is to be used, a variable (or variables) must be selected for monitoring. In a new process, there may be many different quality characteristics to monitor. However, as the process becomes more stable, the number of moni- tored characteristics will most likely be reduced.
Sometimes the variable of interest is the most critical dimension of the prod- uct. Contractual requirements with a customer sometimes specify the variable(s) to be monitored via a control chart. If the root cause of the assignable variation is known, an input variable, such as voltage or air pressure, may be monitored. It is possible to monitor several variables on separate control charts. But it is also use- ful to monitor two or more characteristics using a single control chart (multivariate control chart). Ultimately, the selection of the quality characteristic to be charted depends on experience and judgment.
F.4. rational Subgrouping
The selection of samples is important in the construction of control charts. The method used to select samples for a control chart must be logical or rational. In general, rational subgrouping involves selecting samples such that if assignable causes of variation are present in the system, there should be a greater probability of variation between successive samples while the variation within the sample is kept small.
Samples frequently consist of parts that are produced successively or con- secutively by the same process, to minimize the within- sample variation. The next sample is chosen later so that any process shifts that have occurred will be
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displayed on the chart as between- sample variation. Choosing the rational sub- group requires care to make sure the same process is producing each item.
In some instances it is more appropriate to select the sample over the entire interval since the last sample was chosen. This approach to rational subgrouping is effective in detecting shifts that may occur between samples taken consecutively. The sample represents all units produced since the last sample was taken. In gen- eral, the subgroup is a random sample of units selected over the entire interval since the last subgroup was selected.
Caution should be used when interpreting control charts where the sub- groups are units randomly selected over an interval. It is possible to make even an out- of-control process appear to be in control simply by increasing the interval between selected units.
F.5. Control Charts
Control charts are the most common tool for monitoring a quality characteris- tic of interest. Walter A. Shewhart introduced the concept of control charts in the 1920s. Because of his work, several control charts monitoring a single qual- ity characteristic of interest are referred to as Shewhart control charts. Control charts can be used for monitoring individual observations or subgroups. Differ- ent types of control charts are used for continuous versus discrete data, but all of the charts can be used to monitor for changes or trends in the quality character- istics of interest.
In this section, we present control charts for variables data and attributes data, including the following:
• x– and R control charts
• x– and s control charts
• Individuals control charts
• Fraction nonconforming control charts
• Control charts for nonconformities
For each of these control charts, distribution assumptions must be satisfied. For example, the x– charts are based on the assumption that x– follows a normal dis- tribution. The Shewhart control charts are sensitive to this assumption. If the normality assumption is violated, the overall performance of these charts can be very poor and result in incorrect signals.
Control limits are calculated based on data from the process. Formulas for control limits and examples of each are given in this section. The formulas are repeated in Appendix A. Several constants are needed in the formulas. These appear as subscripted letters, such as A2. The values of these constants are given in Appendix B. When calculating control limits, it is prudent to collect as much data as practical. Many authorities specify at least 25 samples. The examples in the fol- lowing sections use fewer samples for simplicity. It is desirable for the sample size to be held constant if possible.
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F.5.a. Variables Control Charts
The most commonly used control charts for variables (continuous) subgroup data are the x– and R chart and the x– and s chart. The x– chart monitors the mean of the process while the R chart and s chart monitor the process variability.
F.5.a.i. x– and R Charts
Suppose there are m subgroups each of size n chosen at random from a particular process (see Table 6.38). The sample mean and range for each subgroup are also given in Table 6.38.
The statistic x is the grand average and is the best estimate of the true process mean μ. R is the average range and will be used to estimate the process variability and to construct control charts. The upper control limit (UCL), center line (CL), and lower control limit (LCL) for the x– control chart are
UCL
CL
LCL
= +
=
= −
x A R
x
x A R
2
2
(6.138)
The UCL, CL, and LCL for the R control chart are
UCL
CL
LCL
=
=
=
D R
R
D R
4
3
(6.139)
A2, D3, and D4 are constants that depend on the sample size n. They can be found in Appendix B. Derivations of these constants can be found in Montgomery (2013).
Table 6.38 General notation for subgroup data.
Subgroup, i Measurements x–i Ri = x(max i) – x(min i)
1 x11, x21, . . . , xn1 x –
1 R1
2 x12, x22, . . . , xn2 x –
2 R2
3 x13, x23, . . . , xn3 x –
3 R3
.
.
.
.
.
.
.
.
.
.
.
.
m x1m, x2m, . . . , xnm x –
m Rm
x x
m
i i
m
= = ∑
1 R R
m
i i
m
= = ∑
1
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ExaMpLE 6.67
The turnaround time for complete blood count analysis from the laboratory to the emer- gency room at a local hospital is an important quality characteristic to be monitored. Turnaround times were recorded over 20 days in a one-month period. Four specimens were randomly selected per day and the turnaround times (in minutes) recorded. The times as well as the subgroup averages and ranges are given in Table 6.39. The grand average and average range are given in the last row of Table 6.39.
Table 6.39 Turnaround time data for x– and R charts.
Day x1 x2 x3 x4 x –
i Ri
1 83 49 65 78 68.75 34
2 81 77 75 76 77.25 6
3 71 67 44 58 60.00 27
4 92 53 93 74 78.00 40
5 75 58 90 51 68.50 39
6 70 79 87 49 71.25 38
7 74 50 68 45 59.25 29
8 80 66 75 64 71.25 16
9 80 63 72 81 74.00 18
10 90 77 92 64 80.75 28
11 75 51 89 74 72.25 38
12 64 65 88 59 69.00 29
13 97 57 88 76 79.50 40
14 84 62 55 68 67.25 29
15 76 63 70 66 68.75 13
16 62 68 66 55 62.75 13
17 73 77 91 83 81.00 18
18 65 65 84 46 65.00 38
19 73 64 84 71 73.00 20
20 75 88 65 93 80.25 28
x– –
= 71.39 R –
= 27.05
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From Appendix B with n = 4, we find A2 = 0.729, D3 = 0, and D4 = 2.282. The control limits for the x– control chart (using Equation (6.138)) are
UCL
CL
= + = + =
= =
x A R
x
2 71 39 0 729 27 05 91 11
71 3
. . ( . ) .
. 99
71 39 0 729 27 05 51 672LCL = − = − =x A R . . ( . ) .
The control limits for the R control chart (using Equation (6.139)) are
UCL
CL
LCL
= = =
= =
=
D R
R
D R
4
3
2 282 27 05 61 73
27 05
. ( . ) .
.
== =0 27 05 0( . )
The x– and R control charts for turnaround times are displayed in Figure 6.30. There are no points that plot outside the control limits on either chart. There also
do not appear to be any obvious patterns on the x– control chart. The process appears to be in control. More discussion of interpretation of control charts is provided in section F.6.
50
60
80
70
90 UCL = 91.09
x–– = 71.39
LCL = 51.68 2 201816141210864
S am
pl e
m ea
n
Sample
0
15
45
30
60 UCL = 61.71
R– = 27.05
LCL = 0 2 201816141210864
S am
pl e
ra ng
e
Sample
Figure 6.30 x– and R control charts for turnaround times.
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F.5.a.ii. x– and s Charts
Whenever possible, the sample standard deviation should be used instead of the range in estimating the process variability. When the sample size n is large, say n > 10, the sample standard deviation is a better estimate of the true pro- cess standard deviation than the range. As n increases, the range R loses sta- tistical efficiency and becomes less precise. The sample standard deviation is also a better estimator for the process standard deviation for nonconstant sam- ple sizes.
The development of the x– and s control chart is similar to development of the x– and R control chart. In this case, the subgroup standard deviation is calculated instead of the range. Suppose xi1, xi2, . . . , xin represent a sample of size n for any subgroup i. The formula for the sample standard deviation of subgroup i is
s
x x
ni
ij j
n
= −
− = ∑( )2
1
1 (6.140)
The average standard deviation for all m subgroups is
s
s
m
i i
m
= = ∑
1
(6.141)
The control limits and center line for the x– control chart are then
UCL
CL
LCL
= +
=
= −
x A s
x
x A s
3
3
(6.142)
The control limits and center line for the s chart are
UCL
CL
LCL
=
=
=
B s
s
B s
4
3
(6.143)
where A3, B3, and B4 are constants that depend on the sample size n. They can be found in Appendix B.
Further interpretation of these and other control charts is presented in sec- tion F.6.
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ExaMpLE 6.68
Reconsider the turnaround time data from the previous example. Instead of the range for each day (subgroup), the standard deviation is calculated. The subgroup averages will not change. The sample standard deviations and average standard deviations are given in Table 6.40.
Table 6.40 Turnaround time data for x– and s charts.
Day x1 x2 x3 x4 x –
i si
1 83 49 65 78 68.75 15.20
2 81 77 75 76 77.25 2.63
3 71 67 44 58 60.00 11.97
4 92 53 93 74 78.00 18.81
5 75 58 90 51 68.50 17.52
6 70 79 87 49 71.25 16.38
7 74 50 68 45 59.25 13.94
8 80 66 75 64 71.25 7.54
9 80 63 72 81 74.00 8.37
10 90 77 92 64 80.75 13.00
11 75 51 89 74 72.25 15.73
12 64 65 88 59 69.00 12.94
13 97 57 88 76 79.50 17.29
14 84 62 55 68 67.25 12.37
15 76 63 70 66 68.75 5.62
16 62 68 66 55 62.75 5.74
17 73 77 91 83 81.00 7.83
18 65 65 84 46 65.00 15.51
19 73 64 84 71 73.00 8.29
20 75 88 65 93 80.25 12.69
x– –
= 71.39 s– = 11.97
Continued
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The control limits and center line for the x– control chart (using Equation (6.142)) are
UCL
CL
= + = + =
= =
x A s
x
3 71 39 1 628 11 97 90 88
71 3
. . ( . ) .
. 99
71 39 1 628 11 97 51 903LCL = − = − =x A s . . ( . ) .
The control limits and center line for the s chart (using Equation (6.143)) are
UCL
CL
LCL
= = =
= =
=
B s
s
B s
4
3
2 266 11 97 27 12
11 97
. ( . ) .
.
== =0 011.97( . )
where A3 = 1.628, B3 = 0, and B4 = 2.266, from Appendix B with n = 4. The x – and s control
charts are displayed in Figure 6.31.
50
60
80
70
90 UCL = 90.87
x–– = 71.39
LCL = 51.90 2 201816141210864
S am
pl e
m ea
n
Sample
0
10
20
30 UCL = 27.12
s– = 11.97
LCL = 0 2 201816141210864
S am
pl e
st an
da rd
de vi
at io
n
Sample
Figure 6.31 x– and s control charts for complete blood count analysis turnaround times.
The process appears to be in control since there are no obvious trends or patterns and points plot within the control limits on both charts.
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F.5.a.iii. Individuals Control Charts
Many practical applications exist in which the subgroup consists of a single observa- tion (n = 1). Examples include very slow processes or processes in which the mea- surement is very expensive to obtain, such as with destructive tests. An individuals control chart for variable data is appropriate for this type of situation.
The individuals control chart uses the moving range of two successive sub- groups to estimate process variability (see Montgomery [2013] for a detailed dis- cussion of moving range and individuals control charts in general). The moving range is given by
MRi = |xi – xi–1| (6.144)
For m subgroups of size n = 1 each, m – 1 moving ranges are defined as MR2 = |x2 – x1|, MR3 = |x3 – x2|, . . . , MRm = |xm – xm–1|. The average moving range is simply
MR
MR
m
i i
m
= −
= ∑
2
1 (6.145)
Division is done by m – 1 since only m – 1 moving range values are calculated (there is no moving range for subgroup 1). Control charts are constructed for the individual observations (individuals chart) and the moving range of the sub- groups (MR chart).
The control limits and center line of the x (or individuals) control chart are
UCL
CL
LCL
= +
=
= −
x MR d
x
x MR d
3
3
2
2
(6.146)
where d2 is a constant that depends on the number of observations used to calcu- late the moving range for each subgroup (i.e., n = 2). Values for d2 can be found in Appendix B. The control chart for individuals is constructed by plotting the actual observation xi, the control limits, and the center line against the subgroup (or time) order.
The control limits and center line for the moving range control chart are
UCL
CL
LCL
=
=
=
D MR
MR
D MR
4
3
(6.147)
where D3 and D4 are constants found in Appendix B for n = 2. The moving range control chart is constructed by plotting the m – 1 moving ranges, the control limits, and the center line against the subgroup (or time) order.
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ExaMpLE 6.69
Packages of a particular instant dry food are filled by a machine and weighed. The weights (in ounces) for 15 successive packages have been collected and are displayed in Table 6.41. The engineer wishes to determine whether the filling process is indeed in control.
The moving ranges are calculated using Equation (6.144). To illustrate, consider the first moving range at subgroup 2:
MR2 = |x2 – x1| = |19.92 – 19.85| = 0.07
The remaining moving ranges are calculated accordingly and are given in Table 6.41. The control limits and center line for the individuals chart with moving ranges of size 2 (using Equation (6.146)) are
UCL
CL
LCL
= + =
=
=
19 954 3 0 39
1 128 20 991
19 954
19
. . .
.
.
.. . .
.954 3 0 39
1 128 18 917− =
Table 6.41 Weights for dry food packages.
Bottle Weight (xi ) Moving range
1 19.85 —
2 19.92 0.07
3 19.93 0.01
4 19.26 0.67
5 20.36 1.10
6 19.96 0.40
7 19.87 0.09
8 19.80 0.07
9 20.40 0.60
10 19.98 0.42
11 20.17 0.19
12 19.81 0.36
13 20.21 0.40
14 19.64 0.57
15 20.15 0.51
x– = 19.954 M —
R –
= 0.39
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The control limits and center line for the moving range chart (using Equation (6.147)) are
UCL
CL
LCL
= =
=
= =
3 267 0 39 1 274
0 39
0 0 39 0
. ( . ) .
.
( . )
The control charts for individual observations and for the moving range are displayed in Figure 6.32.
Examining these control charts, the process does not appear to be out of statistical control.
19.0
19.5
20.5
20.0
21.0 UCL = 20.991
x– = 19.954
LCL = 18.917 21 151412 13111097 86543
In di
vi du
al v
al ue
Observation
0.0
0.3
0.9
0.6
1.2 UCL = 1.274
MR = 0.39
LCL = 0 21 151412 13111097 86543
M ov
in g
ra ng
e
Observation
Figure 6.32 I and MR control charts for package weights.
It is important to note that the moving range control chart cannot be inter- preted in the same way as the R chart presented earlier, with respect to patterns or trends. Patterns or trends identified on the moving range chart do not necessarily indicate that the process is out of control. The moving ranges are correlated. There is a natural dependency between successive MRi values.
F.5.b. Attributes Control Charts
Attributes control charts are used for discrete or count data. In many scenarios the quality characteristic of interest is simply a classification of the measurement into a single category. For example, manufactured products may be measured but clas- sified only as defective/nondefective, conforming/nonconforming, or pass/fail. Other situations may involve monitoring the number of nonconformities on an item. For example, billing statements may be examined for errors such as incorrect name, missing information, and incorrect amounts or type of service identified. Variables control charts are not appropriate for many of these situations. Control
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charts for data that can be classified are attributes control charts. We discuss the fol- lowing attributes control charts:
• Fraction nonconforming control charts (p charts)
• Number nonconforming control charts (np charts)
• Control charts for nonconformities (c and u control charts)
Control charts for nonconformities are similar to those for the number of non- conforming items, discussed in the previous subsection. The p chart and np chart represent the fraction of nonconforming items. When the variable of interest is the number of nonconformities per unit, the p and np charts are not appropriate. For p and np charts, we noted that the number of nonconforming units could not exceed the number of units being investigated in the subgroup, that is, X ≤ n. For monitoring nonconformities, there is no such restriction. In this case, nonconfor- mities are counted per unit. There could be an infinite (countably infinite) number of nonconformities on a unit or units. More than one of these errors may occur on any one unit. Control charts for nonconformities are the c chart and the u chart.
F.5.b.i. The p Chart
For the fraction nonconforming control charts, the quality characteristic of interest can be placed into one of exactly two categories. These categories may be pass/ fail, conforming/nonconforming, and so on. For simplification, the term “noncon- forming” will be used as a general reference regardless of the final categories to be used. The notation to be used is as follows:
• n—number of items examined (lot size, sample size).
• m—number of subgroups.
• X—number of nonconforming items found in the sample of size n, where X ≤ n.
• p—probability that any one item of interest will be nonconforming. This parameter is often unknown and must be estimated.
• p̂—sample fraction nonconforming. By definition, from Equation (6.64), this is
p̂ X n
=
and is calculated for each of the m subgroups.
• p —average fraction nonconforming. By definition,
p
p
m
i i
m
= = ∑ ˆ
1
(6.148)
and is an estimate of p, defined above.
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The p chart is used to monitor the proportion nonconforming directly. The con- trol limits and center line (when p is unknown) are
UCL
CL
LCL
= + −
=
= − −
p p p
n
p
p p p
n
3 1
3 1
( )
( )
(6.149)
Note that if the LCL is computed to be less than zero, it is set at zero. The control limits, center line, and individual sample fraction noncon-
forming p̂i are plotted against the subgroup number m. If any of the fraction nonconforming lie outside the control limits, the process is considered out of control. Patterns or trends would also be an indication of possible out- of-control situations.
It is not necessary that the sample sizes be equal for all subgroups. For exam- ple, suppose surgeries that result in surgical site infections are monitored at a par- ticular hospital. The number of surgeries performed each month is examined, and those resulting in surgical infections are recorded. Typical data for a 12-month period are given in Table 6.42.
Table 6.42 Surgical site infection rates.
Month Surgeries Surgical infection
1 57 8
2 62 6
3 66 1
4 57 2
5 69 2
6 63 6
7 55 10
8 56 6
9 54 9
10 62 3
11 65 4
12 69 5
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ExaMpLE 6.70
A small bank collects data on the number of weekly account activities that are recorded in error. Over a 12-week period 1000 account activities are randomly selected and exam- ined for the number that are in error. The bank would like to monitor the proportion of errors being committed, by establishing control charts.
The fraction in error p̂i for each week must be computed. The fraction in error for each week is given in Table 6.43 for n = 1000. The average fraction in error p– is found (using Equation (6.148)) to be
p p i
i= == ∑ ˆ
.1
12
12 0 01042
Table 6.43 Errors in account activities.
Week Number of errors p̂i
1 6 0.006
2 11 0.011
3 4 0.004
4 10 0.010
5 5 0.005
6 30 0.030
7 9 0.009
8 8 0.008
9 12 0.012
10 7 0.007
11 12 0.012
12 11 0.011
The control limits and center line for the p chart (using Equation (6.149)) are
UCL = + −
= + −
p p p
n 3
1 0 01042 3
0 01042 1 0 01042 1
( ) .
. ( . ) 0000
0 020052
0 01042
3 1
0 010
=
=
= − −
=
.
.
( ) .
CL
LCL p p p
n 442 3
0 01042 1 0 01042 1000
0 00078− −
= . ( . )
.
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The resulting p chart is displayed in Figure 6.33. There is a single point that plots beyond the upper control limit. This point should be investigated to determine if it is truly an unusual point. If it is found to be unusual and the assignable cause identified, the point can be removed and the center line and control limits recalculated. Suppose in this case that the cause for the outlier in week 6 was identified and a revised control chart con- structed. The revised control chart is shown in Figure 6.34. Notice that the control limits and center line have been updated while the fraction in error for week 6 is still plotted on the graph. On a revised control chart, the removed point is used only as a placeholder.
0.000
0.005
0.010
0.015
0.020
0.025
0.030
21 3 4 5 6 7 8 9 10 11 12
P ro
p o
rt io
n
Week
UCL = 0.02005
p– = 0.01042
LCL = 0.00078
1
Figure 6.33 p chart for errors in account activities.
0.000
0.005
0.010
0.015
0.020
0.025
0.030
21 3 4 5 6 7 8 9 10 11 12
P ro
p o
rt io
n
Sample
UCL = 0.01741
p– = 0.00864
LCL = 0
1
Figure 6.34 Revised p chart for errors in account activities.
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The sample size is variable, and there are two ways to calculate the control limits:
• Use the same formulas for the control limits given earlier, using the average sample size as an estimate for n
• Use the actual sample sizes and construct varying control limits
For the surgical infection rates, varying control limits were used. The control chart is displayed in Figure 6.35.
F.5.b.ii. The np Chart
The np chart is a variation of the p chart, with the actual number of nonconform- ing items plotted on the chart. The np chart and the p chart for the same problem will provide identical information. That is, if the p chart indicates that a process is out of control, then the np chart will also indicate that the same process is out of control. One of the reasons the np chart is an attractive alternative to the p chart is ease of interpretation.
The average fraction nonconforming p is the only value that must be esti- mated before constructing the control limits. It can be found without having to calculate the sample fraction nonconforming values ( p̂i). For the np chart, the aver- age fraction nonconforming can be calculated as
p
X
mn
i i
m
= = ∑
1
(6.150)
Figure 6.35 p chart for surgical site infection rate using varying sample sizes.
0.00
0.05
0.10
0.15
0.20
3 4 51 2 6 7 8 9 10 11 12
P ro
p o
rt io
n
Month
UCL = 0.1847
p– = 0.0844
LCL = 0
Tests performed with unequal sample sizes
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The control limits and center line are then
UCL
CL
LCL
= + −
=
= − −
np np p
np
np np p
3 1
3 1
( )
( )
(6.151)
The control limits, center line, and number of nonconforming items Xi are plot- ted against the subgroup. Interpretation of the np chart is identical to that of the p chart.
ExaMpLE 6.71
Reconsider the accounts in error from the previous example. The average fraction in error was found to be p– = 0.01042. The control limits and center line for the np control chart (using Equation (6.151)) are
UCL = + − = +np np p3 1 1000 0 01042 3 1000 0 01042( ) ( . ) ( . ))( . ) .
( . ) .
1 0 01042 20 05
1000 0 01042 10 42
− =
= = =CL np
LLCL = − − = −np np p3 1 1000 0 01042 3 1000 0 0104( ) ( . ) ( . 22 1 0 01042 0 787)( . ) .− =
The np control chart is displayed in Figure 6.36. As with the p chart, the np chart indi- cates that the process is out of statistical control.
0
5
10
15
20
25
30
21 3 4 5 6 7 8 9 10 11 12
S a m
p le
c o
u n
t
Week
UCL = 20.05
n–p– = 10.42
LCL = 0.78
1
Figure 6.36 np control chart for errors in account activities.
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F.5.b.iii. The c Chart
If the subgroup size n is constant from subgroup to subgroup, the c chart is an appropriate control chart for nonconformities. For the c chart:
• n = number of units inspected, sample size (this can be size n = 1 or greater)
• m = number of subgroups
• X = number of nonconformities per unit inspected or per subgroup
• c = average number of nonconformities:
c
X
m
i i
m
= = ∑
1
(6.152)
The control limits and center line for the c chart are
UCL
CL
LCL
= +
=
= −
c c
c
c c
3
3
(6.153)
ExaMpLE 6.72
Billing statements for a local hospital are being examined for errors. Twenty billing state- ments are randomly chosen each day over a 24-day period and examined for missing information, incorrect amounts, and wrong type of service identified. The number of errors (nonconformities) is given in Table 6.44.
Table 6.44 Errors on hospital billing statements.
Day Number of
errors Day Number of
errors
1 4 13 10
2 18 14 13
3 14 15 3
4 7 16 12
5 7 17 17
6 8 18 13
7 16 19 9
8 6 20 17
9 10 21 9
10 12 22 9
11 9 23 6
12 8 24 10
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The average number of errors (using Equation (6.152)) is
c X
m
i i
m
= = + + +
== ∑
1 4 18 10 24
10 29 ...
.
The control limits and center line for the c chart (using Equation (6.153)) is
UCL
CL
LCL
= + = + =
= =
= −
c c
c
c
3 10 29 3 10 29 19 91
10 29
. . .
.
33 10 29 3 10 29 0 67c = − =. . .
The c chart is displayed in Figure 6.37. The process appears to be in statistical control.
0
5
10
15
20
2 4 6 8 10 12 1614 18 20 22 24
S am
pl e
co un
t
Day
UCL = 19.91
c– = 10.29
LCL = 0.67
Figure 6.37 c chart for number of billing errors.
F.5.b.iv. The u Chart
The c chart monitors the number of nonconformities. The u chart, on the other hand, monitors the average number of nonconformities. Like the p and np charts, the resulting c and u charts for constant sample size will provide identical results. It is not necessary for the sample size to be constant from subgroup to subgroup for the u chart. Let ui be the average number of nonconformities for the ith sub- group (i = 1, 2, . . . , m), where
u
X ni
i=
(6.154)
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Also, let u represent the overall average number of nonconformities per unit, that is,
u
u
m
i i
m
= = ∑
1
(6.155)
The control limits and center line for the average number of nonconformities are
UCL
CL
LCL
= +
=
= −
u u n
u
u u n
3
3
(6.156)
The control limits, center line, and ui are plotted on the control chart against the subgroup.
ExaMpLE 6.73
Reconsider the errors on billing statements example discussed in Example 6.72. The average number of nonconformities ui for each day, in which 20 billing statements were collected, is given in Table 6.45.
Table 6.45 Billing statement errors for a 24-day period.
Day Number of
errors ui Day Number of
errors ui
1 4 0.20 13 10 0.50
2 18 0.90 14 13 0.65
3 14 0.70 15 3 0.15
4 7 0.35 16 12 0.60
5 7 0.35 17 17 0.85
6 8 0.40 18 13 0.65
7 16 0.80 19 9 0.45
8 6 0.30 20 17 0.85
9 10 0.50 21 9 0.45
10 12 0.60 22 9 0.45
11 9 0.45 23 6 0.30
12 8 0.40 24 10 0.50
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The overall average number of nonconformities per unit (using Equation (6.155)) is then
u u
m
i i
m
= = + + +
== ∑
1 0 20 0 90 0 50 24
0 515 . . ... .
.
The control limits and center line for the u chart (using Equation (6.156)) are
UCL
CL
LCL
= + = + =
= =
u u n
u
3 0 515 3 0 515
20 0 996
0 515
. .
.
.
== − = − =u u n
3 0 515 3 0 515
20 0 033.
. .
The u control chart is displayed in Figure 6.38. Again, the process does not appear to be out of statistical control.
0.0
0.4
0.2
0.6
0.8
1.0
2 4 6 8 10 12 1614 18 20 22 24
S a m
p le
c o
u n
t p
e r
u n
it
Day
UCL = 0.996
u– = 0.515
LCL = 0.033
Figure 6.38 u chart for billing statement errors.
F.5.c. Cumulative Sum Control Charts
So far, only Shewhart control charts for monitoring various processes have been presented. Shewhart control charts are known to be poor at detecting small shifts in the process mean because they are based only on the current observation (see Gan [1991], Hawkins [1981, 1993], Montgomery [2013], and Woodall and Adams [1993]). An alternative to the use of Shewhart control charts is the cumulative sum (CUSUM) control chart. The CUSUM control chart has been shown to be more sen- sitive to small shifts in the process because it is based on both the current observa- tion and the most recent past observations.
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Page (1961) first introduced the cumulative sum control chart. The control chart plots the cumulative sums of deviations of the observations from a target value. That is, the CUSUM control chart plots the quantity
C xi j
j
i
= −( ) = ∑ 0
1
µ
(6.157)
against the subgroup i where
Ci is the cumulative sum of deviations up to and including the ith sample
x– j is the mean of the jth sample
μ0 is the target value for the process average
As long as the process average remains at the target value μ0, then the cumula- tive sums Ci will be approximately zero. If the process shifts away from the target mean, then Ci will increase in absolute value.
Since the CUSUM chart uses information from the current and recent past observations, it can detect small shifts in the process more quickly than a standard Shewhart chart. CUSUM control charts can be used for subgroup data or individu- als data. In addition, there have been applications for both variables and attributes data. The two- sided tabular CUSUM control chart for individuals is presented here. One- sided CUSUM charts can be constructed if the interest is in a particular direction, downward or upward, but not necessarily both.
The tabular form of the two- sided CUSUM chart involves two statistics, Ci + and
Ci −. Ci
+ represents the cumulative sum of deviations above the target mean and is referred to as the one- sided upper CUSUM. Ci
− is the cumulative sum of deviations below the target mean and is referred to as the one- sided lower CUSUM. Ci
+ and Ci
− are calculated as
= −C x K Ci + ++ +max 08 i 0, ( )µ 8i–1 (6.158)
C x Ki i − = − −min8 80 0, (µ )) +
−Ci–1 (6.159)
where xi is the ith observation. Ci + and Ci
− are initially set at C0 + = C0
– = 0. The con- stant K is a reference value and calculated as
K =
−µ µ1 0 2
(6.160)
where μ0 is the target mean and μ1 is the out- of-control mean that we are interested in detecting. If μ1 is unknown, we can let K = kσ, where σ is the process standard deviation and k is some constant chosen so that a particular shift is detected. To illustrate, suppose a shift from a target of 1.5 standard deviations is important to detect. That is, we want to detect a shift from μ0 to μ0 – 1.5σ or to μ0 + 1.5σ. In this case, K = 1.5σ. If the process standard deviation is not known, it must be estimated from the data provided.
The values of Ci + and Ci
− for each sample are plotted on a two- sided CUSUM control chart. If either value plots outside a stated decision interval (–H, H), the process is considered out of control. H should be chosen after careful consider- ation. There are many possible values for H, but a common setting is H = 5σ. It has been shown that this decision value results in a low false- alarm rate for the process under study. For further discussion of the design of CUSUM control charts, see Hawkins (1993) or Woodall and Adams (1993).
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ExaMpLE 6.74
Packages of a particular instant dry food are filled by a machine and weighed. The weights (in ounces) for 24 successive packages have been collected and are displayed in Table 6.46. The target mean weight is μ0 = 20 ounces. From past experience, it is believed that the process standard deviation is σ = 0.20 ounces. If the process mean shifts from this target by one-half of the process standard deviation, then the filling process is deemed out of control. The engineer would like to design a two-sided CUSUM control chart and deter- mine whether the process is indeed in control at the target μ0 = 20 ounces.
Table 6.46 Weights for dry food packages with custom values.
Package Weight (xi) C i + C i
–
1 20.26 0.16 0
2 19.97 0.03 0
3 19.76 0 –0.14
4 19.72 0 –0.32
5 19.69 0 –0.53
6 19.85 0 –0.58
7 19.96 0 –0.52
8 20.03 0 –0.39
9 20.06 0 –0.23
10 19.71 0 –0.42
11 19.68 0 –0.64
12 19.94 0 –0.6
13 20.3 0.2 –0.2
14 19.77 0 –0.33
15 20.4 0.3 0
16 19.98 0.18 0
17 19.91 0 0
18 20.18 0.08 0
19 20.08 0.06 0
20 20.05 0.01 0
21 20.2 0.11 0
22 19.9 0 0
23 19.95 0 0
24 20.12 0.02 0
Continued
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Some of the known or assumed quantities are:
μ0 = 20 ounces
σ = 0.20 ounces
K = 0.5σ = 0.5(0.20) = 0.10
H = 5σ = 5(0.20) = 1
The CUSUM values Ci + and Ci
– are compared to the decision interval (–H, H) = (–1, 1). If any cumulative sum falls outside the interval (–1, 1), the process is considered to be out of control. To illustrate these calculations, consider the first observation, x1 = 20.26 ounces. Initially, C0
+ = C0 – = 0, and as previously shown, K = 0.1. The first cumulative sums
(using Equations (6.158) and (6.159), respectively) are
C x K C1 1 0 00
0 20 26 20 0
+ += − + + = − +
max
max
, ( )
, . ( .. )
, .
.
1 0
0 0 16
0 16
+[ ] = [ ] =
max
µ
C x K C11 0 00
0 20 26 20 0
− −= − − + = − −
min
min
, ( )
, . ( .. )
, .
.
1 0
0 0 36
0
+[ ] µ
= −[ ] =
min
The remaining CUSUMs can be calculated similarly, but it is recommended that the calculations be done using a spreadsheet package or modern statistical package. Notice that both cumulative sums are within the decision interval (–1, 1), so the process has not signaled out of control at this point. The CUSUM chart is shown in Figure 6.39.
The CUSUMs plot well within the decision interval, so there does not appear to have been a shift of 0.5σ from the target value of 20 ounces.
–1.0
–0.5
0.0
0.5
1.0
2 4 6 8 10 12 1614 18 20 22 24
C u
m u
la ti
v e s
u m
Sample
UCL = 1
0
LCL = –1
Figure 6.39 CUSUM chart for package weight.
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F.5.d. Exponentially Weighted Moving Average and Moving Average Control Charts
In this section we discuss the exponentially weighted moving average (EWMA) control chart and the moving average (MA) control chart. They are similar in that both use a summary statistic involving past observations.
F.5.d.i. Exponentially Weighted Moving Average Control Chart
The EWMA control chart, like the CUSUM, is a good chart for detecting small shifts in the process mean parameter. The EWMA control chart was first intro- duced by Roberts (1959). The EWMA statistic is defined as
zi = λxi + (1 – λ)zi–1 (6.161)
where λ is a weight and 0 < λ ≤ 1, xi is the current observation, and zi–1 is the previ- ous EWMA statistic. Initially, z0 = μ0, the process target mean. If the process target mean is not known, then x– is used as the initial value.
Like the CUSUM, the EWMA includes information from recent past obser- vations as well as the current observation xi. Control limits can be placed on the values of zi. If one or more of the zi values fall outside the control limits, then the process is considered to be out of statistical control. The steady-state control limits for the EWMA for large values of i are
UCL
LCL
= + −
= − −
λ λ
λ λ
0
0
2
2
L
L
c c
c c
σ
σ
µ
µ
(6.162)
where L is the width of the control limits. The values of L and λ can significantly impact the performance of the chart. Small values of λ work well in practice (0.05 ≤ λ ≤ 0.25) with values of L between 2.6 ≤ L ≤ 3 (see Crowder [1989], Lucas and Saccucci [1990], and Montgomery [2013]).
Patterns and trends on the CUSUM and EWMA control charts do not neces- sarily indicate an out- of-control process. The statistics plotted on the charts are cor- related since they are functions of not only the current observation but also recent past observations. As such, patterns can be expected even with an in- control process.
ExaMpLE 6.75
Reconsider the 24 packages being filled by a machine and weighed in the previous exam- ple. The package weights and EWMA values are displayed in Table 6.47. The target mean weight is μ0 = 20 ounces. From past experience, it is believed that the process standard deviation is σ = 0.20 ounces. We would like to construct an EWMA control chart for these data using λ = 0.10 and L = 2.7. The EWMA statistic is
z x z
x z
x
i i i
i i
i
= + −
= + −
=
−
−
l l( )
. ( . )
.
1
0 10 1 0 10
0 10
1
1
++ −0 90 1. zi
Continued
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Package Weight (xi) z
1 20.26 20.026
2 19.97 20.0204
3 19.76 19.99436
4 19.72 19.966924
5 19.69 19.9392316
6 19.85 19.93030844
7 19.96 19.9332776
8 20.03 19.94294984
9 20.06 19.95465485
10 19.71 19.93018937
11 19.68 19.90517043
12 19.94 19.90865339
13 20.3 19.94778805
14 19.77 19.93000924
15 20.4 19.97700832
16 19.98 19.97730749
17 19.91 19.97057674
18 20.18 19.99151906
19 20.08 20.00036716
20 20.05 20.00533044
21 20.2 20.0247974
22 19.9 20.01231766
23 19.95 20.00608589
24 20.12 20.0174773
To illustrate the calculation of the EWMA statistic for each observation, consider the first observation, x1 = 20.26 ounces. If we initialize the process using z0 = μ0 = 20, z1 is found to be
z x z1 010 10 0 90
0 10 20 26 0 90 20
20 02
= +
= +
=
. .
. ( . ) . ( )
. 66
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The remaining EWMA statistics are calculated similarly. Table 6.47 includes all of the EWMA values. The control limits and center line (using Equation (6.162)) are
UCL = + −
= + −
λ λ0 2
20 2 7 0 2 0 10
2 0 10 L . ( . )
. .
=
=
= − −
= −
20 12
20
2 20 2 70
.
. (
CL
LCL λ λ
L 00 2 0 10
2 0 10 19 88. )
. .
. −
=
b
b
b
b
σ
σ
µ
µ
The control limits and the EWMA statistics zi are plotted on the EWMA control chart in Figure 6.40. The process appears to be in control since all EWMA statistics fall within the control limits.
19.90
19.95
20.00
20.05
20.10
20.15
0 5 10 15 20 25
z i
Sample
UCL = 20.12
CL = 20.00
LCL = 19.88
Figure 6.40 EWMA control chart for package weight.
F.5.d.ii. Moving Average Control Chart
The MA control chart is similar to the EWMA in that it uses a moving average of a certain span (not necessarily consecutive observations). However, the mov- ing average is an unweighted average of the observations. Suppose there are n observations, x1, x2, . . ., xn, selected from the process with mean μ0 and standard deviation σ. Furthermore, suppose a moving average of span w is of interest. The moving average statistic at time i can be written as
MA
x x x wi
i i i w= + + +− − +1 1...
(6.163)
The values of MAi are plotted on a control chart with control limits and center line:
UCL
CL
LCL
= +
=
= −
µ σ
µ
µ σ
0
0
0
3
3
w
w
(6.164)
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The MA chart may be suitable for the following situations:
• When data are collected periodically or when it may take some time to produce a single item
• When it is desirable to dampen the effects of overcontrol
• When it is necessary to detect shifts in the process that are smaller than what a Shewhart chart can detect
F.5.e. Choosing a Control Chart
The choice of which control chart to use in a particular problem depends on the process under investigation. Figure 6.41 includes some guidelines for control chart selection. Shewhart control charts are easy to construct and interpret, but are less effective at detecting small shifts in the process parameter. It has also been shown that Shewhart control charts are very sensitive to the assumption of normality (see Borror, Montgomery, and Runger [1999]). That is, if the underlying distribution of the process is non- normal, then Shewhart charts can often signal out- of-control when in fact the process is in control. CUSUM and EWMA control charts are quite robust to departures from normality and are better at detecting small shifts in the process than the Shewhart charts.
Figure 6.41 Some guidelines for univariate control chart selection. Source: D. C. Montgomery, Introduction to Statistical Quality Control, 7th ed. (Hoboken, NJ: John Wiley & Sons, 2013).
Are process data autocorrelated?
Variables or attributes?
Shift size
CUSUM EWMA
CUSUM EWMA
CUSUM EWMA using p
CUSUM EWMA using c, u; time between events
x (individuals) MR
x–, R x–, S
Shift size
Shift size
Shift size
c, up, np
Sample size
Data type
Is there an adjustment variable?
YesNo
Use feedback control with an
adjustment chart or another EPC procedure
or EPC/SPC
Fit ARIMA; apply standard control charts (EWMA, CUSUM, x, MR)
to either residuals or original data or
use moving center line EWMA or use a model- free approach
AttributesVariables
n = 1n > 1
SmallLarge SmallLarge
Defects (counts)Fraction
SmallLargeSmallLarge
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F.6. Control Chart analysis
A critical tool in the analysis of charted data is the process log. The process log may be a separate document or it may be maintained as part of the control chart itself. Entries in the log should include all changes in the process and its envi- ronment, including maintenance, raw materials, adjustments, tooling, fixturing, and so on.
Each of the control limit formulas discussed in the previous section uses data from the process. Although it is not always obvious from the formulas, the upper and lower limits are placed at ±3σ from the average. The use of three- sigma lim- its is a direct result of the underlying assumption of normality. It can be shown that if the underlying distribution is normal, then approximately 99.7% of all the data will lie within three standard deviations of the mean. Therefore, an observa- tion that falls beyond three standard deviations from the mean would be flagged as unusual since the probability of this occurring is 0.003 and may be an indica- tion of an out- of-control process. Since the Shewhart control charts are based on the normality assumption, it is common to use three standard deviations in the construction of the control limits for these charts. For the EWMA control chart, if L = 3 and λ = 1, then the control limits would reduce to the standard Shewhart control limits. But it has been shown that values other than L = 3 and λ = 1 can result in well- performing control charts, especially for detecting small shifts in the process parameter.
It should be noted that the probability of a point falling inside or outside three standard deviations is somewhat theoretical because no process runs as if its output were randomly selected numbers from some historical distribution. It is enough to say that when a point falls outside the control limits, the probability is quite high that the process has changed. When the probability is very high that a point did not come from the distribution used to calculate the control limits, the process is said to be out of statistical control. Unfortunately, this is often abbrevi- ated to “out of control,” which seems to imply some wild action on the part of the process. In reality, the out- of-statistical-control condition is often very subtle and would perhaps not be detected without the control chart. This, in fact, is one of the main values of the control chart: it detects changes in a process that would not otherwise be noticed. This may permit adjustment or other action on the process before serious damage is done.
One of the hazards of using a control chart without proper training is the ten- dency to react to a point that is not right on target by adjusting the process, even though the chart does not indicate that the process has changed. If an adjustment is made whenever a point is not exactly on target, it may tend to destabilize a stable process. In the ideal situation, a process should not need adjustment except when the chart indicates it is out of statistical control. W. Edwards Deming (1986) states that “the function of a control chart is to minimize the net economic loss from . . . overadjustment and underadjustment.”
A number of events are very unlikely to occur unless the process has changed, and thus serve as statistical indicators of process change. The lists of rules that reflect these statistical indicators vary somewhat from textbook to textbook, but two of the most widely used lists of rules are the eight rules used by the software package Minitab and the six rules listed by the AIAG in its SPC manual.
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The eight Minitab rules are as follows:
1. One point more than 3σ from the center line (either side)
2. Nine points in a row on the same side of the center line
3. Six points in a row, all increasing or all decreasing
4. Fourteen points in a row, alternating up and down
5. Two out of three points more than 2σ from the center line (same side)
6. Four out of five points more than 1σ from the center line (same side)
7. Fifteen points in a row within 1σ of the center line (either side)
8. Eight points in a row more than 1σ from the center line (either side)
In the third edition of its SPC manual, the AIAG provides a list of special cause criteria that is identical to Minitab’s list except for rule 2, which says:
2. Seven points in a row on one side of the center line
The AIAG SPC manual emphasizes that “the decision as to which criteria to use depends on the process being studied/controlled.” CQEs may find it useful to gen- erate additional tests for particular situations. If, for instance, an increase in values represents a safety hazard, it would not be necessary to wait for the specified num- ber of successively increasing points to take action. The ±3σ location for the control limits is somewhat arbitrary and could conceivably be adjusted based on the eco- nomic trade- off between the costs of not taking action when an out- of-control con- dition occurs and the costs of taking action when an out- of-control condition has not occurred. In general, moving the control limits up and down can be a source of additional problems, and it would be better in most cases to put that energy into reducing variation.
Sensitizing rules should always be used with caution. Although sensitiz- ing rules can improve a Shewhart chart’s ability to detect small shifts, they can seriously degrade the performance of the chart when the process is indeed in control. Control chart performance is often measured by the average run length (ARL), which is defined as the number of cycles, time periods, or samples that elapse before the process signals out- of-control. If the process is in control, we want the ARL to be large. If the process is out of control, a small ARL is desir- able. When several sensitizing rules are used simultaneously on a control chart, the in- control ARL can become unacceptably small. For example, suppose that independent process data are being monitored using a standard Shewhart con- trol chart. For an in- control process, the ARL is approximately 370. However, the Shewhart control chart with Western Electric rules (Western Electric [1956]) has an in- control ARL of approximately 91 (see Champ and Woodall [1987]). Thus, even if the process is in statistical control, the sensitizing rules may lead to more false alarms than the standard Shewhart control chart with no sensitiz- ing rules.
The important issue, of course, is not the exact wording of the rules so much as the action that takes place once the unusual event has occurred. The first step
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always should be to ascertain that the point is calculated and plotted correctly. If possible, a double check should be made on the measurement itself. For variables charts, the range section should be analyzed first. Increases in the range values represent increased variation between the readings within an individual sample. Possible causes include bearings, tooling, or fixtures. In the case of cutoff opera- tions, for instance, if the part is pushed against a backstop for measurement, the backstop could have become “rubbery.” Changes in the averages chart represent some sort of shift in the process. Frequent causes are tool wear, changes in raw materials, and changes in measurement systems or process parameters such as machine settings, voltages, pneumatic pressure, and so on. It is useful to construct a list of things to check when certain chart characteristics occur. Such a list can come from a discussion among experienced personnel as well as from data from a process log.
In some cases the events on the out- of-control lists represent improved situa- tions. For instance, the process is considered out of control if too many points are in the middle third of the control limit area. Recall that the control chart tests are used to help determine whether the current values come from the distribution that was used to calculate the control limits. If too many points are grouped around the center line, the points probably come from a different distribution. The process should be investigated to determine what changed and to see whether this change can be perpetuated. If a log is maintained for the process, it may be possible to find changes that correspond to the time that the improvement occurred. Experience is the best teacher when it comes to chart interpretation. Efforts should be made to document a body of knowledge about each process.
Finally, note that a control chart is really a graphical hypothesis test. The null hypothesis is that the process has not changed, and as each point is plotted, the chart is examined to determine whether there is sufficient evidence to reject the null hypothesis and conclude that the process has changed. The significance level varies somewhat with the chart test employed.
For an overview of research on control charting methods for process moni- toring and improvement, see Woodall and Montgomery (1999, 2014). Also, see Woodall (2017) for an in- depth review of Phase I SPC analysis, which includes discussion aimed at bridging the gap between theory and practice. Vining (2009) describes the differences between Phase I and Phase II analysis in SPC.
Additional references on SPC can be found in Capizzi (2015); Chakraborti, Human, and Graham (2008); Hawkins and Wu (2014); and Trip and Does (2010). For a detailed case study using SPC, see Godfrey, Russell, and Betz- Stablein (2016), where the authors present an application of control charts to monitor chronic kid- ney failure in patients.
F.7. Pre- Control Charts
A pre-control chart is sometimes used in place of a control chart or until sufficient data are collected to construct a control chart. An important difference between pre- control charts and control charts is that the upper and lower pre- control limits are calculated from the tolerance limits rather than from data from the process.
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Thus, the pre- control chart is not statistical in the sense that the distribution of the current process is not being compared with some historic distribution.
A fairly standard way to construct the pre- control limits is to multiply the value of the tolerance (upper specification limit – lower specification limit) by 0.25. Then subtract the resulting value from the upper specification limit to form the upper pre- control limit, and add it to the lower specification limit to form the lower pre- control limit.
As parts are measured, their values are compared with the pre- control limits and appropriate action is taken based on rules such as these:
1. If the first part is outside the specification limits, adjust the process
2. If a part is inside the specification limits but outside pre- control limits, measure the next part
3. If two successive parts are outside pre- control limits, adjust the process
4. If five successive parts are inside pre- control limits, consider switching to less frequent measuring
There has been much debate about the use of pre- control charts. Some of the advantages and disadvantages of pre- control charts follow.
Advantages
• Pre-control charts are easy to implement and interpret.
• Pre-control charts can be very useful in initial setup operations in determining whether product being produced is centered between the tolerances.
Disadvantages
• Pre-control does not provide information about how variability can be reduced if necessary or how the process can be brought back into control.
• Pre-control charts should only be used for processes whose process capability ratio (to be discussed in the next section) is greater than one. If the capability of the process is very poor, then pre- control charts will signal that the process should be stopped and assignable causes found. But low capability does not necessarily indicate that any assignable causes are actually present. That is, unnecessary tampering will most likely occur in this case.
• The small sample sizes used in pre- control will greatly reduce the chart’s ability to detect moderate to large shifts.
The material presented here can be found in greater detail in Ledolter and Burrill (1999) and in an article by Ledolter and Swersey (1997). Various authors provide additional rules. The principal advantage of pre-control is that it is simpler than standard control charts. The main disadvantage is that it is not statistically based.
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ExaMpLE 6.76
The specification for a dimension is 5.000 ± 0.010. The tolerance is 0.020, so 25% of the tolerance is 0.005. Therefore, the upper pro-
cess control (PC) limit would be placed at 5.005 and the lower PC limit at 4.995, as indi- cated in Figure 6.42.
5.010 Upper specification limit
5.005 Upper PC limit
5.000
4.995 Lower PC limit
4.990 Lower specification limit
A B C D E F G H
Figure 6.42 Example of a pre-control chart.
The actions to be taken at each of the lettered points in Figure 6.42 are:
A. Adjust process
B. Measure another part
C. Measure another part
D. Measure another part
E. Adjust process
F. Adjust process
G. Measure another part
H. Measure another part
When the pre-control rules indicate that the process should be adjusted, there is not necessarily a high probability that the process has changed. This may lead to overadjustment and decreased stability of the process. For this reason, there is some controversy over the use of pre-control, with Montgomery (2013) stat- ing, “This author believes that pre-control is a poor substitute for standard control charts and would never recommend it in practice.”
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F.8. Short- run Statistical Process Control
The control charts presented to this point apply to processes that are considered long, continuous production runs. These charts are not appropriate for short production runs. Short production runs are commonplace and include processes that produce built- to-order product or quick turnaround production. Short- run control charts should be considered when data are collected infrequently or aperiodically. They may be used with historical target or current target values, attribute or variable data, and individual or subgrouped averages. Standardized control charts are commonly used to monitor short production runs. A simple illustration for attribute data will be presented. For complete details on short production runs, see Montgomery (2013).
The short- run control charts for attribute data are actually standardized control charts. The attribute for the control chart of interest is standardized and plotted on a control chart. To illustrate, consider the standardized value using the number of nonconformities (i.e., c chart). The standardized value is
Z
c c
c i
i= −
(6.165)
which follows a standard normal distribution. The following properties of all stan- dardized control charts apply:
• Each data point is standardized
• The standardized random variable Zi has a standard normal distribution
• The center line for all standardized charts is zero
• The control limits for all of the standardized charts are –3 and 3
ExaMpLE 6.77
Nonconformities are counted on 10 printed circuit boards. The boards come from a short production run. The nonconformities are given in Table 6.48.
Table 6.48 Number of nonconformities for printed circuit boards.
Printed circuit board
Number of nonconformities
1 2 3 4 5 6 7 8 9
10
4 0 1 3 6 3 1 0 5 2
Total 25
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A short-run c chart is appropriate for this situation. To construct the short-run con- trol chart on the number of nonconformities, we first calculate the average number of nonconformities c–, then calculate the standardized values.
For this problem, c– = 25/10 = 2.5 (where there are a total of 25 nonconformities and 10 boards). The standardized values are then found (using Equation (6.165)) as
Z c
i i= − 2 5 2 5
. .
To illustrate, the standardized value for the first circuit board is
Z c
1 1 2 5
2 5 4 2 5
2 5 0 95=
− =
− =
. .
. .
.
The remaining standardized values are calculated similarly. The short-run c chart is shown in Figure 6.43. The process does not appear to be out of statistical control.
–2
–1
1
2
21 4 6 9873 5 10
z
Board
3 3
–3 –3
0 0
Figure 6.43 Short-run c chart for printed circuit boards.
There are a number of methods for constructing control charts for short produc- tion runs. The EWMA and CUSUM control charts can be very effective in this situ- ation. See Hawkins and Olwell (1998) for more details on the CUSUM approach for short production runs.
g. ProCESS and PErForManCE CaPaBiLiTy This section addresses four aspects of process and performance capability: process capability studies, process performance versus specifications, process capability indices, and process performance indices.
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g.1. Process Capability Studies
The purpose of a capability study is to determine whether a process is capable of meeting certain requirements. Capability of a process can be evaluated through determination of a probability distribution, its shape, center, and spread. Tools such as histograms, probability plots, and stem- and-leaf plots can be used to eval- uate process capability without having stated specification limits for the quality characteristic of interest.
Process capability is often investigated with respect to given specifications. In theory, a capability study should be performed for every product dimension and every quality characteristic. In practice, however, people familiar with a process usually are able to identify the few characteristics that merit a full capability study, i.e., those characteristics that experience has shown to be difficult to hold to speci- fication. For example, suppose a customer requires certain process outputs to be 45–55, such as:
• The arrival time for a delivery vehicle must be between 45 and 55 minutes after the hour
• Manufactured pumps must produce between 45 psi and 55 psi
• The plating thickness must be from 45 to 55 mm
In these instances, the 45–55 requirement is called the specification, which typi- cally is inclusive of the endpoints.
Bothe (1997) identifies six major activities as parts of a process capability study:
1. Verifying process stability
2. Estimating process parameters
3. Measuring process capability
4. Comparing actual capability to desired capability
5. Making a decision concerning process changes
6. Reporting the results of the study with recommendations
These six areas are not unique and may require several different methods to com- plete any one activity. For example, control charts and designed experiments can be implemented to estimate process capability. In addition, Montgomery (2013) recommends the use of histograms and probability plotting in addition to process capability ratios (presented in this section) as techniques useful in determining the capability of a process.
The first step in conducting a capability study is to verify that the process is stable. A stable process can be thought of as a process without special causes of variation present. Process stability can be determined by using a control chart. The process is considered to be stable if the chart shows that no special causes are present after an appropriate number of points have been plotted. A key phrase in the previous sentence is “appropriate number of points.” Although authorities
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disagree on the number of points needed, 20–30 points are commonly used. How- ever, the more points you plot, the higher the confidence you can have in the sta- bility conclusion.
The second step in conducting a capability study is to determine whether it is reasonable to assume that the process data come from a normal distribution. To do this, a normal probability plot or histogram could be constructed using the original readings (not the averages) from the control chart. If the histogram looks normal, with most points grouped around a single peak and fairly symmetric tails on each side, you may assume that the data constitute a sample drawn from an approximately normal population. Using a normal probability plot, we conclude that the normality assumption is satisfied if the data fall along a straight line. Again, the more data you use, the higher the confidence you can have in this conclusion. The normality assumption is absolutely necessary in order for the results of a pro- cess capability study (process capability ratios, discussed next) to be considered valid. If the data are non- normal, a transformation to induce normality may be necessary. Kotz and Lovelace (1998) also discuss process capability indices that can be used for non- normal distributions. For information about a hypothesis test to check if data are normally distributed, refer to Devore (2016) or Montgomery and Runger (2013).
If the data are normally distributed, the next step is to use the normal distribu- tion to estimate process capability. The most common method is to use the data from a control chart to estimate μ and σ. The sampling plan is then the same as that used for the control chart. Once the chart exhibits statistical control, the values of x and R calculated from the control chart are used in the capability analysis formulas. This process is similar when using x and s charts, and individuals and MR charts.
g.2. Process Performance versus Specifications
In this section we investigate the capability of a process in relation to specification limits. The capability of a process could be described by the fraction of units that fall outside the specification limits. To estimate the capability, we need to estimate the process standard deviation, σ, for an in- control process. Depending on the type of control chart used to monitor the process, σ is estimated as
σ̂ = R
d2 (6.166)
for an x– and R chart,
σ̂ = s
c4 (6.167)
for an x– and s chart, and
σ̂ = MR
d2 (6.168)
for an individuals and MR chart. To illustrate, consider Example 6.78.
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ExaMpLE 6.78
A dimension has specifications of 2.125 ± 0.005. Data from the process indicate that the distribution is normally distributed, and the x– and R control chart indicates that the pro- cess is stable. The control chart used a sample of size five and it is found that x–
– = 2.1261
and R – = 0.0055. We wish to determine what fraction of the manufactured product will
have this particular dimension outside the specification limits.
Solution: Let X represent the dimension of the quality characteristic of interest. What we are look- ing for is the fraction of the manufactured product that will have this particular dimen- sion outside the specification limits; this can be written as 1 – (2.120 < X < 2.130). Since X is normally distributed, we can use the standard normal distribution to determine this fraction. The best point estimate for μ is x–
– = 2.1261. The point estimate for process stan-
dard deviation σ is given by Equation (6.166) and computed as
ˆ . .
.σ = = =R d2
0 0055 2 326
0 00236
The constant d2 can be found in Appendix B for n = 5. The estimated fraction that does conform to specifications (using Equations (6.20)
and (6.23)) is
P X P X x
2 120 2 130 2 120 2 1261
0 00236 . .
. . . ˆ
< <( ) = − < − σ
<< −
= − < <( )
2 130 2 1261 0 00236
2 58 1 65
. . .
. .P Z
== <( ) − < −( ) = −
=
P Z P Z1 65 2 58
0 9505 0 0049
0 9456
. .
. .
.
c c
Therefore, the fraction that is nonconforming is 1 – 0.9456 = 0.0544. Approximately 5.44% of the products will fall outside specification for this quality characteristic.
G.2.a. Control Limits, Specification Limits, and Natural Tolerance Limits
It should be noted that there is a significant difference between control limits, specification limits, and natural tolerance limits. Control limits are determined by the natural tolerance of the process, while specification limits are determined externally—usually by management, engineers, customers, and so on. There is no relationship between specification limits and control limits. Suppose we are moni- toring the sample mean x– where the population of interest is normally distributed with mean μ and standard deviation σ. The different limits could be written as follows:
Natural limits: μ ± 3σ (from Equation (6.67))
Control limits: x n
± 3 σ (6.169)
Specification limits: [LSL, USL] (determined externally)
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G.2.b. Defective Parts per Million Opportunities
In section G.3 we discuss several metrics of process capability for variables data. When a process is measured as attribute data, however, alternate metrics must be used. One of the most frequent measures for process performance is parts per mil- lion (ppm) defective. The ppm measures the number of defective/nonconforming parts in one million parts produced.
Another metric used to measure process capability for attributes is defects per million opportunities (DPMO), which is estimated as
DPMO = (1,000,000)
Total number of defects
(Number of units)(Number of opportunities/unit) d d
(6.170)
An opportunity is defined as the number of potential chances within a unit for a defect to occur (Montgomery 2013). DPMO differs from ppm in that ppm mea- sures the number of nonconforming parts while DPMO measures the number of nonconformities. These metrics are the same when the number of opportuni- ties per unit is equal to one. DPMO is useful because it can be used to compare the quality of different products, provided that the number of opportunities for defects has been thoroughly determined (Kubiak 2009).
ExaMpLE 6.79
Consider the hospital billing data in Example 6.72, where 20 billing statements were ran- domly selected over a 24-day period. Suppose that each billing statement has 12 oppor- tunities for a mistake (a defect) to occur. Estimate DPMO for this scenario.
There were 245 total defects in this time period and (20)(24) = 480 billing statements examined. Using Equation (6.170),
DPMO (480)(12)
= (1,000,000) = 42,534.72245b b
For additional information on common process performance metrics, see Montgomery (2013) and Kubiak (2009).
g.3. Process Capability indices
Various capability indices have been developed to try to quantify process capability in a single number. The stability and normality requirements discussed earlier must be met for these measures to be effective. Four such indices are Cp, Cr, Cpk, and Cpm.
G.3.a. Cp and Cr
Cp compares the tolerance (the width of the engineering specifications) with the natural process tolerance. Cp is given by
C
USL LSL p =
− 6σ
(6.171)
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where LSL is the lower specification limit and USL is the upper specification limit. The true process standard deviation σ is usually unknown and must be estimated from the sample data. We can use the sample standard deviation s or Equation (6.166) if control charts are used in the analysis. The estimate of Cp is
ˆ
ˆ C
USL LSL p =
− 6σ
(6.172)
Consider Example 6.78, which involved a quality characteristic with specifica- tion limits set at 2.125 ± 0.005 and σ̂ = 0.00236. Our estimate of Cp, using Equa- tion (6.172), is
ˆ . .
( . ) .C
USL LSL p =
− =
− =
2 130 2 120 6 0 00236
0 706 ˆ6σ
Generally, it is desirable for Cp > 1. Based on this analysis, the process does not appear to be capable.
The Cr measure is simply the inverse of Cp, that is, Cr = 1/Cp. An estimate of Cr, which is also referred to as the PTR, is
Ĉ
USL LSLr =
− ˆ6σ
(6.173)
A simple interpretation of Cr is the percentage of the tolerance (or specification band) that is used up by the process. Consider our example again where it was found that Ĉ p = 0.706. Then Ĉr = 1/0.706 = 1.416. This value can be interpreted that the process uses 141.6% of the specification band, giving further evidence that this is not a capable process. This ratio is sometimes referred to as the capability ratio, and smaller values are better.
G.3.b. Cpk
The capability measure Cpk penalizes a process whose mean is off center. Cpk takes into account process centering and is given by
C
USL L SL pk =
− − min ,
µ σ
µ σ3 3
; ;
(6.174)
It is desirable to have a value of Cpk > 1, which indicates that the process exceeds the stated minimum requirement. An estimate of Cpk is
ˆ min ,C
USL LSL pk =
− −x x; ; ˆ3σˆ3σ
(6.175)
For our example, using Equation (6.175),
ˆ min ,
min . .
C USL LSL
pk = − −
= −
x x
2 130 2 11261 3 0 00236
2 1261 2 120 3 0 00236( . )
, . .
( . ) −
=
=
min . , .
.
0 551 0 862
0 551
; ;
; ;
66
ˆ3σˆ3σ
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Historically, a Cpk value of 1.0 or larger was considered capable. This would be equivalent to stating that the natural process limits lie inside the tolerance limits. More recently, quality requirements have become more stringent, and many cus- tomers require Cpk values of 1.33, 1.66, or 2.00.
G.3.c. Cpm
The previous measure, Cpk, was developed to take into account centering of the process. However, studies have shown that a large value of Cpk does not necessar- ily indicate that the location of the process mean is centered between the LSL and USL. The Cpm metric was developed to provide a better measure of centering and is given by
C pm =
USL LSL−
6 (µ – T)2 + σ 2 (6.176)
An estimate is then
ˆ ˆ
C pm = USL LSL−
6 (x – T)2 + σ 2
(6.177)
where T is the process target. Again, the estimates of μ and σ are obtained from control charts.
G.3.d. Interpreting Process Capability Ratios
The assumptions underlying Cp, Cr, Cpk, and Cpm are critical to accurately inter- pret their respective values and the capability of the process. As discussed at the beginning of this section, the process must be stable and the population normally distributed in order to correctly interpret process capability. Proper interpretation of Cp requires the process mean to be centered between the LSL and USL as well. Several studies have discussed the implications of violating these assumptions. In general, it has been shown that these indices are highly sensitive to their assump- tions. See, for example, Somerville and Montgomery (1996).
When the process is not centered, Cpk < Cp. Therefore, Cp is said to measure the potential capability of a process while Cpk measures the actual capability. A process is considered capable when the process capability ratios are greater than 1, and incapable when they are less than 1. A process with Cp = 1 and both an LSL and a USL results in 2700 ppm defective. A process with Cp = 1.50 and both an LSL and a USL results in 7 ppm defective (Montgomery 2013). Therefore, the larger the capability ratio, the better. Six sigma processes require process capabil- ity ratios equal to 2 (Perez-Wilson 1997). Table 6.49 displays recommended mini- mum values of process capability ratios for different processes.
The point estimators for the capability indices given in this section have some degree of error or variability associated with them. It has been recommended that confidence intervals on the process capability indices be constructed to quantify the precision associated with the point estimators. The reader is encouraged to see Kotz and Lovelace (1998) for complete details of the point estimators for and confidence intervals on these and other capability indices.
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g.4. Process Performance indices
Performance indices provide a picture of current process operation and have been used for comparison and prioritization of improvement efforts. Two such perfor- mance indices are Pp and Ppk. The performance indices have been recommended for use when the process is not in statistical control. The formulas for Pp and Ppk are equivalent to those for Cp and Cpk, respectively, except that the sample standard deviation (Equation (6.9)) is used instead of σ.
Some practitioners recommend the use of Pp and Ppk when the process is not in control. This is a somewhat controversial position because an out- of-control process is by definition unpredictable. Montgomery (2013) states, “The process performance indices Pp and Ppk are actually more than a step backwards. They are a waste of engineering and management effort—they tell you nothing.” Wheeler (2004) disagrees with Montgomery and uses Pp and Ppk to calculate what he refers to as the effective cost of production. The reader is encouraged to see Kotz and Lovelace (1998) for more discussion on performance and capability indices.
In general, the longer the time span over which the data are collected, the more valid the capability analysis. The analysis of data collected over a few hours can provide information about the process during those hours and may be useful for comparison purposes during process improvement efforts. Using control charts for process capability allows for the evaluation of both short- term and long- term process capability. For example, x– and R charts provide both instantaneous vari- ability and variability over time.
Once again, verifying the normality of a process is important. If the underlying distribution is not normal, the indices described in this chapter may not be valid. Various transformations and alternative indices have been proposed when the dis- tribution is non- normal. See Kotz and Lovelace (1998), Luceño (1996), Montgom- ery (2013), and Rodriquez (1992) for details on dealing with non- normality and process capability.
Table 6.49 Recommended minimum values of the process capability ratio.
Two-sided specifications
One-sided specifications
Existing processes 1.33 1.25
New processes 1.50 1.45
Safety, strength, or critical parameter, existing process 1.50 1.45
Safety, strength, or critical parameter, new process 1.67 1.60
Source: D. C. Montgomery, Introduction to Statistical Quality Control, 7th ed. (Hoboken, NJ: John Wiley & Sons, 2013).
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H. dESign and anaLySiS oF ExPEriMEnTS Experiments are an essential part of research and process and product develop- ment. It is important to correctly design and implement any experiment to obtain statistically valid results. All experiments can be considered “designed” experi- ments, but some of them may be designed poorly. Positive results can be achieved when a statistically designed experiment is developed and implemented correctly. Some of the results of a good experimental design include the following:
• Improvement in process yield
• Reduction in process variability (closer conformance to nominal or target requirements is often achieved)
• Reduction in design and development time
• Reduction in operation costs
The purpose of conducting a statistically designed experiment is to gain as much relevant information as possible with a minimum amount of cost (cost includes time, money, resources, and so on). Therefore, it is important to construct and carry out an efficient designed experiment. An efficiently designed experiment is one that includes the minimum number of runs and minimizes the amount of resources, personnel, and time utilized. Most statistically designed experiments are efficient and economical. Experiments that are not statistically designed are often expensive and inefficient and can often result in a waste of resources.
Before discussing the actual design and implementation of valid experiments, some important terminology must be introduced.
H.1. Terminology
This section provides definitions for several important terms. Figure 6.44 depicts the general process of a system.
In experimental design, the dependent variable or response, y (as in Figure 6.44), is the result or outcome of interest of the experiment, for example, yield of a pro- cess, time to complete a task, and taste score.
Figure 6.44 General system process. Source: D. C. Montgomery, Design and Analysis of Experiments, 9th ed. (New York: John Wiley & Sons, 2017).
ProcessInputs
x1 x2 xp
z1 z2 zq
y
. . .
. . .
Output
Uncontrollable factors
Controllable factors
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In experimental design, the independent variables, x’s (as in Figure 6.44), some- times referred to as treatments or factors, are chosen by the experimenter or prac- titioner to determine what effect, if any, they will have on the outcome of the experiment. Examples include the following:
• Type of gasoline (such as standard, plus, or super)
• Condensation temperature and its effect on yield
• Carbonation level in a test of a soft- drink taste
• Supplier of raw material in a manufacturing process
Factors can be quantitative (e.g., temperature, amount of fertilizer per acre) or quali- tative (e.g., technician, different additives, supplier, or type of keyboard).
There may be more than one factor under investigation in any one experi- ment. In addition, factors can take on one of a number of roles. For example, con- trol factors are process inputs to be controlled in actual production. These factors can be adjusted in practice to affect the output of a process. Noise factors, on the other hand, z’s (as in Figure 6.44), can be controlled during the experiment but are allowed to vary naturally in actual production. These factors are difficult to control in practice and can introduce variability into the response of interest. Understand- ing the effect of noise factors on the response can aid in reducing this variability in practice while not completely removing it. Examples of noise factors include humidity within a manufacturing plant, ambient temperature, and how a product is actually used in practice.
Levels in experimental design refers to the levels of the factors—for example, temperature levels of 200°C, 300°C, and 400°C; cooking times of one hour or two hours; two suppliers, A and B; and percent additive of 0.2%, 0.5%, and 0.8%.
A treatment in experimental design refers to a combination of the levels of each factor assigned to an experimental unit. This is sometimes called a treatment com- bination. To illustrate, consider an experiment on the breaking strength of a mate- rial. Two factors of interest are the machine (M1, M2, M3) on which the material is produced and the technician (T1, T2) using the machine. One treatment com- bination would be technician 2 using machine 1 (T2M1). “Treatment” is a term left over from the early days of experimental design and its roots in agricultural experimentation.
Factorial designs are those where all treatment combinations of the factors are carried out. Suppose an experiment involves three factors, A, B, and C, with three, five, and two levels investigated, respectively. A full- factorial design would con- sist of 3 × 5 × 2 = 30 treatment combinations.
Error in experimental design has several meanings. In any experimental sit- uation, error could represent errors in experimentation, errors of measurement, variation in materials or factors in general, or the effect of noise factors on the response, for example. Experimental error is the variability that is observed when a treatment combination is repeated, that is, replicated.
The objective of a designed experiment is to generate knowledge about a product or process. The experiment seeks to find the effect a set of independent variables has on a set of dependent variables. Mathematically this relationship can be denoted y = f(x) + ε, where x is an independent variable and y is the dependent variable (although there will most likely be more than one independent variable).
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For example, suppose a machine operator who can adjust the feed, speed, and coolant temperature wishes to find the settings that will produce the best surface finish. The feed, speed, and coolant temperature are the independent variables or fac- tors. Surface finish is the dependent variable or response and its value depends on the values of the independent variables. Independent variables may also be thought of as input variables, and dependent variables as output variables. There may be additional independent variables, such as the hardness of the material or humid- ity of the room, that have an effect on the dependent variable. These factors are considered noise factors since they may induce variability in the surface finish but cannot necessarily be controlled in actual production. In this example, the experi- mental design may specify that the speed will be set at 1300 rev/min for part of the experiment and at 1800 rev/min for the remainder. These values are referred to as the levels of the speed factor. The team decides to test each factor at two levels, as follows:
• Feed (F): 0.01 and 0.04 in/rev
• Speed (S): 1300 and 1800 rev/min
• Coolant temperature (C): 100°F and 140°F
A full- factorial design for the three factors will be used. A full- factorial experiment tests all possible combinations of levels and factors, using one run for each com- bination. The total number of combinations is given by LF, where F represents the number of factors of interest, each with L levels. In this situation, the number of treatment combinations is 23 = 8. The team develops a data collection sheet listing those eight experiments, with room for recording five replicates (n = 5) for each run (see Table 6.50).
As the data are collected, the values are recorded as shown in Table 6.51. These data are also referred to as the response values since they show how the process or product responds to various treatments.
Table 6.50 A 23 full-factorial data collection table.
Run F S C 1 2 3 4 5
1 0.01 1300 100
2 0.01 1300 140
3 0.01 1800 100
4 0.01 1800 140
5 0.04 1300 100
6 0.04 1300 140
7 0.04 1800 100
8 0.04 1800 140
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Note that the five values for a particular run are not all the same. This may be due to drift in the factor levels, variation in the measurement system, and/or the influence of noise factors. The variation observed in the readings for a particular run is referred to as experimental error. If the number of replications is decreased, the calculation of experimental error is less accurate, although the experiment has a lower total cost. If all the factors that impact the dependent variable are included in the experiment and all measurements are exact, replication is not needed and a very efficient experiment could be conducted. Thus, the accurate determination of experimental error and cost are competing design properties.
Once the data are collected as shown in Table 6.51, it may be useful to find the average of the five replication responses for each run. These averages are shown in Table 6.52.
Table 6.51 A 23 full-factorial data collection table with data.
Run F S C 1 2 3 4 5
1 0.01 1300 100 10.1 10.0 10.2 9.8 9.9
2 0.01 1300 140 3.0 4.0 3.0 5.0 5.0
3 0.01 1800 100 6.5 7.0 5.3 5.0 6.2
4 0.01 1800 140 1.0 3.0 3.0 1.0 2.0
5 0.04 1300 100 5.0 7.0 9.0 8.0 6.0
6 0.04 1300 140 4.0 7.0 5.0 6.0 8.0
7 0.04 1800 100 5.8 6.0 6.1 6.2 5.9
8 0.04 1800 140 3.1 2.9 3.0 2.9 3.1
Table 6.52 A 23 full-factorial data collection table with run averages.
Run F S C Average surface finish reading
1 0.01 1300 100 10
2 0.01 1300 140 4
3 0.01 1800 100 6
4 0.01 1800 140 2
5 0.04 1300 100 7
6 0.04 1300 140 6
7 0.04 1800 100 6
8 0.04 1800 140 3
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H.2. The Process of designing and analyzing an Experiment
There is a general process of designing and analyzing an experiment. Table 6.53 provides the seven steps associated with planning, designing, conducting, and analyzing an experiment. In this section we provide some details on the steps in this process.
H.2.a. Planning and Organizing a Designed Experiment
Planning and organizing a designed experiment is just as important as conducting the experiment and analyzing the results. Important steps in planning and orga- nizing experiments are the first three steps in Table 6.53.
The team making these types of choices and decisions should include engi- neers, technicians, management, customers, statisticians, and others who have firsthand knowledge of and experience with the process under study. It is impor- tant to ensure that the experiment is conducted as planned. Errors that occur as the experiment is carried out or errors in the measurements could deliver invalid results.
When preparing to conduct an experiment, we first ask, “What question are we seeking to answer?” In the example illustrated in the previous section, the objective was to find the combination of process settings that minimizes the sur- face finish reading.
The objective of a designed experiment is considered the goal of the experi- ment. Recognition of and statement of the problem is the first step in designing a successful experiment. Although stating the objective of the problem may seem obvious, it is not always given due consideration in the initial stages of planning the experiment.
The response (or responses) of interest is the outputs to be measured. The responses should represent all aspects of quality, productivity, and functional- ity. What factors might significantly affect the response? In most processes we could measure a very large number of variables of which only a few have any real impact on the response. Initially, many factors should be included and screening
Table 6.53 Guidelines for designing an experiment.
1. Recognition of and statement of the problem H Pre-experimental planning2. Selection of the response variable* 3. Choice of factors, levels, and ranges*
4. Choice of experimental design
5. Performing the experiment
6. Statistical analysis of the data
7. Conclusions and recommendations
Source: D. C. Montgomery, Design and Analysis of Experiments, 9th ed. (New York: John Wiley & Sons, 2017). *In practice, steps 2 and 3 are often done simultaneously or in reverse order.
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experiments carried out to eliminate those factors that do not significantly affect the response. One task in designing an experiment is to maximize the chance of including the significant variables in the design and leaving out those that have little impact.
The levels of the factors should also be given serious consideration. The span or scope of the experimental conditions will have an impact on one’s ability to determine the significance of a factor. For example, should the range of tempera- ture be from 100°C to 200°C or from 125°C to 175°C for a particular problem? If the range is too narrow, important effects could be completely missed.
Once the objective of the experiment has been determined and factors and levels selected, an appropriate measurement system is chosen. The measurement method is determined by the response that has been decided on. For example, if the outcome measured is placed into one of several possible categories (categorical data), the response that will be modeled or used in the analysis would be quite dif- ferent than if the measured outcome is continuous. The measurement system must be appropriate for the type of response of interest and can only be determined by people familiar with the process and output. Regardless of the type of response, methods exist that can adequately address these issues. This is discussed in sec- tion H.3 of this chapter.
H.2.b. Choice of Design
Once the objective of the experiment has been decided on, the factors, levels, and responses determined, and the method of measurement chosen, the next step is to choose the type of design to be used. This is step 4 in Table 6.53. The choice of design will depend on the previous steps (stating the objective, choosing factors, levels, and responses, and determining the measurement method). Other impor- tant considerations include the size of the design that is acceptable, the number of replicates, the run order of the design, and whether blocking is involved. Many standard statistical packages aid the practitioner in determining an appropriate design. In choosing the appropriate design, the objective of the experiment should always be kept in mind. Therefore, rather than designing a massive experiment involving many variables and levels, it is usually best to begin with more mod- est screening designs whose purpose is to determine the variables and levels that need further study.
H.2.c. Analysis of Results
Designed experiments, when conducted properly, can lead to very reliable results that provide insight into the important factors and optimal level settings. Properly designed experiments and the results of appropriate analysis easily lend them- selves to sequential experimentation for more detailed understanding and model- ing of the process. The analysis of the results involves some very straightforward but important steps:
• Exploratory and graphical analysis. Simple plots and tables of the data can provide insight into the process.
• Model fitting. Mathematical models of the form y = f(x) + ε are built and provide a relationship between the response and the independent variables.
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• Fine-tuning the model. Not all independent variables will be significantly related to the response. Several analysis steps can be taken to remove terms from the fitted model that have no significant effect on the response.
• Model diagnostics. Assumptions should be verified. The use of plots (such as residual plots) is useful in this step.
• Refining the model. This step is necessary if any of the assumptions are violated. Model refitting may be necessary, or a new form of the model investigated.
In the next several sections, various basic designs and analysis techniques are pre- sented. Complete details on these and other aspects of experimental designs can be found in Montgomery (2017).
H.3. design Principles
Once the experiment is planned and carried out, and the outcomes recorded, appropriate analysis is necessary to make final decisions on factors, factor settings, and prediction. Some analysis techniques are described at the end of this section.
H.3.a. Randomization
Randomization in experimental design is the ordering of the treatment combina- tions in a sequence that will reduce the effect of uncontrolled variables that might affect the dependent variable. Randomization will reduce the effect of unwanted nuisance factors that are not part of the experiment but may influence the results.
Returning to the surface finish example given earlier, there are eight treat- ments with five replications per treatment. This produces 40 tests or treatments. The tests from this design should be performed in random order. This is referred to as a completely randomized design. For the surface finish example, suppose the machine used in the process has some temperature effect; that is, machine tem- perature increases the longer the machine is running and can possibly affect the surface finish. Furthermore, suppose the treatment combinations are carried out in the order they appear in Table 6.50. If machine temperature does have an effect on surface finish, and the factor “feed rate” is found to be statistically significant, we cannot be sure whether the significant effect is really due to the change in feed rate or due to the temperature of the machine. These two factors could very well be confounded. Confounding in experimental design is the term used to signify that the effect of one independent variable is indistinguishable from the effect of another independent variable or combination of independent variables (interactions). The 40 tests in the surface finish example may be randomized in two possible ways:
1. Number the tests from 1 to 40 and randomize those numbers to obtain the order in which tests are performed. This is referred to as a completely randomized design.
2. Randomize the run order, but once a run is set up, make all five replicates for that run.
Although it usually requires more time and effort, the first method is better. To see that this is true, suppose time of day is a noise factor such that products made
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before noon are different from those made after noon. By randomizing, the time effect of when the product is made is minimized. In this way, if significant effects of the factors are identified, we are more confident that the effect is due to the changes made in the controllable factors rather than outside, often uncontrollable, factors.
H.3.b. Replication
In experimental design, replication is the repetition of the basic experiment. This involves a complete reset of the factor levels and repeating the experiment. Repli- cation provides an estimate of experimental error and leads to more precise esti- mates of the factor effects. It should be noted that multiple measurements of a treatment combination do not necessarily constitute replication. There is a signifi- cant difference between true replication and repeated measures.
H.3.c. Blocking/Local Control of Error
There are many instances when a factor may affect the response of interest but it is not a factor in which we are interested. These factors are often referred to as nui- sance factors. For example, suppose the 40 tests in the surface finish example cannot be conducted during one shift, but must be carried out over two shifts. In addition, it is believed that the shift may have an effect on surface finish. The team would be concerned about the impact the shift difference could have on the results.
Randomization can often reduce the effects of a nuisance factor when there is no way of controlling this factor in practice. If the nuisance factor is known and can be controlled for purposes of experimentation, then the factor can be taken into account during testing. A technique called blocking can be used to reduce variability transmitted by a nuisance factor. By removing the influence of this factor, the statis- tical analysis is more likely to reveal whether the factor of interest is truly significant or not. The simplest form of blocking is pairing, used to compare two dependent samples (see section D.3.c of this chapter for discussion of paired comparisons).
Blocking is one form of R. A. Fisher’s concept of local control of error. In gen- eral, local control refers to grouping experimental units in such a way that units within the group are homogeneous. This type of control aids in eliminating the variability or noise due to inactive or extraneous factors. Local control also includes the use of covariates when blocking is not possible in an experiment.
H.4. one- Factor Experiments
One-factor experiments were first introduced in section D.5 of this chapter, as part of the discussion on ANOVA. In that presentation, the analysis approach for a sin- gle factor with several levels (we also referred to this as comparing several treat- ments) was outlined. The same ANOVA approach is used when analyzing data from designed experiments.
We now introduce a slightly different form of the model that was given previ- ously. The reader will often see both of these models in the literature, so both are presented in this handbook.
H.4.a. One- Factor Experiments without Blocking
As discussed in section D.5 of this chapter, if we are interested in comparing more than two levels for a single factor, we must use an appropriately designed
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experiment and analyze it with ANOVA techniques. It is assumed that the experi- ment has been carried out as a completely randomized design.
We showed the general form of a model for a response of interest y involv- ing a levels (or treatments), each with n replicates, in section D (Equation (6.98)), which is called the means model. An alternative way to write a model for the response uses an effects model:
yij = μ + τi + εij for i = 1, 2, . . . , a and j = 1, 2, . . . , n (6.178)
where
yij is the jth observation from the ith factor level
μ is the overall mean (Note: μi = μ + τi)
τi is the parameter representing the effect of the ith factor level
εij is the error associated with the jth observation from the ith factor level
In a one- factor design and corresponding experiment, we are trying to determine whether there is a significant difference among the a factor levels. Since τi repre- sents the ith factor level and we assume that factor levels do not affect the response differently from one another, our null hypothesis to be tested is
H0: τ1 = τ2 = . . . = τa = 0
against the alternative
Ha: τi ≠ 0 for at least one i
The ANOVA approach presented in section D.5 of this chapter applies to the one- factor experimental design.
H.4.b. One- Factor Experiments with Blocking
Often in one- factor experiments there is a nuisance factor that may have some influence on the results. This factor is considered a blocking factor and should be included when carrying out the design and analyzing the results. We are not inter- ested in determining whether the levels of the blocking factor are significantly dif- ferent, but the factor should be included in the experimental design nonetheless. In general, for randomized block designs:
• The blocking factor is not modeled as being involved in an interaction with the treatments. Including the blocking factor reduces its effect on the response. This will allow for estimation of factor (independent variable)–level effects.
• We are interested in determining whether the factor levels are statistically significantly different, but we are not interested in determining whether the levels of the blocking factor are statistically significantly different. There is already some reason to believe that the blocking factor would influence the results.
• If we do not include the blocking variable in a designed experiment and it should be included, we could reach incorrect conclusions about the factor we are investigating.
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Suppose we have a levels of the factor of interest (or treatments), b levels of the blocking factor, and a response denoted y. A general model for the response of interest is
yijk = μ + τi + βj + εijk for i = 1, 2, . . . , a, j = 1, 2, . . . , b, and k = 1, 2, . . . , n (6.179)
where
yijk = the response for the kth observation at the ith factor level and the jth level of the blocking factor
μ = overall mean
τi = the parameter representing the effect of the ith level of the factor of interest
βj = the parameter representing the effect of the jth level of the blocking factor
εijk = the error associated with the ith level of the factor of interest and the jth level of the blocking factor
In a randomized block design and corresponding experiment, we are trying to determine whether there is a significant difference among the a levels of the factor of interest. The null hypothesis of interest is
H0: τ1 = τ2 = . . . = τa = 0
against the alternative
Ha: τi ≠ 0 for at least one i
A general display of the data is given in Table 6.54, where
• The totals are the sum across that particular row or column
• y.. is the sum of all the observations in the entire experiment
• The total number of observations is given by N = abn
Note that in this illustration there is exactly one observation per cell (i.e., a single replicate, n = 1, so N = ab).
The following formulas are necessary to carry out an analysis for any random- ized block design with one treatment of interest and one blocking factor. The same notation as presented earlier will be used here to maintain consistency. In random- ized block designs, the total sum of squares can be partitioned as follows:
SST = SSFactor + SSBlock + SSE (6.180)
Table 6.54 Randomized block general data table.
Blocks Treatment
totalsTreatment 1 2 . . . b
1 2 . . . a
y11 y12 . . .
ya1
y12 y22 . . .
ya2
. . .
. . . . . .
. . .
y1b y2b . . .
yab
y1. y2.
ya. ya.
Block totals y.1 y.2 . . . y.b y..
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The total sum of squares SST is given by
SST ij
j
b
i
a
ij j
b
i
y y y y N
= −( ) = − == == ∑∑ ∑.. ..
2
11
2 2
11
aa
∑
(6.181)
where yij is a single observation. Note that this is equivalent to yijk with k = 1. The total degrees of freedom are N – 1.
The sum of squares due to different factor levels SSFactor (referred to as SSTreatments in section D.5 of this chapter) is a portion of SST that represents the variability explained by or due to the factor levels themselves:
SSFactor = −( ) = −
= = ∑ ∑b y y
b y
y Nii
a
i i
a
. .. . ..2
1
2
1
21
(6.182)
The degrees of freedom for the factor of interest are a – 1. The sum of squares due to different levels of the blocking factor SSBlock is the portion
of the SST that represents the variability explained by or due to the different block levels themselves:
SSBlock = −( ) = −
= = ∑ ∑a y y
a y
y Njj
b
j j
b
. .. . ..
2
1
2
1
21
(6.183)
The degrees of freedom for the blocking factor are b – 1. The error sum of squares SSE is that portion of the SST that represents the inher-
ent variability and can be found by subtraction (using Equation (6.180)):
SSE = SST – SSFactor – SSBlock
The degrees of freedom for error are (a – 1)(b – 1). As before, the sums of squares will be converted into mean square quantities
and the appropriate test statistics calculated. An ANOVA table can be constructed to summarize the test. Note that Equations (6.181), (6.182), and (6.183) assume n = 1 as shown in Table 6.54.
The ANOVA table for a randomized block design is shown as Table 6.55. As with the completely randomized design, reject the null hypothesis if F0 > Fα,a–1,(a–1)(b–1),
Table 6.55 ANOVA table for a randomized block design.
Source of variation SS df MS F p-value
Factor SSFactor a – 1 MS SS
Factor Factor= −a 1
F E
0 = MS
MS Factor
P(F > F0)
Block SSBlock b – 1 MS SS
Block Block= −b 1
Error SSE (a – 1)(b – 1) MS SS
E = − − E
a b( )( )1 1
Total SST N – 1
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where α is the level of significance, the numerator degrees of freedom are a – 1, and the denominator degrees of freedom are (a – 1)(b – 1). In addition, p-values can be used to make a statistical decision. If the p-value is less than α, reject the null hypothesis.
ExaMpLE 6.80
Four washing solutions are to be compared to study their effectiveness in retarding bac- teria growth on a particular type of produce. The analysis is conducted in a lab and the experiment is carried out over a three-day period. The results are recorded and given in Table 6.56.
Table 6.56 Bacterial growth data.
Day
Solution 1 2 3
1 21 11 12
2 22 21 13
3 31 17 21
4 15 12 8
In this experiment:
• The treatment or factor is the washing solution. The goal is to determine if there is a statistically significant difference between the four types of solution.
• The blocking factor is “day” because it is believed that the day on which the measurements are taken is a source of variability. It is not a goal to determine if there is a statistically significant difference between the days. We believe that the day does make a difference and that is the reason we used it as a block.
• We are not interested in interactions between the washing solution and the day.
There are three levels of the blocking factor “day,” so b = 3. There are four levels of wash- ing solution, thus a = 4. An analysis of variance will be conducted to determine whether there is any statistically significant difference between washing solutions. We will test using the level of significance α = 0.05. The hypotheses of interest are:
H0: τ1 = τ2 = τ3 = τ4 = 0
Ha: τi ≠ 0 for at least one i = 1, 2, 3, 4
The appropriate sums of squares and degrees of freedom can be found using the for- mulas given previously. The resulting ANOVA table is given in Table 6.57. The analysis of variance was carried out using a reliable statistical software package.
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The p-value is less than our stated level of significance, so the null hypothesis is rejected and we conclude that there is a statistically significant difference between the four washing solutions. The type of washing solution appears to have a significant effect on retarding bacteria growth. Multiple comparison techniques can be carried out to determine which washing solution is better at retarding (minimizing) bacteria growth.
Table 6.57 ANOVA table for the washing solution example.
Source of variation df SS MS F p-value
Washing solution
4 – 1 = 3 218.0 218 0 3
72 67 .
.= 72 67 11 08
6 56 . .
.= 0.025
Day 3 – 1 = 2 171.5 171 5 2
85 75 .
.=
Error (4 – 1)(3 – 1) = 6 66.5 66 5 6
11 08 .
.=
Total 12 – 1 = 11 456.0
Blocking can be very important in a designed experiment. If there is an indi- cation that an underlying (nuisance) factor exists that will influence the response of interest, then it should be included in the experiment as a blocking factor. If a nuisance factor is influencing the response and it is not included in the designed experiment as a blocking factor, then the final conclusion (reject or not reject the null hypothesis) could be incorrect. Consider the washing solution example again, but without “day” as the blocking factor.
ExaMpLE 6.81
Suppose “day” was not included in the experiment as the blocking factor in the previous example. The experiment would contain only the factor of interest (washing solution) and becomes a one-factor experimental design. The data table would look like Table 6.58:
Table 6.58 Bacterial growth data without blocking.
Solution Growth
1 21 11 12
2 22 21 13
3 31 17 21
4 15 12 8
Continued
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where
• There is one factor, the washing solutions with four levels, that is, a = 4
• There are three observations for each type of solution, that is, n = 3
The hypotheses of interest are the same as if the blocking factor had been included:
H0: τ1 = τ2 = τ3 = τ4 = 0
Ha: τi ≠ 0 for at least one i = 1, 2, 3, 4
Using the same level of significance, α = 0.05, we can calculate the sum of squares and construct the ANOVA table for a one-factor experimental design (completely randomized design). Here, a = 4, n = 3, and N = an – 1 = 12 – 1 = 11. The ANOVA table is given in Table 6.59.
Since the p-value is quite large (0.139 > 0.05), we fail to reject the null hypothesis. Thus we do not have enough evidence to conclude that there is a significant difference between the four washing solutions.
Table 6.59 ANOVA table for the washing solution example without blocking.
Source of variation df SS MS F p-value
Washing solution
4 – 1 = 3 218.0 218 0 3
72 67 .
.= 72 67 29 8
2 44 . .
.= 0.139
Error 4(3 – 1) = 8 238.0 238 0 8
29 8 .
.=
Total 12 – 1 = 11 456.0
The two previous examples demonstrate the importance of considering a pos- sible blocking factor. If there is some underlying factor influencing the response and this factor is not taken into consideration, then the results of the experimental design could be incorrect. If there is any doubt about whether a nuisance factor is influencing the results, then it would be in the experimenter’s best interest to include the factor in the experiment and carry out a randomized block design. It is important to note that once the experiment has been carried out with blocking included, we cannot reanalyze the experiment as if it were not blocked.
Finally, a residual analysis should be conducted in order to test the assump- tion that the observations are normally and independently distributed with equal variance across factor levels. If there is a serious violation of one or more of the assumptions, a transformation may be necessary. Other methods for dealing with violated assumptions can be found in Devore (2016), Montgomery (2017), Mont- gomery and Runger (2013), or Vining and Kowalski (2011).
H.5. Factorial Experiments
In full- factorial experiments all possible combinations of the levels of factors are investigated. The two- factor factorial was introduced in section D.5.b of this chap- ter when discussing the two- way ANOVA. Consider an experiment that involves exactly two factors of interest, A and B, where there are a levels of factor A, b levels
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of factor B, and n replicates at each combination of A and B. The general model that would describe the response of interest y is given as
yijk = μ + τi + βj + (τβ)ij + εijk for i = 1, 2, . . . , a; j = 1, 2, . . . , b; and k = 1, 2, . . . , n
(6.184)
where
yijk = the kth response at the combination of the ith level of A and the jth level of B
μ = the overall mean effect
τi = the parameter for the effect of the ith level of A
βj = the parameter for the effect of the jth level of B
(τβ)ij = the parameter for the effect of the ijth level of the interaction between A and B
εijk = the error
We are interested in the following hypotheses:
H0: τ1 = τ2 = . . . = τa = 0 (the effect for each level of A is zero)
Ha: τi ≠ 0, for at least one i
H0: β1 = β2 = . . . = βb = 0 (the effect for each level of B is zero)
H1: βj ≠ 0, for at least one j
H0: (τβ)11 = (τβ)12 = . . . = (τβ)ab = 0 (no significant interaction between A and B)
Ha: (τβ)ij ≠ 0 for at least one i and one j
The sums of squares, degrees of freedom, mean squares, test statistics, and p-values can be calculated using a reliable statistical software package. The result- ing ANOVA table would look like Table 6.60.
Table 6.60 ANOVA table for two-factor factorial experiment.
Source of variation df SS MS F p-value
Factor A a – 1 SSA MS SS
A A
a =
− 1 F A
E 0 =
MS MS
P(F > F0)
Factor B b – 1 SSB MS SS
B B
b =
− 1 F B
E 0 =
MS MS
P(F > F0)
AB interaction (a – 1)(b – 1) SSAB MS SS
AB AB
a b =
− −( )( )1 1 F AB
E 0 =
MS MS
P(F > F0)
Error ab(n – 1) SSE MS SS
E E
ab n =
−( )1
Total abn – 1 SST
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It should be noted that the ANOVA approach is not the only method for test- ing the significance of the effects and interactions. If each factor has exactly two levels, it is common to examine the results of t-tests on the coefficients represent- ing each factor and interaction. To build a model as recommended, t-tests can be very useful.
H.5.a. Randomized Block Designs
Before presenting the details of factorial designs at two levels, we will make a dis- tinction between a randomized block design and a two- factor full- factorial design. For example, in an experiment involving two factors A and B, we would be inter- ested in the following:
• Differences between the levels of factor A
• Differences between the levels of factor B
• Whether a significant interaction between A and B exists
When is a design that involves two factors a randomized block design and when is it a factorial design? The distinction between the two types of designs comes from the information that is to be gained from conducting the experiment. For instance, consider an experiment that involves two factors. The type of design to use can be determined if the following remarks are kept in mind.
Use a randomized block design if:
• There is no interest in whether significant differences exist between levels of one of the factors (this is the blocking factor). There is some belief that this factor influences the result, and by blocking we minimize its influence.
• There is little likelihood of an interaction between the two factors involved in the study.
Use a factorial design if:
• There is an interest in determining the differences between the levels of both factors
• There is an interest in determining whether a significant interaction between the two factors exists
Of course, there are other issues to consider before conducting the experiment. In addition, it is quite possible to have an experiment that involves more than one factor of interest and one or more blocking factors. Designs sometimes used for experiments involving one independent variable and two blocking factors are referred to as Latin square designs.
H.5.b. Two- Level Factorial Designs
A special type of factorial design that receives a great deal of attention is a design where all factors are run at exactly two levels. If there are k factors, we say the design is a 2k factorial design. The number of experimental runs (or observations) is 2k. For example, consider the surface finish illustration given earlier. There are
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three factors of interest: feed rate, speed, and coolant temperature, each at two levels. A full- factorial design consists of 23 = 8 runs or treatment combinations.
The two levels of each factor can be coded as –1 and 1, the low and high levels of each factor, respectively. Consider an experiment with two factors A and B, each at two levels, shown in Table 6.61.
There would be a total of four combinations, and we would like to determine whether A, B, or AB is significant. The combinations can be written in Table 6.62. To obtain the column for the levels of the AB interaction, multiply column A and column B.
Suppose there are two replicates for each run and the average response is calculated. Geometrically, we can display the data on a square for the factors as shown in Figure 6.45.
Table 6.61 Coded factor levels.
A B
Low –1 –1
High 1 1
Table 6.62 Combinations for terms in a two-factor interaction model.
Run A B AB Responses
1 –1 –1 1 y11, y12, . . . y1n
2 –1 1 –1 y21, y22, . . . y2n
3 1 –1 –1 y31, y32, . . . y3n
4 1 1 1 y41, y42, . . . y4n
Figure 6.45 All possible combinations of two factors A and B, with two levels each.
1
B
–1
–1 –1A
24.5
25.5
20.0
27.5
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Each corner of the square represents a run or treatment combination. The val- ues at the corners represent the average response of each combination. For exam- ple, at the low level of A and the high level of B the average response is 24.5.
Standard analysis techniques can be applied to the special case of factors with only two levels. We will examine the estimation of the main factors and interac- tion using an example. The effect estimate is calculated as the average response at the high (+1) level minus the average response at the low (–1) level. The main effect provides a measure of how each individual factor (main factors such as A, B, and C) affects the response as we move from one level of the factor to the next. The estimated effect for each factor is simply the difference in the average response at the high level of the factor and the average response at the low level of the factor. The term “estimated effect” is denoted by ee. Main effects plots can be useful for examining the change in the factor effects from the low to high levels. See Montgomery (2017) for further details.
ExaMpLE 6.82
In an article by Lee and Awbi (2004), the authors discuss the effect internal partitioning of office space has on room air ventilation. In the design of modern office buildings, it is important to consider the air quality in a room. For office buildings it is desirable to construct a highly energy-efficient building, often with an open-space floor plan. With open-space construction, internal partitions are introduced to design the office to fit the current needs of the company. With internal partitioning, the layout can easily be restructured for different occupants. However, the air ventilation system is designed for open-space rooms. When interferences are introduced (such as office furniture, wall partitions, and so on) the air quality can be significantly affected. In the study on the effect of internal partitioning on room air quality, three factors are of interest: partition location (A), partition height (B), and gap underneath (C). The partition locations are chosen at 40% and 60% of the room length from the left end of the room. The partition heights are chosen as 60% and 80% of the room height. The factor “gap underneath” represents the space between the floor and the bottom of the partition. Gap is set at 0% of the room height and 10% of the room height.
One response of interest is ventilation effectiveness yv, a scaleless quantity that is a function of contamination concentration. Larger values of yv indicate better ventilation effectiveness. The tests are conducted on a small scale model test room with the length, width, and height of the room measured in meters. The factors and their levels are given in Table 6.63.
Table 6.63 Factor levels for ventilation experiment.
Factor Low level (–1) High level (+1)
Partition length (A) 40% 60%
Partition height (B) 60% 80%
Gap underneath (C) 0% 10%
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Suppose a similar experiment was conducted using these factors to test their effect on ventilation effectiveness. The design used was a 23 factorial in two replicates, with results given in Table 6.64 (factors are coded). A complete randomization of the treat- ments for all 16 runs was carried out.
Table 6.64 Partitioning effect on ventilation effectiveness.
Treatment A B C yv
1 –1 –1 –1 2.227, 1.874
2 1 –1 –1 2.134, 2.252
3 –1 1 –1 1.470, 1.404
4 1 1 –1 2.091, 2.270
5 –1 –1 1 2.073, 1.825
6 1 –1 1 2.162, 2.480
7 –1 1 1 1.615, 1.558
8 1 1 1 2.157, 2.169
Graphically, we can display the results using the average response for each treat- ment. Figure 6.46 displays a cube plot for the three factors: partition length (A), partition height (B), and gap underneath (C).
1
B
–1 –1
C
1
–1 1A
2.0505
2.1805
1.5865
1.9490
2.1630
2.3210
1.4370
2.1930
Figure 6.46 Cube plot for partition length, partition height, and gap underneath.
Continued
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Consider the estimated effect of gap underneath. The average response at the high level (+1) of C (gap underneath) is
C+ = + + + + + +
1
2 073 1 825 2 162 2 480 1 615 1 558 2 1. . . . . . . 557 2 169 8
2 005 +
= .
.
The average at the low level (–1) of C is
C− = + + + + + +
1
2 227 1 874 2 134 2 252 1 470 1 404 2 0. . . . . . . 991 2 270 8
1 965 +
= .
.
The estimated effect of C (gap underneath) on ventilation effectiveness is then
ee( ) . . .C C C= − = − =+ −1 1 2 005 1 965 0 04
The estimated effect for C shows that as the gap underneath the partition is changed from 0% to 10%, the average ventilation effectiveness increases by 0.04. The estimated effects of partition length and partition height are calculated similarly and found to be 0.46 and – 0.29, respectively. Main effects plots for partition length, partition height, and gap underneath are displayed in Figure 6.47.
It appears that there is a significant difference between the levels of factor A and the levels of factor B, but not necessarily between the levels of factor C.
1.8
2.2
2.1
2.0
1.9
–1 1 –1 1
A B
1.8
2.2
2.1
2.0
1.9
–1 1
C
M e a n
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Figure 6.47 Main effects plot for the air quality example.
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Using this definition of main effect, the larger the absolute value of the main effect, the more influence that factor has on the quality characteristic. It is possible that the perceived difference between high and low results is not statistically sig- nificant. This would occur if the experimental error were so large that it would be impossible to determine whether the difference between the high and low levels is due to a real difference in the dependent variable or due to experimental error. This may be determined by using ANOVA procedures and/or t-tests.
For analysis of data from an experiment, the null hypothesis is that changing the factor level does not make a difference in the dependent variable. The α risk is the probability that the analysis will show that there is a significant difference when there is not. The β risk is the probability that the analysis will show that there is no significant difference when there is. The power of the experiment is defined as 1 – β, so the higher the power of the experiment, the lower the β risk. In general, a higher number of replications or a larger sample size provides a more precise estimate of experimental error, which in turn reduces the β risk.
An interaction in experimental design describes the change in the response when two or more factors are interdependent. Interactions are discussed in some detail in section D and also further in this section with respect to factorial designs.
Interactions may exist between the factors of interest, and this interaction effect must be determined as was done with the main effects. The interactions in our example include the two- factor interactions and the three- factor interaction: partition length by partition height (AB), partition length by gap underneath (AC), partition height by gap underneath (BC), and partition length by partition height by gap underneath (ABC).
Again, we want to find the average difference in the response between the high level and the low level of each interaction. What is considered a “high” and “low” level for an interaction? The levels of the interactions are simply the results of the levels of the main effects. In coded form we can label the high and low levels of each interaction simply by multiplying the levels of each factor involved in the interaction. For example, if A is set at its low level (A = –1) and B is set at its high level (B = 1), then the corresponding level of the interaction AB would be –1 (since –1 × 1 = –1). This is simply a label that is convenient for determining the low and high levels of each interaction and is a result of the geometry of the design. The table for the main effects and interactions for the 23 full- factorial design is given in Table 6.65. Notice that any column multiplied by itself results in a column of +1’s only. When a column consists of 1’s only, it is called the identity column and denoted I. For example, A × A = I.
The estimated effects of the interactions can be easily calculated. See Table 6.65 for the rows associated with AB at the high and low levels. For example, the aver- age response at the high level of the interaction AB (partition length and partition height) is
AB+ = + + + + + +
1
2 227 1 874 2 091 2 270 2 073 1 825 2. . . . . . .1157 2 169 8
2.08575 +
= .
The average response at the low level of AB is
AB− = + + + + + +
1
2 134 2 252 1 470 1 404 2 162 2 480 1. . . . . . .6615 1 558 8
1.884375 +
= .
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The estimated effect of the AB interaction is then
ee(AB) = AB+1 – AB–1 = 2.08575 – 1.884375 = 0.2014
The remaining interaction effects can be estimated similarly. The estimated effects for all of the main effects and interactions are given in Table 6.66.
Interaction plots, discussed in section D of this chapter, are often useful for examining the two- factor interactions. The interaction plots for our example are given in Figures 6.48 through 6.50.
Based on the interaction plot in Figure 6.48, it is possible that a significant interaction exists between partition length (A) and partition height (B) because the lines are not parallel. Figure 6.49 indicates that there does not appear to be an interaction between partition length (A) and gap underneath (C). There appears to be a weak interaction between partition height (B) and gap underneath (C), which is indicated by the slightly nonparallel lines in Figure 6.50. However, the plots are somewhat subjective, and more statistically based evidence is needed.
We can determine whether the effects are statistically significant using the ANOVA approach, a model- fitting approach, or both. We will first look at the ANOVA approach.
Table 6.65 Main effect and interaction table for the ventilation factorial design.
Treatment A B C AB AC BC ABC yv
1 –1 –1 –1 1 1 1 –1 2.227, 1.874
2 1 –1 –1 –1 –1 1 1 2.134, 2.252
3 –1 1 –1 –1 1 –1 1 1.470, 1.404
4 1 1 –1 1 –1 –1 –1 2.091, 2.270
5 –1 –1 1 1 –1 –1 1 2.073, 1.825
6 1 –1 1 –1 1 –1 –1 2.162, 2.480
7 –1 1 1 –1 –1 1 –1 1.615, 1.558
8 1 1 1 1 1 1 1 2.157, 2.169
Table 6.66 Estimated effects for the air quality example.
Factor Estimated effect (ee) Factor Estimated effect (ee)
A 0.459 AC 0.016
B –0.287 BC 0.026
C 0.040 ABC –0.099
AB 0.201
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Figure 6.48 Interaction plot for partition length and partition height.
2.3
2.2
2.1
2.0
1.9
1.8
1.7
1.5
1.6
–1 B
1
M e a n
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1
–1
A
Figure 6.49 Interaction plot for partition length and gap underneath.
2.3
2.2
2.1
2.0
1.9
1.8
1.7 –1
C 1
M e a n
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1
–1
A
In the ANOVA approach, sums of squares are calculated for each effect and the experimental error. The degree of freedom for factors with exactly two levels (–1, +1) is 1 (the number of levels minus 1 as with all factorial designs). The sums of squares for each factor can be found easily using the estimated effects and can be shown to be
SSFactor = n2 (k–2)ee2 (6.185)
where n is the number of replicates and k is the number of factors. Test statistics can be calculated for each factor and each interaction. All of this information can
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be summarized in an ANOVA table. The ANOVA table for our example is given in Table 6.67, where degrees of freedom for error are 2k(n – 1) and total degrees of freedom are n2k – 1. Based on the results of the ANOVA, it appears that factors A and B and the interaction AB are significant. This is evident by the small p-values for each of these terms. Based on these results, the analysis should be carried out again but with only the significant terms included. The results for the new analysis are given in Table 6.68. Again, the main effects of A and B and the interaction AB are statistically significant. From the ANOVA in Table 6.68 we also see that σ̂ 2 = MSE = 0.01832, which is less than the previous value using the full model (see Table 6.67).
We can determine which levels of the factors will result in large values for ventilation effectiveness. More details on determining these levels are provided later in this chapter.
Figure 6.50 Interaction plot for partition height and gap underneath.
2.15
2.10
2.05
2.00
1.95
1.90
1.80
1.85
–1 C
1
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1
–1
B
Table 6.67 ANOVA table for the ventilation example.
Source df SS MS F p-value
A 1 0.84135 0.84135 39.48 0.000
B 1 0.32862 0.32862 15.42 0.004
C 1 0.00628 0.00628 0.29 0.602
AB 1 0.16221 0.16221 7.61 0.025
AC 1 0.00098 0.00098 0.05 0.836
BC 1 0.00278 0.00278 0.13 0.727
ABC 1 0.03930 0.03930 1.84 0.211
Error 8 0.17048 0.02131
Total 15 1.55199
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We now present the model- fitting approach. The model- fitting approach is simply the regression analysis procedure described in section E.2 of this chapter. It is used to fit a model relating the dependent variable (response) and independent variables (factors). Multiple linear regression was briefly introduced in section E.3 of this chapter.
Model fitting (regression analysis) and ANOVA are not necessarily separate approaches. An ANOVA is often reported when the regression approach is used to study the effects of the factors and interactions on the response. A model relat- ing the independent variables and dependent variable (response) can be given by Equation (6.135), reproduced below
y = β0 + β1x1 + β2x2 + . . . + βkxk + ε
where y is the response of interest and x1, x2, . . . , xk represent the independent variables. In our factorial designs, the independent variables are our factors such as A, B, and C and the interactions among the factors AB, AC, BC, and ABC. To illustrate, we can let x1 represent factor A, x2 represent factor B, and so on. Note that the convention to let x1x2 represent the AB interaction, for example, is used. The coefficients βi on each term can be tested using t-tests, as done in section E of this chapter. The null hypothesis of interest is H0: βi = 0 for all i. Results of the t-test for the air quality example are given in Table 6.69.
The “Effect” column displays the estimated effects for the main effects and interactions. From the p-values for the t-tests, we again conclude that partition length (A), partition height (B), and the interaction between the two factors (AB) are significant. The analysis should be rerun involving only the terms found sig- nificant. A model relating ventilation effectiveness to partition length, partition height, and the interaction can now be fit. The column labeled “Coef” provides the estimates of the coefficients in the regression model
ˆ . . . .y x x x x= + − +1 9851 0 2293 0 1433 0 10071 2 1 2
where x1 represents partition length, x2 represents partition height, and x1x2 rep- resents the interaction between the two factors. It should also be noted that the coefficient estimates are one- half of the estimated effects. The fitted model above is in coded form. That is, if we want to make predictions for certain levels of the factors, we would use the notation (–1, 1) to plug into the equation. For example,
Table 6.68 ANOVA table for the ventilation example, with only statistically significant terms.
Source df SS MS F p-value
A 1 0.84135 0.84135 45.93 0.000
B 1 0.32862 0.32862 17.94 0.001
AB 1 0.16221 0.16221 8.86 0.012
Error 12 0.21982 0.01832
Total 15 1.55199
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if we wanted to predict the ventilation effectiveness for the low levels of A and B, we would let x1 = –1 and x2 = –1 in our fitted model:
ˆ . . . .
.
y x x x x= + − +
=
1 9851 0 2293 0 1433 0 1007
1 98
1 2 1 2
551 0 2293 1 0 1433 1 0 1007 1 1
2
+ − − − + − −
=
. ( ) . ( ) . ( )( )
.00
The model can also be written in terms of the actual levels of the factors. It is recommended that the model fitting be done using a reliable statistical software package. The model using the actual levels can be shown to be
ŷ = 5.366 − 0.048A − 0.065B + 0.001AB
The predicted value when partition length and partition height are at their low levels is found by replacing A and B with the actual levels of the factors. For A at its low level (40) and B at its low level (60), the predicted value is
ŷ = 5.366 − 0.048A − 0.065B + 0.001AB = 5.366 − 0.048(40) − 0.065(60) + 0.001(40)(60) = 2.0
The difference between this estimate and the one from the model in coded units is strictly due to round- off error. Either model can be used to fit the data. In addition, the actual levels in this example are left as percentage values such as 40 and not converted to decimal form such as 0.40. This was only by choice. Using a statistical package we could have stated the lower level and upper level of partition length as 0.40 and 0.60, respectively, but we chose to use 40 and 60. The same main effects and interaction would still be found to be significant.
Once the model has been refined so that it contains only those terms that are statistically significant, the three assumptions of normality, independence, and
Table 6.69 t-tests for factors and interactions for the air quality example.
Term Effect Coef SE Coef t p-value
Constant — 1.9851 0.03649 54.39 0.000
A 0.4586 0.2293 0.03649 6.28 0.000
B –0.2866 –0.1433 0.03649 –3.93 0.004
C 0.0396 0.0198 0.03649 0.54 0.602
AB 0.2014 0.1007 0.03649 2.76 0.025
AC 0.0156 0.0078 0.03649 0.21 0.836
BC 0.0264 0.0132 0.03649 0.36 0.727
ABC –0.0991 –0.0496 0.03649 –1.36 0.211
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constant variance should be investigated. If the order in which the treatments were carried out was not recorded, the independence assumption will be difficult to verify. Hopefully, by randomizing all 16 runs while all extraneous factors are held constant there is no significant problem with dependency. Again, it is desirable to actually be able to check this assumption.
Recall that the residuals are defined as ei = yi – ŷ i, where the predicted values are found as shown previously. The 16 residuals can be calculated and analyzed through residual plots. The normal probability plot of the residuals is shown in Fig- ure 6.51. The residuals appear to generally fall along a straight line, so the normality assumption does not appear to be violated. The residuals plotted against the signifi- cant factors are displayed in Figure 6.52 and Figure 6.53. There does not appear to be a problem with constant variance across the factor level because the vertical spread of the residuals for each factor level is approximately the same in both figures.
Figure 6.51 Normal probability plot of the residuals for the air quality example.
99
95
90
80 70 60 50
10
20 30 40
5
1 –0.3 –0.2 –0.1 0.0 0.1 0.2
Residual 0.3
P e rc
e n
t
Figure 6.52 Residuals plotted against levels of factor A (partition length).
0.2
0.1
–0.1
0.0
–0.2 –1.0 –0.5 0.0 0.5
A 1.0
R e s id
u a l
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Now that the significant terms are identified and the necessary assumptions have been shown to be satisfied, the next step is to determine the optimal settings for the significant factors. In our example, a goal is to maximize the ventilation effectiveness. There are several ways to determine these settings. We will discuss two graphical methods.
A useful graphical display of the fitted model is a contour plot. The contour plot for our fitted model in coded form is shown in Figure 6.54. A contour plot dis- plays constant values of the predicted response (contours) over the range of the significant factors. Notice that the contour lines are curved; this is the result of a significant interaction between the two factors. For our example, we see that the lower right- hand corner of the contour plot displays higher values of ventilation
Figure 6.53 Residuals plotted against factor B (partition height).
0.2
0.1
–0.1
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–0.2 –1.0 –0.5 0.0 0.5
B 1.0
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Figure 6.54 Contour plot for the air quality example.
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effectiveness. If the goal is to maximize ventilation effectiveness, it appears that factor A (partition length) should be set at its high level (60%) while factor B (parti- tion height) should be set at its low level (60%).
Main effects plots and interaction plots are also useful graphical displays of the results. Since the two main effects found significant are involved in a signifi- cant interaction, it is the interaction plot that should be examined. In fact, if the factors are involved in a significant interaction but only the main effects plots of these factors are examined, it is possible to choose less than optimal settings of the factors. Recall the interaction plot of factors A and B displayed in Figure 6.48. On this figure we see that the highest ventilation effectiveness occurs at the low level of B (partition height) and high level of A (partition length).
The reader is encouraged to consult Devore (2016); Montgomery and Runger (2013); Montgomery, Runger, and Hubele (2010); Vining and Kowalski (2011); and Myers, Montgomery, and Anderson- Cook (2016) for more details on methods for determining acceptable levels of the significant factors.
H.5.c. 2k Designs with a Single Replicate
Often it is not possible or economical to obtain more than a single replicate for a designed experiment (i.e., n = 1). When this is the case, it is not possible to test the significance of all the effects. There is no internal estimate of error since there is no replication. Specifically, there are no degrees of freedom left over for error in order to estimate the process variability σ 2. The total degrees of freedom for a design with a single replicate are 2k – 1. Each main effect and interaction is given one degree of freedom. For example, suppose that in our ventilation effectiveness example we have only one replicate. The total number of runs would be eight, and the total degrees of freedom would be 2k – 1 = 8 – 1 = 7. Furthermore, there are three main effects (A, B, and C) and four interactions (AB, AC, BC, and ABC). Since every effect has one degree of freedom, all of the degrees of freedom are used. There are no degrees of freedom for error. t-tests and the ANOVA method cannot be carried out.
To address this issue, several approaches can be employed. These approaches are often based on the sparsity-of-effects principle. That is, an assumption is made that some higher- order interactions are negligible (orders higher than two- factor interactions), and the system being investigated is believed to be dominated by the main effects and the low- order interactions. Under this assumption the degrees of freedom for the higher- order interactions are pooled into error degrees of freedom. Any sums of squares these interactions may have had get pooled into error sums of squares.
If there is any indication that one or more of the higher- order interactions are significant, then the pooling approach is not appropriate. A different method of analysis that is often used is examination of a normal probability plot of the esti- mated effects. This approach was suggested by Daniel (1959) and is available in most statistical software packages. Effects that are not significant (or are negli- gible) are said to be normally distributed with mean zero and variance σ 2. When plotted on a normal probability plot, estimated effects that are negligible will tend to fall along a straight line. The negligible effects are pooled into error and the degrees of freedom assigned to error.
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Consider the air quality scenario presented earlier. Suppose that only one replicate was obtained for each of the eight runs. Data typical of this experiment are shown in Table 6.70.
Table 6.70 A single replicate of the air quality example.
Treatment A B C AB AC BC ABC yv
1 –1 –1 –1 1 1 1 –1 2.135
2 1 –1 –1 –1 –1 1 1 2.015
3 –1 1 –1 –1 1 –1 1 1.520
4 1 1 –1 1 –1 –1 –1 1.999
5 –1 –1 1 1 –1 –1 1 1.998
6 1 –1 1 –1 1 –1 –1 2.103
7 –1 1 1 –1 –1 1 –1 1.624
8 1 1 1 1 1 1 1 2.135
The estimated effects for the main factors and all of the interactions can still be cal- culated using the formulas given previously. The estimated effects are then plotted on a normal probability plot (sometimes a standardized value of the effects will be plotted). The normal probability plot of the estimated effects is displayed in Figure 6.55.
99
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Standardized effect
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Figure 6.55 Normal probability plot of the estimated effects for the air quality example.
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The normal probability plot of the effects (the effects have been “standardized”) indicates that the main effects A and B and the interaction AB may be significant. An analysis could then be carried out on A, B, and AB. An analysis is conducted on the three terms with all other terms pooled into error. The results of t-tests on the effects are shown in Table 6.71.
Model fitting and residual analysis can be completed similarly to the case of more than one replicate. An analysis of variance can also be conducted with A, B, and the AB interaction as the only terms in the model.
Table 6.71 t-test results for the air quality example.
Term Effect Coef SE Coef t p-value
Constant — 1.9411 0.02953 65.72 0.000
A 0.2438 0.1219 0.02953 4.13 0.015
B –0.2432 –0.1216 0.02953 –4.12 0.015
AB 0.2512 0.1256 0.02953 4.25 0.013
The methods and procedures outlined in this section can be used for any num- ber of factors at two levels each. A drawback to the use of 2k full- factorial designs is that the design size becomes prohibitively large as the number of factors increases. For example, even if there are only seven factors each at two levels, the number of experimental runs would be 27 = 128, without replication. It is not unusual, espe- cially in screening experiments, to have six, seven, or more factors of interest being investigated. In these cases, it is often very useful to run experiments involving fractions of the full- factorial design. These designs are commonly referred to as fractional factorial designs and are discussed in the next section.
H.6. Two- Level Fractional Factorial Experiments
Fractional factorial designs are those where only a fraction of the full- factorial design is used. Fractional factorial designs are economic alternatives to the full- factorial designs as the number of factors increases. Screening experiments often involve a large number of factors, so full- factorials are not always practical or economical.
Half-fractions of a 2k are designs that consist of half of the standard 2k design. Half- fractions are usually denoted 2k–1 (one-half of the 2k = 2k/2 = 2k–1).
ExaMpLE 6.84
Consider an experiment with six factors each at two levels. A full-factorial design con- sists of 26 = 64 combinations or runs. A full 26 experimental design is in many instances prohibitively large. However, 32 experimental runs may be more economical. In this case, we can choose one-half of the runs from the full 26. The resulting design is referred to as a 26 –1 design. Runs selected from the full-factorial are not chosen at random. More on this later in this section.
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In some instances, a half- fraction may still be too large and impractical. It may be more economical to use designs that are one- fourth, or possibly one- eighth, the size of the full- factorial. The general notation for a fractional factorial design is denoted as 2k–p, where 1/2p represents the fraction of the full- factorial.
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A screening experiment is going to be conducted involving 10 factors each at two levels. A full-factorial design with a single replicate would still require 210 = 1024 runs. The experimenters can afford to do no more than 40 runs initially for screening. A frac- tion of the full-factorial that could be used and still meet the size requirements would be a 210 – 5 design, which would require only 32 experimental runs (210 – 5 = 25 = 32).
When conducting an experiment with k factors, we are interested in not only the significance of each factor but also the interactions between the fac- tors. If a full- factorial could be implemented, then all main effects of interest and all two- factor interactions are fully estimable. However, when employing fractional factorial designs, the design size is reduced, and not all interactions of interest may be estimable separately from main effects or other interactions. Some of the interactions and/or main effects may be confounded or aliased with one another, making it difficult to determine which factor or interaction is truly significant.
The identity column will be very useful in fractional factorial designs. If any column in the design is multiplied by the identity column, the result is the origi- nal column. For example, A × I = A. The ABC interaction is a generator and would be used to generate the column for one of the main factors. ABC is often referred to as a word.
An important characteristic of fractional factorial designs is the defining rela- tion. The defining relation is one that contains all possible “words” whose signs do not change in the experiment. For example, from Table 6.72 we see that the interaction column ABC consists of all +1’s. Therefore, ABC would be a word in the defining relation. Since it is the only column with the signs unchanged, it is the only word in the defining relation. In this problem, our defining relation would be I = ABC. All aliases are found through the defining relation. For example, the alias for factor A is
A ∙ I = A ∙ ABC = BC
Therefore, factor A is aliased with the BC interaction. The other aliases are found similarly:
B ∙ I = B ∙ ABC = AC
C ∙ I = C ∙ ABC = AB
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A study is going to be conducted involving three factors each at two levels. Suppose we can only afford to conduct four treatments and not the eight that would make up a full 23 factorial design. A half-fraction of the 23 design would seem like a natural choice and would be called a 23 –1 design. A 23 –1 design and all columns for the interactions are given in Table 6.72.
Table 6.72 Main effects and interactions table for a 23–1 design.
Run A B C AB AC BC ABC
1 –1 –1 1 1 –1 –1 1
2 1 –1 –1 –1 –1 1 1
3 –1 1 –1 –1 1 –1 1
4 1 1 1 1 1 1 1
Notice that the column for factor C and the column for the AB interaction are iden- tical. We would say that factor C is aliased or confounded with the AB interaction, that is, C = AB. Also notice in the table that the column for the ABC interaction contains only the high level of the interaction. We would say that ABC is equal to the identity column (I = ABC).
The configuration in Table 6.72 guarantees that none of the main factors have identical columns (therefore they are not aliased or confounded with one another). But main effects are aliased with two- factor interactions. If it is believed that the AB interaction may be significant, then a different design (with more runs) would have to be used.
The 23–1 design is said to be of resolution III. Resolution III designs are those where main effects are aliased with two- factor interactions. More on this later in this section.
Resolution IV or higher designs are desirable, since they guarantee that the main effects will be clear of (not aliased with) other main effects and two- factor inter- actions. The obvious drawback to resolution IV designs is that two- factor interac- tions are aliased with other two- factor interactions. Suppose we carry out the 24–1 design, analyze the results, and determine that all main effects and the two- factor interaction AB are found to be statistically significant. With resolution IV designs, we do not know for sure that AB is truly significant or if the two- factor interaction it is aliased with (here AB = CD) is significant. There are methods for breaking these aliases that involve adding a subset of new experimental runs. See Box, Hunter, and Hunter (2005) or Montgomery (2017) for more details on breaking these aliases (also referred to as “de-aliasing”).
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A study is going to be conducted involving four factors each at two levels. Suppose we can only afford to conduct eight treatments and not the 16 that would make up a full 24 factorial design. A half-fraction of the 24 design would seem like a natural choice. The half-fraction of the 24 is the 24 –1 design and would contain 23 = 8 runs. One possible frac- tion is displayed in Table 6.73.
Table 6.73 Half-fraction of a 24 factorial design.
Run A B C D
1 –1 –1 –1 –1
2 1 –1 –1 1
3 –1 1 –1 1
4 1 1 –1 –1
5 –1 –1 1 1
6 1 –1 1 –1
7 –1 1 1 –1
8 1 1 1 1
In this study a full 23 design was constructed for factors A, B, and C. Column D was generated from the three-factor interaction ABC, that is, D = ABC. In this exam- ple ABC is the generator and the defining relation is I = ABCD. This design is said to be of resolution IV. Resolution IV designs are those where main effects are aliased with three-factor interactions, and two-factor interactions are aliased with other two-factor interactions. Using the defining relation I = ABCD we can obtain all of the aliases. For the main effects:
A = BCD
B = ACD
C = ABD
D = ABC
For the two-factor interactions:
AB = AB ∙ I = AB ∙ ABCD = CD
AC = AC ∙ I = AC ∙ ABCD = BD
AD = AD ∙ I = AD ∙ ABCD = BC
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The following are some of the properties for experimental design:
1. Resolution III designs have main effects confounded with two- factor interactions
2. Resolution IV designs have main effects confounded with three- factor interactions and two- factor interactions confounded with each other
3. Resolution V designs have two- factor interactions confounded with three- factor interactions only
ExaMpLE 6.88
Consider an experiment involving six factors where only 16 runs can be used. A full- factorial design in 16 runs is a 24 design. Our design is referred to as a 26 –2 fractional fac- torial design.
We could construct a 24 full-factorial design for four of the six factors, but the remaining two factor columns would have to be generated. Let A, B, C, D, E, and F represent the six factors. Suppose a full-factorial design is constructed for A, B, C, and D. It can be shown that two generators needed for E and F could be E = ABC and F = BCD. The resulting defining relation would be I = ABCE = BCDF = ADEF. The last “word,” ADEF, is found by multiplying the two original generators, ABCE and BCDF (see Box, Hunter, and Hunter [2005] or Montgomery [2017] for more details). The resolution of this design is IV.
In general, the resolution of a design can always be determined from a complete defining relation. By definition, the resolution of a design is equal to the length of the smallest word in the defining relation. For example, consider a 27–2 design with factors A, B, C, D, E, F, and G. The complete defining relation for this design using the generators F = ABCD and G = ABDE can be shown to be
I = ABCDF = ABDEG = CEFG
The length of the smallest word is four, so the design is of resolution IV. There are numerous approaches and methods involving fractional factorial
designs. The reader is encouraged to see Box, Hunter, and Hunter (2005); Ledolter and Swersey (2007); and Montgomery (2017) for complete details and examples of full and fractional factorial designs and their applications.
H.7. designed Experiments and Statistical Control
There has been considerable debate about the use of designed experiments in indus- try if the process under investigation is not known to be in statistical control. Some researchers have argued that the process must be in statistical control before con- ducting legitimate industrial experiments, while others have argued that statistical control is not necessary (see Bisgaard [2008]). Research by R. A. Fisher first pub- lished in 1925 showed that statistical control was not a prerequisite for implementing
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designed experiments when replication, blocking, and randomization were key components of the experimentation. Arguments have been made to the effect that Fisher’s results, while applicable in agricultural experiments, do not apply in indus- trial settings.
Anyone involved in conducting experiments should read the Quality Engi- neering article by Søren Bisgaard (2008). Statistical control and designed experi- ments are discussed in detail by Bisgaard, with discussion of his article provided by G. Geoffrey Vining, Thomas P. Ryan, George E. P. Box, Donald J. Wheeler, and Douglas C. Montgomery.* The article and discussions are a must- read for practi- tioners and researchers alike and provide numerous references for further reading. Simpson, Listak, and Hutto (2013) also provide recommendations for planning and assessing well- designed experiments.
*These discussions are found in Quality Engineering (vol. 20, no. 2, 2008) as follows: Vining, 151–53, doi:10.1080/08982110701866198; Ryan, 154–57, doi:10.1080/08982110801894892; Box, 158–59, doi:10.1080/08982110801890148; Wheeler, 160–64, doi:10.1080/08982110801924509; Montgomery, 165–68, doi:10.1080/08982110801894900; Bisgaard (rejoinder), 169–76, doi:10.1080/ 08982110801973118.
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ISO 31000:2009 defines risk management as “coordinated activities to direct and con- trol an organization with regard to risk.” Quality and risk management are integral parts of any organization. While there is overlap between the two, we differentiate quality management from risk management based on goals and objectives. ISO standards define quality as the “degree to which a set of inherent characteristics fulfills requirements” and risk as “the effect of uncertainty on objectives.”
Quality places great emphasis on variability and the impact of variability on performance. Thus, quality management emphasizes creating products and ser- vices with desirable characteristics from a customer standpoint. Risk manage- ment, on the other hand, is focused on the evaluation of events that may impact any objective within an organization. ISO 31000:2009 lists several risk manage- ment principles. These state that risk management:
• Creates and protects value
• Is an integral part of all organizational processes
• Is part of decision making
• Explicitly addresses uncertainty
• Is systematic, structured, and timely
• Is based on the best available information
• Is tailored
• Takes human and cultural factors into account
• Is transparent and inclusive
• Is dynamic, iterative, and responsive to change
• Facilitates continual improvement of the organization
As discussed in Chapter 2, section B, the new revision of ISO 9001:2015 empha- sizes risk- based thinking in quality management. The effective management of quality, as described in Montgomery (2013), involves three main activities: quality planning, quality assurance, and quality control/improvement. Risk management is similar in that both planning and control are integrated into the process. Analo- gous to “quality assurance,” the management of risk involves an evaluation/ assessment aspect. In this chapter, we discuss risk management from three stand- points: risk oversight, risk identification and prioritization, and risk mitigation
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and control. All of the material in Part VII of the Body of Knowledge (BoK) is contained within this chapter; however, the order has been changed to reflect risk management practice.
a. riSK ovErSigHT Various degrees of mental disposition toward uncertainty within certain individu- als, groups, or an organization can have a large influence on risk oversight. Weber, Blais, and Betz (2002) explain risk attitude as “a descriptive label for the shape of the utility function presumed to underlie a person’s choice.” Examples of risk attitudes include being risk averse, risk tolerant, risk neutral, and risk seeking. Psychologists and economists recognize that these labels are fairly subjective. In an attempt to create a more objective measure of risk attitude, expected values of perceived riski- ness have been suggested. Weber (1997, 1998) shows that riskiness can be treated as a variable that is a weighted sum of expected benefit and perceived risk. If the perceived risk of a variable outweighs the expected benefit, risk managers should change their plans accordingly. Regardless of risk attitude, the key components of risk oversight are the risk management process as well as planning and oversight, which are discussed this section.
a.1. risk Management Process
Risk management is a constant, continuous process and generally involves three basic phases: identification, assessment, and mitigation/control. These three phases are often used iteratively as risks are identified and controlled, mitigated, or accepted. See, for example, Figure 7.1. In addition to these three phases, how- ever, it is important to plan the risk management process before beginning the identification phase. There are several types of risks that may need to be managed. Technical risks of a product or process need to be managed during the design, manufacture, and operation of products, for example. There may also be business risks that need to be managed. Other examples of risk management include proj- ect risk management (one of the most common applications of risk management) and enterprise risk management. Project management is discussed in Chapter 1, section B.2.d. The path of this process, and the order of the iterations, is dependent on the application as well as the industry.
For example, risk management may be incorporated into the design stage of a product or service, or it may be incorporated during the production stage. The steps involved in the risk management process and the order in which they are executed are sometimes controlled by industry- specific guidelines or regulations. For example, consider the FDA’s approach to quality risk management. Its docu- mentation characterizes risk management into the following four phases, which are similar to the phases typically applied in the manufacturing industry, as dis- cussed in the beginning of this section (Rodriguez-Perez 2012):
Phase 1: Determining acceptable levels of risk (risk acceptability criteria)
Phase 2: Risk analysis, which includes identifying and quantifying risks
Phase 3: Risk evaluation, in which the estimated risks are compared to the risk acceptability criteria
Phase 4: Risk control and monitoring
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The planning phase of risk management is discussed in the next subsection. Risk assessment includes identification, analysis, and evaluation of risks and is dis- cussed in detail in section B. Risk mitigation (the process of attempting to minimize the impact of a risk should it occur), risk control (documenting and monitoring risks), and risk acceptance are discussed in section C.
a.2. Planning and oversight
Risk management should be integrated across an organization regardless of a cen- tralized or decentralized approach. Depending on the company culture, risk can be treated from a defensive or offensive approach. Historically, risk management teams have taken a proactive approach, which falls into the defensive category. The offensive approach to risk allows risk events to be treated as potential oppor- tunities for success rather than failure. We will look at risk events in this chapter from both the proactive and reactive standpoints. It is up to the individual or orga- nization to decide which approach is best.
A decentralized risk management approach involves treating risk as a disci- pline or process. For example, every product or process could incorporate a risk assessment that is used to identify, analyze, and monitor risk. Alternatively, risk could be managed from a single department that oversees all risk manage- ment activities across the organization.
Figure 7.1 Steps common to a risk management process. Source: Reproduced by permission from P. R. Garvey, Analytical Methods for Risk Management (Boca Raton, FL: Taylor & Francis Group, 2009).
Risk events and their relationships are defined
Consequences may include cost, schedule, technical performance impacts, as well as capability or functionality impacts
Decision-analytic rules applied to rank-order identified risk events from “most-to-least” critical
Risk events assessed as medium or high criticality might go into risk mitigation planning and implementation; low critical risks might be tracked/monitored on a watch-list
Probabilities and consequences of risk events are assessed
Identify risks
Assess probability and consequence
Watch- listed risks
Reassess existing risk events and
identify new risk events
Assess risk
criticality
Risk mitigation
1. Risk identification
2. Risk impact assessment
3. Risk prioritization
analysis
4. Risk mitigation planning,
implementation, and progress
monitoring
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Regardless of the way in which risk management is incorporated into an orga- nization, it is important to have a plan or policy in place. Planning is crucial for setting ground rules and objectives, identifying key stakeholders, and selecting key methods for risk analysis. Every step of the risk management process should be decided and documented during the planning phase. For more information on risk planning and risk management, see Wang and Roush (2000).
B. riSK aSSESSMEnT There are three main components of risk assessment: identification, analysis, and evaluation. These components collectively help establish a form of risk syntax or statements. A generic form of risk syntax is the following:
Because of <defined cause>, <an uncertain event> may occur, which would lead to <effect on the objective>.
Once a risk has been identified and analyzed, expected outcomes can be pos- tulated, followed by events that may occur as a result. In this section, we provide tools that will help in the process of creating risk statements and outcomes.
B.1. identification
The purpose of risk identification is to identify all possible risks, whether current risks, possible future risks, risks that are not currently under the organization’s control, or risks that may occur due to results of an accumulation of factors or steps in the process. Several approaches that have proven to be effective in identi- fying potential risk events are discussed here: environmental stress screening, fault tree analysis, hazard and operability analysis, failure modes and effects analysis, and failure mode effects and criticality analysis. While these tools are effective at identifying risks, a cause- and-effect diagram (discussed in Chapter 5, section A) is often a useful tool to start the risk identification process.
B.1.a. Environmental Stress Screening
Environmental stress screening (ESS) is a process designed to precipitate nascent defects into detectable failures by use of environmental stresses applied to hard- ware. ESS is most efficient when used at the lowest practical level of hardware. When used at the part level, ESS is often called burn- in.
The most frequently used environments for ESS are temperature cycling and random vibration. Other environments, such as shock, altitude, humidity, and so on, can be used based on the product type and its intended use conditions. Experi- ence shows that the sequence of the application of environments has been found to play a minor role in the effectiveness of ESS. The following two conditions are necessary when applying ESS:
1. The product’s design limit should not be exceeded.
2. More severe environments should be applied at the lower levels of the hardware so that screening environments become less severe with increasing levels of hardware complexity. This will cause failures at the lower levels, where it is less costly to replace or repair.
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The Environmental Stress Screening of Electronic Hardware (ESSEH) committee of the Institute of Environmental Sciences has compiled data from throughout the electronics industry and made the following recommendations (https://www. the-ies.org):
• The optimal number of thermal cycles for electronics is 10. This number is obtained based on experience; scientific or mathematical methods are needed to obtain the “true” optimum for given test conditions and constraints.
• Random vibration is a more efficient screen than sinusoidal vibration.
• The preferred random vibration profile is the naval material command (NAVMAT) profile, which covers the frequency spectrum from 20 hertz to 2000 hertz with an overall acceleration of six grams. Studies show that the vast majority of failures with this profile occur in the first 10 minutes of the test.
It is important that a baseline experiment be conducted and analyzed during ESS to determine the optimum screening parameters.
B.1.b. Fault Tree Analysis
Fault tree analysis (FTA), a top- down approach, is a technique for analyzing com- plex systems to determine potential failure modes and probabilities of their occurrences. The technique was originated by H. A. Watson of Bell Telephone Laboratories to analyze the Minuteman Launch Control System. Many fault trees may be required to assess all potential failures of a process or system. The fol- lowing steps are required in order to develop fault trees (Pilot 2002; Dhillon and Singh 1981):
1. Define the undesired event (top level failure).
2. Thoroughly understand the system and its intended use to determine the possible reasons for the failure to occur.
3. Continue to break down each element in the fault analysis to lower levels. Determine the relationships that can cause the fault conditions.
4. Finalize the fault tree and logical relationships among the inputs. The tree must end in a basic event (human, hardware, or software).
5. Evaluate the probability of occurrence for the lowest level elements and then calculate the probability from the bottom of the fault tree to the top.
FTA requires the construction of a fault tree diagram that represents the system conditions symbolically. This requires definition of the fault tree symbols. Such symbols include, for example, AND gate, OR gate, basic fault event, and priority AND gate. Gates describe the logic between events. The most common events are intermediate events and basic events. Basic events are the lowest- level elements and are represented by circles in the fault tree. Intermediate events are represented by rectangles in the fault tree.
The AND gate denotes that the output event occurs if and only if all the input events occur. Its symbol is shown in Figure 7.2. The OR gate denotes that the out- put event occurs if any of the input events occur. Its symbol is shown in Figure 7.3.
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n t Exhaustive listings of fault tree symbols exist in specialized references (see
Dhillon and Singh [1981] or Barlow, Fussell, and Singpurwalla [1975]). Fault tree evaluations can be performed qualitatively or quantitatively. The
qualitative evaluation determines the minimum cut sets (the minimum number of components that cause system failure), while the quantitative evaluation can be done using failure data for each component or event. The latter can be obtained from historical data or by using computer simulations, for example, Monte Carlo simulations. The result of the qualitative evaluation is a set of critical components, and the result of the quantitative evaluation is the probability of the occurrence of the top event. The end goal of FTA is to identify changes in the process or design that would decrease the likelihood of the top event, the failure.
To obtain quantitative results for the top event, assign failure probability, unavailability, failure, and repair rates data to the basic events, provided the fault tree events are redundancy- free. Using these probabilities and Boolean logic, the probability of the top event occurring can be built up from the bottom of the fault tree. Note that this quantitative analysis may be straightforward to do by hand for simple systems, but may require computer software for larger, more complicated systems.
To analyze the fault tree, we can combine the probabilities of the basic events and use Boolean logic to build up the probability of the top event occurring. Fault trees assume that basic events are independent. Therefore, Equations (6.16) and (6.17) can be used to analyze the probability of failure at AND gates in the fault tree. To analyze OR gates, recall from Equation (6.27), P(failure) + P(success) = 1. If an OR gate has two basic events as inputs, E1 and E2, the probability of failure for that gate can be stated as follows:
P(failure) = 1 – P(success)
= 1 – P(E1 and E2 do not fail)
= 1 – P(E1 does not fail)P(E2 does not fail)
= 1 – (1 – P(E1 fails))(1 – P(E2 fails)
If an OR gate has more than 2 events (say n events), this analysis is easily extended
P(failure) = 1 – (1 – P(E1 fails))(1 – P(E2 fails)) . . . (1 – P(En fails))
This method is demonstrated in Example 7.1.
Figure 7.2 AND gate for fault tree analysis. Figure 7.3 OR gate for fault tree analysis.
Output
Inputs
AND gate
Output
Inputs
OR gate
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Alternately, in Chapter 6, section B, we discussed several rules and properties of probability. Equation (6.13), for example, showed that the probability of either event A or event B occurring is P(A ∪ B) = P(A) + P(B) – P(A ∩ B). This rule can be extended for two or more events, a common occurrence when analyzing fault trees. If there are n events E1, E2, . . . , En, the probability of E1 or E2 or . . . or En occurring is defined as
P(E 1 or E
2 or . . . or E
n ) = P(E
i ) – P(E
i and E
j ) +
i 1
n
= ∑
n
1≤i<j≤n ∑
P(E i and E
j and E
k ) – . . . P(E
1 and E
2 and . . . and E
n )
n
1≤i<j<k≤n ∑ ±
(7.1)
This is known as the inclusion- exclusion formula. Once the fault tree is complete, the logic expression for the top event can be found by building from the bottom of the fault tree to the top. Using Equation (7.1) we can determine the probability formula for the top event. By substituting the probabilities of failures for each basic event, we can determine the probability of the top event (Bartlett 2007).
ExaMpLE 7.1
Construct a fault tree of a simple electric lamp. The top event is “no light” when the switch is turned on. This could be caused by:
1. Power failure E1
2. Switch fails to close E2
3. Lamp failure E3
4. Fuse failure E4
Furthermore, the power failure can be attributed to two events: major power failure or a fuse failure. A simple tree of these events is shown in Figure 7.4. Let P(Ei) represent the probability that event Ei occurs. In other words, the probability of a failure of each basic event Ei is denoted as P(Ei). Suppose P(E1) = 0.005, P(E2) = P(E3) = 0.001, and P(E4) = 0.0001. Find the probability that the top event failure (no light) occurs.
First determine the probability of failure of the intermediate event “no power.”
P(no power) = P(E1 or E4) = 1 – (1 – P(E1))(1 – P(E4)) = 1 – (1 – 0.005)(1 – 0.0001) = 1 – (0.995)(0.9999) = 1 – 0.9949005 = 0.0050995
P(no light) = P(E2 or “no power” or E3) = 1 – (1 – P(E2))(1 – P(no power))(1 – P(E3)) = 1 – (1 – 0.001)(1 – 0.0050995)(1 – 0.001) = 1 – (0.999)(0.9949005)(0.999) = 0.00708831
The probability of a failure of no light for this electric lamp and the given estimated prob- abilities for the basic events is 0.007.
Continued
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No light
No power Intermediate event
Power outage Fuse failure
Lamp failureSwitch fails
E2
E1 E4
E3
Figure 7.4 Simple fault tree.
As a system gets more complex, the fault tree complexity increases, result- ing in more complicated probability statements for the top event. In these cases, computer software should be used. Example 7.1 also assumed a fixed probability of failure for each basic event. Other methods to estimate these probabilities are possible. For example, the exponential distribution is often used to estimate failure times, as discussed in Chapter 3, section E. Computer software can also handle these more complex scenarios.
We have discussed fault trees in detail here; however, note that event trees, which were discussed in Chapter 5 and are similar to fault trees, can also work well for risk identification and assessment. Further details on FTA can be found in Tague (2005) or Ruggeri, Kenett, and Faltin (2007).
B.1.c. Hazard and Operability Analysis
Hazard and operability analysis (HAZOP) can also be used to identify operability issues and potential hazards that may lead to unacceptable products, processes, or services. HAZOP can be applied to a wide range of processes, products, personnel, and services. Applications include process HAZOP, human HAZOP, procedure HAZOP, and software HAZOP. The method is based on the assumption that risks occur due to deviations from the intended design or operating plan. The goal of HAZOP is to identify potential risks from these deviations. To achieve this list
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of risks, HAZOP analysis uses “guide words” to systematically identify potential deviations from normal design or operating conditions. When used as part of a cross- functional team, the guide words in a HAZOP analysis can generate ideas as well as discussion of potential risks. Table 7.1 displays a list of common guide words used in HAZOP.
B.1.d. Failure Modes and Effects Analysis
Failure modes and effects analysis (FMEA) is a team- based problem- solving tool intended to help users identify and eliminate or reduce the negative effects of poten- tial failures before they occur in systems, subsystems, product or process design, or the delivery of a service. FMEA can be used as a stand- alone tool or as part of a comprehensive quality program such as ISO 9000, ISO/TS 16949, advanced product quality planning and control plan (APQP), or Six Sigma. Accordingly, this section discusses terminology, theory, mechanics, and applications of FMEA as it applies to product designs, process designs, and systems.
Table 7.1 HAZOP guide words.
Guide word Meaning Examples
No (no, none) None of the design intent is achieved
• Preventive maintenance was not performed
• No sterilization process was performed
More (more of, higher)
Quantitative increase in a parameter
• Overconcentrated drug product • Infusion pump delivering more dosage
than needed
Less (less of, lower) Quantitative decrease in a parameter
• Customer receiving less product than expected (lower pill count or fill volume)
• Drug product underconcentrated
Other than (other) Complete substitution— another activity takes place
• Product mix-ups • Product sterilized with an incorrect
method
Part of Only some of the design intention is achieved
• Device kit missing one component • No detergent used for cleaning, only
water
As well as (more than)
An additional activity occurs
• An additional component was added to the formulation of the batch
Reserve Logical opposite of the design intention occurs
• Sterile injectable drug with bacterial contamination
Early/late Timing is different from the intention
• Detergent added too early (prior to pre-rinse of soiled equipment)
Before/after Step (or part of it) performed out of order
• Components added to the batch in incorrect order
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Tague (2005) provides a list of when FMEA should be used:
• When a process, product or service is being designed or redesigned, after quality function deployment
• When an existing process, product or service is being applied in a new way
• Before developing control plans for a new or modified process
• When improvement goals are planned for an existing process, product or service
• When analyzing failures of an existing process, product or service
• Periodically throughout the life of the process, product or service
A word of caution: FMEA can be a powerful and effective tool for system, subsys- tem, product or process design, or service delivery improvement. However, com- pleting an FMEA has significant costs associated with it. Organizations that may be tempted to follow the results of an FMEA to implement further levels of refine- ment and specificity should conduct a cost/benefit analysis to ensure that FMEA is the proper tool under the given circumstances. Furthermore, FMEA cannot be used to identify combinations of failure modes, even when they are significant.
B.1.d.i. FMEAs Encountered by Quality Engineers
FMEA can be applied to the system, subsystem, design or process, or service delivery levels. A brief synopsis of each FMEA application is as follows:
• System FMEA. A system or subsystem is a collection of elements or components working together to accomplish a desired task or function. FMEA is applied at the system or subsystem level to identify potential failure modes and effects that could negatively impact system or subsystem performance. At the system or subsystem level, FMEA is focused at system or subsystem boundaries where potential failures are most likely to occur. The boundaries of interest for a system or subsystem FMEA include functional (i.e., expected outcomes assuming normal operation) or operational components (i.e., specific outputs expected as compared with tolerances, specifications, and timing).
• Design FMEA. A design—or more accurately, a product design—is a set of specifications that describe all aspects of a product (i.e., major functions, operating parameters and tolerances, materials, dimensions, and so on). FMEA is applied to product designs as early in the product design process as is feasible to identify potential failure modes that could result from a design flaw. Design FMEAs are a normal part of key milestones in the product development process, such as concept reviews, concept approvals, preliminary design reviews, and final design reviews.
• Process FMEA. A process design is a set of specifications that describe all aspects of a process (i.e., functional components, flow rates, process steps, equipment to be used, steps to be performed, operators or
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employees to be involved, etc.). Process design FMEA is applied to process designs at the earliest possible point to identify potential failure modes that could result from a design flaw. Process FMEAs are also a normal part of key milestones in the process development process.
• Service delivery FMEA. A service delivery is the completion of a set of tasks designed to meet one or more customer expectations. Service delivery FMEA is applied to service delivery designs to identify potential failure modes that, if experienced, would result in some level of dissatisfaction from customers. Service delivery FMEAs are also completed as early as possible in the design process and are a normal part of key milestones in the service delivery design process.
In most instances, the practicing QE can be expected to work primarily on design and/or process FMEAs. Accordingly, this chapter focuses on design and pro- cess FMEAs and omits system/subsystem and service delivery FMEAs. Readers are encouraged to reference Stamatis (2003) for a detailed discussion of system/ subsystem and service delivery FMEAs.
B.1.d.ii. Selecting a Standard for FMEA
There are two primary standards for FMEA: the military standard (MIL-STD 1629A) and the Society of Automotive Engineers standard (SAE J1739). Both standards are limited in scope to address only design and process FMEAs. These standards provide general FMEA forms and documents, identify criteria for the quantification of risk associated with potential failures, and provide very gen- eral guidelines on the mechanics of completing FMEAs. MIL- STD 1629A and SAE J1739 may be obtained by contacting the Department of the Navy (http://www. navy.mil) or the Society of Automotive Engineers (http://standards.sae.org), respectively.
Another useful reference is the manual Potential Failure Modes and Effects Analysis, published by the Automotive Industry Action Group (AIAG). This man- ual is available at http://www.aiag.org.
FMEA can also be implemented in other fields, for example, healthcare. The Joint Commission recommends several resources and manuals on FMEA in health- care, which can be found at http://www.jcrinc.com. The reader is encouraged to visit the Joint Commission at http://www.jointcommission.org or the Institute for Healthcare Improvement (IHI) at http://www.ihi.org for more information about FMEA and healthcare research and accreditation.
B.1.d.iii. Planning for an FMEA
Planning for an FMEA involves a series of considerations that include, at a mini- mum, the following:
• Select appropriate applications for the analysis. An FMEA may be authorized by individuals at various levels within an organization or may be required by ISO 9000, QS-9000, APQP, Six Sigma methodologies, internal quality programs, or customer requirements. Whether authorized or required, an FMEA is expensive to complete and should be completed only in those instances where the benefits outweigh the costs.
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• Identify and allocate resources. Resources include FMEA team members and a reporting structure, physical space to conduct the analysis and store documentation, time, and clerical/communications support.
• Define the scope. An FMEA can be conducted at a high level (i.e., the system level) or at a very detailed level (i.e., the component level or service delivery level). Since a high- level FMEA may lead to additional FMEAs at more detailed levels, it is very important to set the scope of the analysis before beginning.
• Establish expectations and deliverables. The team- based nature of completing an FMEA means FMEA team members will have dual or multiple responsibilities and reporting structures in addition to the FMEA team. It is critical, therefore, to clearly define performance expectations for all FMEA team members and to communicate those expectations directly to appropriate supervisory or managerial personnel in reporting structures outside the FMEA team. It is equally important that all FMEA team members understand what deliverables will result from the analysis and their respective roles in developing those deliverables.
• Establish milestones, due dates, and deadlines. Key milestones for an FMEA include receiving authorization for the analysis, establishing a reporting structure, allocating resources (particularly FMEA team members), gathering input for the analysis, completing the analysis, taking and monitoring corrective action, preparing documentation, and completing report- outs and debriefings. To ensure effectiveness, an FMEA should be conducted like a project from the perspective of establishing a schedule specifying due dates and deadlines for each of the major milestones.
• Establish a single point of responsibility. Although FMEA is a team- based analysis, sufficient practical experience supports the idea that assigning responsibility to a cross- functional team rather than a single individual is not the most effective policy. So for a variety of reasons, a single person should be assigned the responsibility of FMEA team leader. That person must have authority to make decisions and allocate resources to complete the FMEA as planned.
B.1.d.iv. FMEA Team Members
The belief that only the one or two people closest to a system, subsystem, product or process design, or service delivery should be assigned to an FMEA violates the very intent of the analysis. FMEA is intended to be completed by team members representing a broad cross section of expertise—technical and nontechnical. For example, an FMEA team should have representation from the following functional groups, as a minimum:
• Design engineering
• Manufacturing engineering
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• Production
• Quality/reliability
• Purchasing/material control
• Sales and marketing
• Customers
It cannot be overemphasized that for an FMEA to be truly effective, the viewpoints and perspectives of every functional group mentioned above must be included— particularly customers. As Palady (1997) explains, “Excluding the customer’s input from the FMEA will result in an incomplete list of the effects and low esti- mates of the severity.”
B.1.d.v. Inputs and Outputs of an FMEA
To prepare for an FMEA, it is necessary to gather information from several sources. These data should be gathered prior to the initial FMEA team meeting in order to maximize the effectiveness of team members’ time. Minimum inputs to an FMEA include the following:
• Process flowchart or functional block diagram
• Design specifications
• Customer requirements/specifications
• Testing data/results
• Data on similar process/design technology
• Warranty data
• Failure/rework data
• Design/configuration change data
• Prior FMEAs
• Results from quantitative analysis (DOE, SPC, process capability assessments, reliability assessments, etc.)
Outputs or deliverables from an FMEA include the following:
• FMEA documentation
• System, subsystem, design, process, and/or service delivery documentation
• Recommendation reports
• Corrective action reports
• Design changes
• Compliance reports
• Debriefings and presentations
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B.1.d.vi. Basic Steps in an FMEA
Complexity in an FMEA is directly related to the number of levels of analysis dic- tated by the situation or team members. At the most fundamental level, however, every FMEA consists of the same basic steps, including:
1. Identify a starting point for the analysis. The starting point will be a system, subsystem, product or process design, or service delivery system of interest.
2. Gather all relevant inputs to support the analysis. Gathering inputs for an FMEA is a milestone to be completed prior to the initial FMEA meeting. It is far more effective, from both cost and efficiency perspectives, to have all team members at meetings participating in the analysis rather than leaving meetings to gather input. Other quality tools are frequently used during the completion of an FMEA. These other quality tools include, but are not limited to, cause- and-effect diagrams, process decision program charts, histograms, Pareto diagrams, run charts, force field analysis, fault tree diagrams, and root cause analysis.
3. Identify potential failure modes, such as:
– Who would be impacted by a failure?
– What would happen in the event of a failure?
– When would the failure occur?
– Where would the failure occur?
– Why would the failure occur?
– How would the failure occur?
4. Quantify the risk associated with each potential failure. Risk assessment is based on severity, occurrence, and detection of a potential failure.
5. Develop a corrective action plan for the most significant risks.
6. Repeat the analysis until all potential failures pose an acceptable level of risk. What constitutes an acceptable risk must be clearly defined by the individual or agent authorizing the FMEA.
7. Document results.
8. Report-out and/or present results.
B.1.d.vii. Design and Process FMEAs
Following the steps previously outlined that describe the planning functions pre- ceding an FMEA, the analysis proceeds as the FMEA team completes appropriate documentation, such as the FMEA form. For purposes of this discussion, one form applicable to either a design or process FMEA will be described. Where the criteria change between a design FMEA and a process FMEA, both criteria will be pro- vided. Figures 7.5 and 7.6 are blank FMEA forms applicable to design and process FMEAs. Each component of the forms is subsequently identified and described.
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Figure 7.5 Blank design FMEA form.
Item
Function
Potential failure mode
Potential effect(s) of
failure
Current design
controls
Responsibility and target completion
date
Potential cause(s)/
mechanism(s) of failure
Recommended action(s)
S e v
C l a s s
O c c u r
D e t e c
R. P. N.
S e v
O c c
D e t
R. P. N.
Action results
Actions taken
System Subsystem Component
Model year(s) vehicle(s) Core team
Design responsibility
Key date
Prepared by
FMEA date (orig.) (rev.)
FMEA number
Page of
Potential Failure Mode and Effects Analysis
(Design FMEA)
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Figure 7.6 Blank process FMEA form.
Process function
Requirements
Potential failure mode
Potential effect(s) of
failure
Current process controls
Responsibility and target completion
date
Potential cause(s)/
mechanism(s) of failure
Recommended action(s)
S e v
C l a s s
O c c u r
D e t e c
R. P. N.
S e v
O c c
D e t
R. P. N.
Action results
Actions taken
Item
Model year(s) vehicle(s) Core team
Process responsibility
Key date
Prepared by
FMEA date (orig.) (rev.)
FMEA number
Page of
Potential Failure Mode and Effects Analysis
(Process FMEA)
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The heading and documentation information of an FMEA is detailed below.
Product or process name Provide the formal and/or commonly used (if different) name for the product or process.
Product or process description
Provide a brief description of the product or process that is meaningful to the FMEA team members.
FMEA number Assign an FMEA number to each FMEA for tracking and documentation purposes. There are no standards for numbering FMEAs; however, a numbering system that links the FMEA to a specific period of time and product/process family is preferred.
Design/process owner Identify the individual or team assigned primary responsibility for the design or process for tracking and documentation purposes. This individual or team is also identified for reference, if needed, during the FMEA.
FMEA team leader Identify the individual assigned primary responsibility for completion of the FMEA for documentation purposes. This individual is also identified so as to establish a point of contact should any stakeholder need information during or after the FMEA.
FMEA team List each member of the FMEA team along with any key responsibilities relative to the FMEA.
FMEA date Provide the date(s) during which the FMEA is completed to help establish a chronology of events. Revision dates should be noted here as well.
FMEA risk assessment Indicate the basis of the risk assessment. The FMEA risk assessment may be based on either actual failures or failure causes. It is important to document the team’s decision to assess risk based on failures or causes to ensure that everyone evaluating the FMEA understands exactly how risk was assessed.
The analysis content and documentation of an FMEA is explained below.
DFMEA part name, number, and function, or PFMEA process function
Identify the product (i.e., part name, number, and function) or process (i.e., functions to be completed as part of the process).
Potential failure mode List each of the potential failure modes associated with the design or process. Design failure modes may include dented, deformed, fractured, loosened, leaking, warped, and so on. Process failure modes may include overheating, inoperable, visual defect, and so on.
Continued
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Potential effect of failure mode
For each potential failure mode, indicate the potential effect on customers or production/process personnel—it is entirely possible to have multiple effects for each potential failure mode.
Severity Indicate the seriousness of the effect of the potential failure using the severity criteria defined in Tables 7.2 and 7.3. Note: The severity rating applies only to the effect of the potential failure.
Classification Classify any special characteristics that may require additional process controls. SAE J1739 identifies classifications that include critical, key, major, and significant.
Potential cause of failure mode
For each potential effect of each failure mode, identify all possible causes—it is entirely possible to have more than one cause for each potential effect.
Occurrence Indicate how frequently each failure is expected to occur using the occurrence criteria defined in Tables 7.4 and 7.5.
DFMEA design verifications or PFMEA process controls
For a design FMEA, identify the actions completed that ensure or verify the adequacy of the design. For a current process FMEA, identify the control currently in place that prevents a failure mode from occurring.
Detection Indicate the ability of design verification or current process controls to detect a potential failure mode in the event that failure actually occurs. Use the detection criteria defined in Tables 7.6 and 7.7.
Risk priority number (RPN)
For each potential failure mode, multiply the severity (S), occurrence (O), and detection (D) assessments together. Since each scale (S, O, and D) ranges from 1 to 10, min(RPN) = 1 and max(RPN) = 1000.
Recommended actions For each potential failure mode, list one or more recommended corrective actions. For further direction and guidance on prioritizing recommended corrective actions, refer to section B.3 of this chapter.
Individual/team responsible and completion date
For each recommended action, assign an appropriate individual or team and an expected completion date.
Actions taken Provide a brief description of the actual actions taken and their respective action dates.
Resulting RPN analysis Following each action taken, reiterate the severity, occurrence, and detection assessments and calculate a new resulting RPN. Actions taken based on RPNs and resulting RPNs continue until the risk assessment for each potential failure is “acceptable” to the customer and/or authorizing agent for the FMEA.
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Table 7.2 Design FMEA severity criteria.
Effect Severity criteria Ranking
Hazardous without warning
Very high ranking when potential failure mode affects safe operation and/or regulation noncompliance. Failure occurs without warning.
10
Hazardous with warning
Very high ranking when potential failure mode affects safe operation and/or regulation noncompliance. Failure occurs with warning.
9
Very high Item or product is inoperable, with loss of function. Customer very dissatisfied.
8
High Item or product is operable, with loss of performance. Customer dissatisfied.
7
Moderate Item or product is operable, but comfort/convenience items inoperable. Customer experiences discomfort.
6
Low Item or product is operable, but with loss of performance of comfort/convenience items. Customer has some dissatisfaction.
5
Very low Certain characteristics do not conform. Noticed by most customers.
4
Minor Certain characteristics do not conform. Noticed by average customers.
3
Very minor Certain characteristics do not conform. Noticed by discriminating customers.
2
None No effect. 1
S × O × D = risk priority number (RPN) Derived from Technical Standard SAE J 1739. Reprinted by permission of The Society of Automotive Engineers (SAE).
Table 7.3 Process FMEA severity criteria.
Effect Severity criteria Ranking
Hazardous without warning
May endanger machine or assembly operator. Very high severity ranking when a potential failure mode affects safe operation and/or involves noncompliance with regulation. Failure will occur without warning.
10
Hazardous with warning
May endanger machine or assembly operator. Very high severity ranking when a potential failure mode affects safe operation and/or involves noncompliance with regulation. Failure will occur with warning.
9
Very high Major disruption to production line. 100% of product may have to be scrapped. Item inoperable, loss of primary function. Customer very dissatisfied.
8
Continued
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Table 7.4 Design FMEA occurrence criteria.
Probability of failure Possible failure rates Ranking
Very high: Failure almost inevitable > 1 in 2 1 in 3
10 9
High: Repeated failures 1 in 8 1 in 20
8 7
Moderate: Occasional failures 1 in 80 1 in 400 1 in 2000
6 5 4
Low: Relatively few failures 1 in 15,000 1 in 150,000
3 2
Remote: Failure is unlikely < 1 in 1,500,000 1
Derived from Technical Standard SAE J 1739. Reprinted by permission of The Society of Automotive Engineers (SAE).
Table 7.3 Process FMEA severity criteria. (Continued)
Effect Severity criteria Ranking
High Minor disruption to production line. A portion of product may have to be sorted and scrapped. Item operable, but at reduced level. Customer dissatisfied.
7
Moderate Minor disruption to production line. A portion of product may have to be scrapped (no sorting). Item operable, but some comfort items inoperable. Customer experiences discomfort.
6
Low Minor disruption to production line. 100% of product may have to be reworked. Item operable, but some comfort items operable at reduced level of performance. Customer experiences some dissatisfaction.
5
Very low Minor disruption to production line. Product may have to be sorted and a portion reworked. Minor adjustments do not conform. Defect noticed by customer.
4
Minor Minor disruption to production line. Product may have to be reworked online, but out of station. Minor adjustments do not conform. Defect noticed by average customer.
3
Very minor Minor disruption to production line. Product may have to be reworked online, but out of station. Minor adjustments do not conform. Defect noticed by discriminating customer.
2
None No effect. 1
Derived from Technical Standard SAE J 1739. Reprinted by permission of The Society of Automotive Engineers (SAE).
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Table 7.5 Process FMEA occurrence criteria.
Probability of failure Possible failure rates Ranking
Very high: Failure almost inevitable. > 1 in 2 1 in 3
10 9
High: Generally associated with processes similar to previous processes that have often failed.
1 in 8 1 in 20
8 7
Moderate: Generally associated with processes similar to previous processes that have experienced occasional failures.
1 in 80 1 in 400 1 in 2000
6 5 4
Low: Isolated failures associated with similar processes. 1 in 15,000 3
Very low: Only isolated failures associated with almost identical processes.
1 in 150,000 2
Remote: Failure is unlikely. No failures associated with almost identical processes.
< 1 in 1,500,000 1
Derived from Technical Standard SAE J 1739. Reprinted by permission of The Society of Automotive Engineers (SAE).
Table 7.6 Design FMEA detection criteria.
Effect Detection criteria Ranking
Absolute Design control will not and/or cannot detect a potential cause/ uncertainty mechanism and subsequent failure mode or there is no design control.
10
Very remote Very remote chance the design control will detect a potential cause/mechanism and subsequent failure mode.
9
Remote Remote chance the design control will detect a potential cause/ mechanism and subsequent failure mode.
8
Very low Very low chance the design control will detect a potential cause/ mechanism and subsequent failure mode.
7
Low Low chance the design control will detect a potential cause/ mechanism and subsequent failure mode.
6
Moderate Moderate chance the design control will detect a potential cause/ mechanism and subsequent failure mode.
5
Moderately high
Moderately high chance the design control will detect a high potential cause/mechanism and subsequent failure mode.
4
High High chance the design control will detect a potential cause/ mechanism and subsequent failure mode.
3
Very high Very high chance the design control will detect a potential cause/ mechanism and subsequent failure mode.
2
Almost Design control will almost certainly detect a potential cause/ certain mechanism and subsequent failure mode.
1
Derived from Technical Standard SAE J 1739. Reprinted by permission of The Society of Automotive Engineers (SAE).
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Figures 7.7 and 7.8, showing examples of design and process FMEAs, have been provided to help guide the reader through an actual analysis.
B.1.e. Failure Mode Effects and Criticality Analysis (FMECA)
MIL-STD 1629A defines two very important terms and concepts with respect to risk assessment. Criticality is “a relative measure of the consequences of a fail- ure mode and its frequency of occurrences.” Criticality analysis is “a procedure by which each potential failure mode is ranked according to the combined influence of severity and probability of occurrence” (MIL-STD-1629A 1980). Note that while this military standard was canceled in 1998, other industry standards for FMEA and FMECA exist—for example, AIAG FMEA-4, SAE J1739, and IEC 60812.
When criticality is considered in an FMEA, the name is changed to failure mode effects and criticality analysis (FMECA). FMECA can be a qualitative or quantitative assessment of risk that leads to a prioritization of corrective action based on sever- ity (S) and occurrence (O) assessments. In the qualitative approach to risk assess- ment in FMECA, risk is categorized as frequent, reasonably probable, occasional, remote, or extremely unlikely. In the quantitative approach to risk assessment in FMECA, failure rate data, failure effect probability data, individual part failure data, and operating time data are required as input to one or more protocols as defined in Military Handbook 217.
Table 7.7 Process FMEA detection criteria.
Effect Detection criteria Ranking
Absolutely impossible
No known controls to detect failure mode. 10
Very remote Very remote likelihood current controls will detect failure mode. 9
Remote Remote likelihood current controls will detect failure mode. 8
Very low Very low likelihood current controls will detect failure mode. 7
Low Low likelihood current controls will detect failure mode. 6
Moderate Moderate likelihood current controls will detect failure mode. 5
Moderately high
Moderately high likelihood current controls will detect failure mode.
4
High High likelihood current controls will detect failure mode. 3
Very high Very high likelihood current controls will detect failure mode. 2
Almost certain
Current controls will almost certainly detect a failure mode. Reliable detection controls are known with similar processes.
1
Derived from Technical Standard SAE J 1739. Reprinted by permission of The Society of Automotive Engineers (SAE).
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Figure 7.7 Design FMEA example.
Item
Function
Potential failure mode
Potential effect(s) of
failure
Current design
controls
Responsibility and target completion
date
Potential cause(s)/
mechanism(s) of failure
Recommended action(s)
S e v
C l a s s
O c c u r
D e t e c
R. P. N.
S e v
O c c
D e t
R. P. N.
Action results
Actions taken
System Subsystem Component
Model year(s)/vehicle(s) Core team
Design responsibility
Key date
Prepared by
FMEA date (orig.) (rev.)
FMEA number
Page of
Potential Failure Mode and Effects Analysis
(Design FMEA) x
01.03/Body closures
199X/Lion 4dr/wagon
T. Fender—Car product dev., Childers—Manufacturing, J. Ford—Assy ops (Dalton, Fraser, Henley assembly plants)
2
5
Body engineering
9X 03 01 ER
3
6
A. Tate—X6412—Body engr
8X 03 22
1234
11
8X 07 14
1
4
7
8
9
10 11 16 19 20 21
22
14
12 13 1715 18
Front door L.H. H8HX-0000-A
• Ingress to and egress from vehicle • Occupant protection from weather, noise, and side impact • Support anchorage for door hardware including mirror, hinges, latch, and window regulator • Provide proper surface for appearance items • Paint and soft trim
Deteriorated life of door leading to:
• Unsatisfactory appearance due to rust through paint over time • Impaired function of interior door hardware
Upper edge of protective wax application specified for inner door panels is too low
Vehicle general durability test vah. T-118 T-109 T-301
Add laboratory accelerated corrosion testing
A Tate-Body Engrg 8X 09 30
Based on test results (Test No. 1481) upper edge spec raised 125mm
7 2 2 286 7 2947Corroded interior lower door panels
Insufficient wax thickness specified
Vehicle general durability testing - as above
Add laboratory accelerated corrosion testing
Conduct Design of Experiments (DOE) on wax thickness
Combine w/test for wax upper edge verification
A Tate Body Engrg 9X 01 15
Test results (Test No. 1481) show specified thickness is adequate. DOE shows 25% variation in specified thickness is acceptable
7 2 2 284 7 196
Insufficient room between panels for spray head access
Drawing evaluation of spray head access
Add team evaluation using design aid buck and spray head
Body Engrg & Assy Ops
Evaluation showed adequate access
7 1 1 74 4 112
Wax application plugs door drain holes
Laboratory test using "worst case" wax application and hole size
None Based on test, 3 additional vent holes provided in affected areas
3 1 21
2 28
Entrapped air prevents wax from entering corner/edge access
Design aid investigation with non-functioning spray head
Add team evaluation using production spray equipment and specified wax
Body Engrg & Assy Ops 8X 11 15
7 1 3 215 8 280
Inappropriate wax formulation specified
Physical and Chem Lab test - Report No. 1265
None2
SAMPLE
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Process function
Requirements
Potential failure mode
Potential effect(s) of
failure
Current process controls
Responsibility and target completion
date
Potential cause(s)/
mechanism(s) of failure
Recommended action(s)
S e v
C l a s s
O c c u r
D e t e c
R. P. N.
S e v
O c c
D e t
R. P. N.
Action results
Actions taken
Item
Model year(s) vehicle(s) Core team
Process responsibility
Key date
Prepared by
FMEA date (orig.) (rev.)
FMEA number
Page of
Potential Failure Mode and Effects Analysis
(Process FMEA)
Front door L.H./H8HX-000-A
199X/Lion 4dr/wagon
A. Tate—Body engrg., J. Smith—OC, R. James—Production, J. Jones—Maintenance
2
5
Body engrg./assembly operations
9X 03 01 ER 9X 08 26 Job #1
3
6
J. Ford—X6521—Assy ops
9X 05 17
1450
11
9X 11 06
1
4
7
8
9
10 11 16 19 20 21
22
14
12 13 1715 18
Manual application of wax inside door
To cover inner door, lower surfaces at minimum wax thickness to retard corrosion
Deteriorated life of door leading to:
• Unsatisfactory appearance due to rust through paint over time • Impaired function of interior door hardware
Manually inserted spray head not inserted far enough
Visual check each hour- 1/shift for film thickness (depth meter) and coverage
Add positive depth stop to sprayer
Automate spraying
MFG Engrg 9X 10 15
Mfg Engrg 9X 12 15
Stop added, sprayer checked on line
Rejected due to complexity of different doors on same line
7 2 5 708 5 2807Insufficient wax coverage over specified surface
Spray heads clogged • Viscosity too high • Temperature too low • Pressure too low
Test spray pattern at start-up and after idle periods, and preventative maintenance program to clean heads
Use Design of Experiments (DOE) on viscosity vs. temperature vs. pressure
Mfg Engrg 9X 10 01
Temp and press limits were determined and limit controls have been installed - control charts show process is in control Cpk=1.85
Automatic spray timer installed - operator starts spray, timer controls shut-off control charts show process is in control Cpk=2.05
7 1 3 215 3 105
2 28
Spray time insufficient Operator instructions and lot sampling (10 doors / shift) to check for coverage of critical areas
Install spray timer
Maintenance 9X 09 15
7 1 7 498 7 392
Spray head deformed due to impact
Preventative maintenance programs to maintain head
None2
SAMPLE
Figure 7.8 Process FMEA example.
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6 /2
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7 1
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9 A
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The key result of an FMECA is a criticality matrix that ranks potential failures with respect to severity. The matrix then identifies a prioritization scheme for cor- rective actions based on the severity of potential failure modes. As displayed in an FMECA criticality matrix, potential failures plotted farther away from the matrix origin on a diagonal line represent higher potential risks of failure, and thus war- rant increased need for corrective action.
ExaMpLE 7.2
Figure 7.9 shows an example of an FMECA for a traveling lawn sprinkler. Note that each hardware item is listed on a separate line. For each possible failure, its effect on the prod- uct is determined. Type of failure is also shown along with estimates for its probability of occurrence and for its seriousness (Gryna, Chua, and Defeo 2007). For additional discus- sion of FMECA, consult MIL-STD-1629A for guidance in completing a criticality assessment.
Designing for Quality
1 = Very low (<1 in 1000) 2 = Low (3 in 1000) 3 = Medium (5 in 1000) 4 = High (7 in 1000) 5 = Very high (>9 in 1000)
Component part number
Worm bearing 4224
Bearing worn Not aligned with bottom housing
Spray head wobbles or slows down
Improve inspection
M 1 4
Zytel 101 Excessive spray head wobble
Spray head wobbles or slows down
Improve worm bearing
M 1 3
Bearing stem 4225
Excessive wear
Poor bearing/ material combination
Spray head wobbles and loses power
Change stem material
M 5 4
Brass Dirty water in bearing area
Spray head wobbles and loses power
Improve worm seal area
M 5 4
Excessive spray head wobble
Spray head wobbles and loses power
Improve operating instructions
M 2 3
Thrust washer 4226
Excessive wear
High water pressure
Spray head will stall out
Inform customer in instructions
M 2 5
Fulton 404 Dirty water in washers
Spray head will stall out
Improve worm seal design
M 5 5
Worm 4527 Excessive wear in bearing area
Poor bearing/ material combination
Spray head wobbles and loses power
Change bearing stem material
M 5 4
Brass Dirty water in bearing area
Spray head wobbles and loses power
Improve worm seal design
M 5 4
Excessive spray head wobble
Spray head wobbles and loses power
Improve operating instructions
M 2 3
Possible failure
Cause of failure
Effect of failure on product
AlternativesT P S
T = Type of failure P = Probability of occurrence S = Seriousness of failure of system H = Hydraulic failure M = Mechanical failure W = Wear failure C = Customer abuse
Product HRC-1
Date Jan. 14, 2017
By S.M.
Figure 7.9 Failure mode effects and criticality analysis.
B.2. analysis
Analysis of risk involves determining both the probability of occurrence and the severity of potential harm of the risk event. Once these values are determined, they
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are combined and assessed in order to go forward with risk control strategies. Probability theory can be used to determine the probability of a risk event. The reader should refer to Chapter 6 for how to compute probabilities of event occur- rence. Additionally, refer to Tables 7.2–7.5 for guidelines on determining sever- ity and occurrence criteria. In this section, we discuss computing the risk priority number and risk matrices as formal techniques for further analyzing and catego- rizing risk events.
B.2.a. Risk Priority Number
For purposes of FMEA, risk has three components: severity, occurrence, and detec- tion. Each of these components is assigned a value, and the values are multiplied to produce the risk priority number (RPN). Refer to Tables 7.2–7.5 for guidelines on how to rate each of these components.
Severity (S) is an indicator of the severity of a failure should the failure occur. Occurrence (O) is an indicator of the likelihood of a failure occurring. Detection (D) is an indicator of the likelihood of detecting a failure once it has occurred. Each of these indicators is described on a 10-point scale. RPN is calculated as:
RPN = (S)(O)(P) (7.2)
From Equation (7.2), the minimum RPN is 1 while the maximum RPN is 1000.
B.2.b. Risk matrix
A risk matrix is a qualitative evaluation method that can assist management in decision making. It can be used to define various levels of risk based on how likely they are to occur and the severity associated with them. The matrix is displayed as a table in which the severity of risk categories are placed in rows, the probability of occurrence placed in columns, and the risk events placed in the appropriate cells. The number of severity and probability categories can be determined by the risk management team and is subjective. The qualitative assessment, where the degree of risk or harm is given as catastrophic, critical, major, minor, or negligible, repre- sents the severity or potential harm.
The Delphi method, originally developed at RAND (Dalkey 1967), could be used as a slightly more formal method for achieving consensus regarding risk severity and probability of occurrence. Delphi is the “set of procedures for eliciting and refining the opinions of a group of people” (Dalkey 1967). The Delphi method requires a group of experts (these could be people in the risk management team and upper- level management within the organization) to anonymously reply to a ques- tionnaire regarding issues of importance and/or likelihood of event outcomes. Sta- tistical representation of the responses is used for representing the “group response.”
Risk matrices should be used carefully and with caution because they require subjective input and interpretation. Their subjective nature means that they can contain errors, have poor resolution, and be used differently by different teams unless the categories are well defined and documented. See, for example, Cox (2008). With this caveat, it is important to note that there are some standard risk matrices used for different contexts within large organizations such as the Depart- ment of Defense (DoD) and NASA.
An example of a risk matrix is shown in Table 7.8. This matrix was created to identify the type and severity of issues affecting a class III medical device.
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Table 7.8 Risk assessment score matrix.
Criterion Category
Risk score
Negligible or low (1)
Medium (2)
High (3)
Safety Critical or catastrophic ✓
Marginal ✓
Negligible ✓
Product classification
Device class I ✓
Device class II ✓
Device class III ✓
Intravenous drug or sterile product ✓
Drug with narrow therapeutic ranges ✓
Other drug products ✓
Reliability or effectiveness
Totally affected ✓
Partially affected ✓
Not affected ✓
Product specification
Final specification failure ✓
Nonfinal specification failure ✓
Specifications are not affected ✓
Product labeling
Final product labels ✓
Nonfinal product labels ✓
No labeling is affected ✓
Frequency or trending
First-time occurrence (isolated event) ✓
Occasional but improving ✓
Occasional but worsening ✓
Frequent ✓
Detectability Not detectable or not detected
Detected by chance ✓ ✓
Detected by the regular process ✓
Regulatory risk
Product can be considered adulterated or misbranded
✓
Product is not adulterated or misbranded ✓
Source: Reprinted by permission from J. Rodriguez-Perez, Quality Risk Management in the FDA-Regulated Industry (Milwaukee, WI: ASQ Quality Press, 2012).
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Class III medical devices are generally the highest- risk devices and are subject to the highest level of regulatory control (US FDA, “About FDA,” http://www.fda. gov/AboutFDA/Transparency/Basics/ucm194438.htm). An example of a class III medical device is a replacement heart valve.
B.3. Evaluation
Risk evaluation is used to make decisions about which risks need to be addressed based on the estimate of metrics identified in risk analysis. This evaluation may be quantitative, qualitative, or both. Once quantitative estimates of the metrics (e.g., RPNs in an FMEA) have been computed, they should be compared with the level of risk or risk criteria already in place. This can be used to determine whether the level of risk is acceptable. Qualitative assessment of risk can include determining the level of severity or potential harm (e.g., catastrophic, critical, or minimal) as in a risk matrix.
A common mistake in assessing FMEA risk is prioritizing corrective action based on the descending order of RPNs. Logic would suggest that the largest RPNs represent the highest risk, which is true, but only to a point. When multi- plying the three risk components together, their importance relative to each other becomes obscured. Consider the following RPNs in Table 7.9, calculated using Equation (7.2).
In each case the resulting RPN is 100, so it is unclear which potential failure to take corrective action on first. There is, however, a generally accepted strategy for taking action on an RPN. Palady (1997) describes this strategy as follows:
1. Eliminate the occurrence
2. Reduce the severity
3. Reduce the occurrence
4. Improve detection
Applying this strategy shows us how to proceed. Eliminating occurrences would, mathematically, reorder the RPNs. Reducing severity next would focus our atten- tion on potential failures 2 and 4. But then what? We still have two potential failures with the same level of risk. Reducing occurrence as the next step in this process focuses our attention on potential failure 4, which had a higher occurrence rating than potential failure 2.
Now our attention can turn to evaluating the remaining potential failures since potential failures 2 and 4 have been ranked as the two most important. Of
Table 7.9 Example RPNs for several potential failures.
S O D RPN
Potential failure 1 2 10 5 100
Potential failure 2 10 2 5 100
Potential failure 3 2 5 10 100
Potential failure 4 10 5 2 100
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the remaining two potential failures, potential failure 1 has the higher occurrence rating and is therefore ranked as the third most important potential failure, and potential failure 3 drops to the least important position by default. The rank order by which the potential failures in the above example should be investigated for corrective action is as follows:
First priority Potential failure 4
Second priority Potential failure 2
Third priority Potential failure 1
Fourth priority Potential failure 3
A common point of confusion arises when considering what is actually rated as part of the risk assessment: the actual failure itself or the cause of a given fail- ure. It is perfectly acceptable to rate either the failure or the cause, as long as the assumption is well documented (on actual FMEA charts, in written correspon- dence, and in all reports/presentations) and everyone on the FMEA team and in the reporting structure is aware of the assumption. Whether rating a failure itself or a cause of that failure, an FMEA should provide consistent results and correc- tive actions.
An FMEA represents an in- depth, objective, quantitative analysis of the risk associated with potential failures that result in the calculation of one or more RPNs. Once RPNs have been calculated and the FMEA team prepares to take corrective action, the analysis necessarily takes on a subjective element as FMEA team members use the risk assessment to guide prioritization of corrective actions.
As was mentioned earlier, the most common practice is to prioritize corrective action based on RPNs. Prioritizing corrective action based solely on RPNs works effectively, however, only as long as there is a “comfortable” difference among the RPN values. When there are clusters of RPN values that are the same, or very close (i.e., within 25–50 points), taking action based on RPNs alone is not straightfor- ward. When there are clusters of RPN values (i.e., grouping of RPN values that are the same or within a 25–50 point range), follow these steps to prioritize corrective action:
1. Rank the RPNs in descending order
2. For those RPNs that cluster within a predefined range, for example, 25–50 points, eliminate occurrence, then reduce severity, then reduce occurrence, then improve detection
3. Plan, take, and monitor corrective action on the largest nonclustered RPNs
4. Plan, take, and monitor corrective action on RPN clusters as defined in step 2
Repeat steps 3 and 4 as needed to address all potential failures identified in the analysis.
As another means of eliminating the subjectivity in prioritizing corrective actions based on RPNs, a method called criticality analysis was developed as part of MIL- STD-1629A.
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C. riSK ConTroL Risk mitigation includes the act of attempting to minimize the risk’s impact on the system, process, or product.
C.1. Mitigation Strategies
A risk mitigation plan is implemented once risks have been identified. Risk miti- gation includes the act of attempting to lessen the risk’s impact on the system, process, or product. Mitigation options include the following:
• Eliminating the risk
• Minimizing the risk
• Accepting the risk
• Monitoring the risk
Eliminating risk means that the risk must be removed completely. This can be in the form of not choosing a particular option during the design phase, fixing a current problem, switching a supplier, and so on. Minimizing or reducing risk can be accomplished by implementing actions that minimize the likelihood of the risk occurring. Methods to minimize risk in the design of a process or product, for example, may be to build a redundant or back- up system. Employing robust parameter design is another way to minimize certain risks. Simulations such as Monte Carlo simulations can also be used to help better understand the likelihood of certain risks occurring during the design phase of a product or process. For more information on Monte Carlo simulations, see Givens and Hoeting (2012) and Robert and Casella (2009).
Risk acceptance is another possible decision in handling risks. Risk acceptance means taking deliberate ownership of a particular risk and not taking action to avoid, control, or mitigate the risk. This choice is not recommended if there are resources available to control or mitigate a given risk. However, it may be more appropriate to accept low- priority risks in order to direct resources to those risks with higher priorities or urgencies. The continuous nature of risk management is important to remember here as accepted risks originally deemed low priority may become more severe over time. Strategies must be in place to monitor and adjust decisions as the risks change.
Finally, there are some risks that may need to be watched or monitored and assessed further at a later time. For example, sometimes assumptions are made during the risk assessment process that may no longer be valid.
C.2. risk Control and documentation
The primary goal of risk control is to maintain the level of risk at or below an agreed upon acceptable level. This determination should be made by key stake- holders and involve the risk management team. Risk control is made up of two key components: risk reduction and risk acceptance. Risk reduction focuses on the avoidance of risk, and risk acceptance focuses on the acceptance of the risk events identified. There are times when a risk treatment does not eliminate a risk, but
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only minimizes it until the risk is at or below an acceptable level. This remaining risk is called residual risk. When controlling risk, a gap occurs when the control in place is not adequate for the risk, or no control has been put in place for a signifi- cant risk. Redundancy in risk control occurs when there is more than one control in place to address a significant risk.
Documentation is an important part of the risk management process. Docu- mentation should include information on the initial risk plan, analysis of risk, mitigation plan, and risk control. Elements of the documentation should include identifying new or upcoming risks, obtaining new information to update risk levels and risk criteria, and assessing the controls in place to make sure they are working and appropriate. Once documented, an ongoing process to maintain information regarding risk should be used and assessed periodically to ensure it is up to date.
Documentation aids in monitoring the risk management process. Controls and risk should be monitored to determine whether risk assessment techniques are being appropriately implemented and the expected results of the risk man- agement process are being maintained. Additionally, it is important to determine whether risk treatments are still effective, which can partially be determined in the auditing and testing phase.
C.3. auditing and Testing
Risk auditing is performed to verify that the known sources of risk are under con- trol. Risk auditing is also often called risk monitoring. Audits are used to verify that risk treatment and mitigation plans are effective. They can also be helpful in identifying new risks. A common method of verification is to examine cur- rent documents and records. This emphasizes the fact that risk management is an ongoing process; documentation resulting from methods such as an FMEA must be continuously maintained and updated as the process and/or treatment plans change. Another way to verify how risks are being controlled is by observing the process in person, interviewing subjects, or testing the process or product to ensure the appropriate mitigation plans are in place (Russell 2013). Goals when testing the process or product include ensuring that risk assessment results match or mirror the actual results. Refer to Chapter 2, section D, for more information on auditing. In addition, ISO 31000:2009, Risk management—principles and guidelines, is an international standard on risk management. While organizations cannot be certified under this standard, it does provide guidance on internal and external audit programs.
For additional information on risk management and risk analysis, see Pinto and Garvey (2012) and Wang and Roush (2000). Luko (2013) discusses common risk management terminology and standards for risk management.
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Variables Charts
x R Averages Chart x A R Range Chand chart: : ± 2 aart D R D R
x s Averages
: LCL UCL
and chart:
= =3 4 CChart x A s Std Dev Chart B s B: . . : LCL UCL± = =3 3 4 ss
Individuals and Moving Range Chart (two-vaalue moving window):
Individuals Chart x: ± 2.. : .66 3 267R Moving Range RUCL
Moving Average
=
and Moving Range (two-value moving window)):
UCLMoving Average x R Moving Range: . :± 1 88 == 3 267. R
attributes Charts
p p p p
n
np np np p
c
chart:
chart: 3
cha
± −( )
± −( )
3 1
1
rrt:
chart:
c c
u u u n
±
±
3
3
appendix a Control Limit Formulas
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appendix b Constants for Control Charts
S u b g ro u p size
n
A2 for median charts
2 1 . 8 8 0 1 . 1 2 8 0 3 . 2 6 7 2 . 6 5 9 0 . 7 9 8 0 3 . 2 6 7 2 . 6 6 0 1 . 8 8 0
3 1 . 0 2 3 1 . 6 9 3 0 2 . 5 7 4 1 . 9 5 4 0 . 8 8 6 0 2 . 5 6 8 1 . 7 7 2 1 . 1 8 7
4 0 . 7 2 9 2 . 0 5 9 0 2 . 2 8 2 1 . 6 2 8 0 . 9 2 1 0 2 . 2 6 6 1 . 4 5 7 0 . 7 9 6
5 0 . 5 7 7 2 . 3 2 6 0 2 . 1 1 4 1 . 4 2 7 0 . 9 4 0 0 2 . 0 8 9 1 . 2 9 0 0 . 6 9 1
6 0 . 4 8 3 2 . 5 3 4 0 2 . 0 0 4 1 . 2 8 7 0 . 9 5 2 0 . 0 3 0 1 . 9 7 0 1 . 1 8 4 0 . 5 4 8
7 0 . 4 1 9 2 . 7 0 4 0 . 0 7 6 1 . 9 2 4 1 . 1 8 2 0 . 9 5 9 0 . 1 1 8 1 . 8 8 2 1 . 1 0 9 0 . 5 0 8
8 0 . 3 7 3 2 . 8 4 7 0 . 1 3 6 1 . 8 6 4 1 . 0 9 9 0 . 9 6 5 0 . 1 8 5 1 . 8 1 5 1 . 0 5 4 0 . 4 3 3
9 0 . 3 3 7 2 . 9 7 0 0 . 1 8 4 1 . 8 1 6 1 . 0 3 2 0 . 9 6 9 0 . 2 3 9 1 . 7 6 1 1 . 0 1 0 0 . 4 1 2
1 0 0 . 3 0 8 3 . 0 7 8 0 . 2 2 3 1 . 7 7 7 0 . 9 7 5 0 . 9 7 3 0 . 2 8 4 1 . 7 1 6 0 . 9 7 5 0 . 3 6 2
E2B4B3C4A3D4D3d2A2
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One-sided tolerance Confidence level
Two-sided tolerance Confidence level
n 0.90 0.95 0.99 n 0.90 0.95 0.99
10 3.532 3.981 5.075 10 3.959 4.433 5.594
11 3.444 3.852 4.828 11 3.849 4.277 5.308
12 3.371 3.747 4.633 12 3.758 4.150 5.079
13 3.310 3.659 4.472 13 3.682 4.044 4.893
14 3.257 3.585 4.336 14 3.618 3.955 4.737
15 3.212 3.520 4.224 15 3.562 3.878 4.605
16 3.172 3.463 4.124 16 3.514 3.812 4.492
17 3.136 3.415 4.038 17 3.471 3.754 4.393
18 3.106 3.370 3.961 18 3.433 3.702 4.307
19 3.078 3.331 3.893 19 3.399 3.656 4.230
20 3.052 3.295 3.832 20 3.368 3.615 4.161
21 3.028 3.262 3.776 21 3.340 3.577 4.100
22 3.007 3.233 3.727 22 3.315 3.543 4.044
23 2.987 3.206 3.680 23 3.292 3.512 3.993
24 2.969 3.181 3.638 24 3.270 3.483 3.947
25 2.952 3.158 3.601 25 3.251 3.457 3.904
30 2.884 3.064 3.446 30 3.170 3.350 3.733
40 2.793 2.941 3.250 40 3.066 3.213 3.518
50 2.735 2.863 3.124 50 3.001 3.126 3.385
appendix C Statistical Tolerance Factors
for at Least 99% of the Population
(“k-Values”)
H1518_Burke_appendix.indd 527 6/29/17 11:52 AM
528
0 0.5000000 0.5000000 500000.0002
0.1 0.5398279 0.4601721 460172.1045
0.2 0.5792597 0.4207403 420740.3128
0.3 0.6179114 0.3820886 382088.6425
0.4 0.6554217 0.3445783 344578.3034
0.5 0.6914625 0.3085375 308537.5326
0.6 0.7257469 0.2742531 274253.0649
0.7 0.7580364 0.2419636 241963.5785
0.8 0.7881447 0.2118553 211855.3339
0.9 0.8159399 0.1840601 184060.0917
1 0.8413447 0.1586553 158655.2598
1.1 0.8643339 0.1356661 135666.1015
1.2 0.8849303 0.1150697 115069.7317
1.3 0.9031995 0.0968005 96800.5495
1.4 0.9192433 0.0807567 80756.71126
1.5 0.9331928 0.0668072 66807.22879
1.6 0.9452007 0.0547993 54799.28945
1.7 0.9554346 0.0445654 44565.43178
1.8 0.9640697 0.0359303 35930.26551
1.9 0.9712835 0.0287165 28716.49286
2 0.9772499 0.0227501 22750.06204
2.1 0.9821356 0.0178644 17864.35742
2.2 0.9860966 0.0139034 13903.39891
2.3 0.9892759 0.0107241 10724.08106
2.4 0.9918025 8.1975289E-03 8197.528869
2.5 0.9937903 6.2096799E-03 6209.679859
2.6 0.9953388 4.6612218E-03 4661.221783
Area to left of ZZ
Area to right of Z
Parts per million right of Z
Continued
appendix D Standard Normal Distribution
for Selected Z-Values
H1518_Burke_appendix.indd 528 6/29/17 11:52 AM
Appendix D Standard Normal Distribution for Selected Z-Values 529
2 . 7 0 . 9 9 6 5 3 3 0 3 . 4 6 7 0 2 3 1 E - 0 3 3 4 6 7 . 0 2 3 0 5 3
2 . 8 0 . 9 9 7 4 4 4 8 2 . 5 5 5 1 9 0 6 E - 0 3 2 5 5 5 . 1 9 0 6 4 2
2 . 9 0 . 9 9 8 1 3 4 1 1 . 8 6 5 8 8 0 1 E - 0 3 1 8 6 5 . 8 8 0 1 4
3 0 . 9 9 8 6 5 0 0 1 . 3 4 9 9 6 7 2 E - 0 3 1 3 4 9 . 9 6 7 2 2 3
3 . 1 0 . 9 9 9 0 3 2 3 9 . 6 7 6 7 1 2 4 E - 0 4 9 6 7 . 6 7 1 2 3 5 6
3 . 2 0 . 9 9 9 3 1 2 8 6 . 8 7 2 0 2 0 8 E - 0 4 6 8 7 . 2 0 2 0 8 0 8
3 . 3 0 . 9 9 9 5 1 6 5 4 . 8 3 4 8 2 5 4 E - 0 4 4 8 3 . 4 8 2 5 3 6 6
3 . 4 0 . 9 9 9 6 6 3 0 3 . 3 6 9 8 0 8 2 E - 0 4 3 3 6 . 9 8 0 8 2 2 9
3 . 5 0 . 9 9 9 7 6 7 3 2 . 3 2 6 7 3 3 7 E - 0 4 2 3 2 . 6 7 3 3 7 3 7
3 . 6 0 . 9 9 9 8 4 0 9 1 . 5 9 1 4 5 7 1 E - 0 4 1 5 9 . 1 4 5 7 1 3 8
3 . 7 0 . 9 9 9 8 9 2 2 1 . 0 7 8 3 0 1 5 E - 0 4 1 0 7 . 8 3 0 1 4 5 4
3 . 8 0 . 9 9 9 9 2 7 6 7 . 2 3 7 2 4 3 4 E - 0 5 7 2 . 3 7 2 4 3 4 2 7
3 . 9 0 . 9 9 9 9 5 1 9 4 . 8 1 1 5 5 1 9 E - 0 5 4 8 . 1 1 5 5 1 8 8 7
4 0 . 9 9 9 9 6 8 3 3 . 1 6 8 6 0 3 5 E - 0 5 3 1 . 6 8 6 0 3 4 6 1
4 . 1 0 . 9 9 9 9 7 9 3 2 . 0 6 6 8 7 1 6 E - 0 5 2 0 . 6 6 8 7 1 5 7 7
4 . 2 0 . 9 9 9 9 8 6 6 1 . 3 3 5 4 0 9 7 E - 0 5 1 3 . 3 5 4 0 9 7 3 3
4 . 3 0 . 9 9 9 9 9 1 5 8 . 5 4 6 0 2 1 2 E - 0 6 8 . 5 4 6 0 2 1 1 9 1
4 . 4 0 . 9 9 9 9 9 4 6 5 . 4 1 6 9 5 3 1 E - 0 6 5 . 4 1 6 9 5 3 0 5 4
4 . 5 0 . 9 9 9 9 9 6 6 3 . 4 0 0 8 0 3 1 E - 0 6 3 . 4 0 0 8 0 3 0 6 2
4 . 6 0 . 9 9 9 9 9 7 9 2 . 1 1 4 6 4 3 4 E - 0 6 2 . 1 1 4 6 4 3 3 7 6
4 . 7 0 . 9 9 9 9 9 8 7 1 . 3 0 2 3 1 5 7 E - 0 6 1 . 3 0 2 3 1 5 6 5 4
4 . 8 0 . 9 9 9 9 9 9 2 7 . 9 4 3 5 2 6 7 E - 0 7 0 . 7 9 4 3 5 2 6 6 9
4 . 9 0 . 9 9 9 9 9 9 5 4 . 7 9 8 6 9 5 5 E - 0 7 0 . 4 7 9 8 6 9 5 4 7
5 0 . 9 9 9 9 9 9 7 2 . 8 7 1 0 5 0 0 E - 0 7 0 . 2 8 7 1 0 5
5 . 1 0 . 9 9 9 9 9 9 8 1 . 7 0 1 2 2 3 1 E - 0 7 0 . 1 7 0 1 2 2 3 1 4
5 . 2 0 . 9 9 9 9 9 9 9 9 . 9 8 3 4 4 0 0 E - 0 8 0 . 0 9 9 8 3 4 4
5 . 3 0 . 9 9 9 9 9 9 9 5 . 8 0 2 2 0 6 6 E - 0 8 0 . 0 5 8 0 2 2 0 6 6
5 . 4 1 . 0 0 0 0 0 0 0 3 . 3 3 9 6 1 2 3 E - 0 8 0 . 0 3 3 3 9 6 1 2 3
5 . 5 1 . 0 0 0 0 0 0 0 1 . 9 0 3 6 3 9 9 E - 0 8 0 . 0 1 9 0 3 6 3 9 9
5 . 6 1 . 0 0 0 0 0 0 0 1 . 0 7 4 6 2 1 7 E - 0 8 0 . 0 1 0 7 4 6 2 1 7
5 . 7 1 . 0 0 0 0 0 0 0 6 . 0 0 7 6 5 3 2 E - 0 9 0 . 0 0 6 0 0 7 6 5 3
5 . 8 1 . 0 0 0 0 0 0 0 3 . 3 2 6 0 5 1 7 E - 0 9 0 . 0 0 3 3 2 6 0 5 2
5 . 9 1 . 0 0 0 0 0 0 0 1 . 8 2 3 5 7 9 3 E - 0 9 0 . 0 0 1 8 2 3 5 7 9
6 1 . 0 0 0 0 0 0 0 9 . 9 0 1 2 1 8 7 E - 1 0 0 . 0 0 0 9 9 0 1 2 2
Area to left of ZZ
Area to right of Z
Parts per million right of Z
Continued
H1518_Burke_appendix.indd 529 6/29/17 11:52 AM
530
0z
p(Z ≤ z)
z –0.09 –0.08 –0.07 –0.06 –0.05 –0.04 –0.03 –0.02 –0.01 –0.00
–3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
–3.4 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003
–3.3 0.0003 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0005 0.0005 0.0005
–3.2 0.0005 0.0005 0.0005 0.0006 0.0006 0.0006 0.0006 0.0006 0.0007 0.0007
–3.1 0.0007 0.0007 0.0008 0.0008 0.0008 0.0008 0.0009 0.0009 0.0009 0.0010
–3.0 0.0010 0.0010 0.0011 0.0011 0.0011 0.0012 0.0012 0.0013 0.0013 0.0013
–2.9 0.0014 0.0014 0.0015 0.0015 0.0016 0.0016 0.0017 0.0018 0.0018 0.0019
–2.8 0.0019 0.0020 0.0021 0.0021 0.0022 0.0023 0.0023 0.0024 0.0025 0.0026
–2.7 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033 0.0034 0.0035
–2.6 0.0036 0.0037 0.0038 0.0039 0.0040 0.0041 0.0043 0.0044 0.0045 0.0047
–2.5 0.0048 0.0049 0.0051 0.0052 0.0054 0.0055 0.0057 0.0059 0.0060 0.0062
–2.4 0.0064 0.0066 0.0068 0.0069 0.0071 0.0073 0.0075 0.0078 0.0080 0.0082
–2.3 0.0084 0.0087 0.0089 0.0091 0.0094 0.0096 0.0099 0.0102 0.0104 0.0107
–2.2 0.0110 0.0113 0.0116 0.0119 0.0122 0.0125 0.0129 0.0132 0.0136 0.0139
–2.1 0.0143 0.0146 0.0150 0.0154 0.0158 0.0162 0.0166 0.0170 0.0174 0.0179
–2.0 0.0183 0.0188 0.0192 0.0197 0.0202 0.0207 0.0212 0.0217 0.0222 0.0228
–1.9 0.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287
–1.8 0.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359
–1.7 0.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446
–1.6 0.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548
–1.5 0.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655 0.0668
appendix e Areas under Standard Normal
Distribution to the Left of Z-Values
H1518_Burke_appendix.indd 530 6/29/17 11:52 AM
Appendix E Areas under Standard Normal Distribution to the Left of Z-Values 531
z –0.09 –0.08 –0.07 –0.06 –0.05 –0.04 –0.03 –0.02 –0.01 –0.00
–1.4 0.0681 0.0694 0.0708 0.0721 0.0735 0.0749 0.0764 0.0778 0.0793 0.0808
–1.3 0.0823 0.0838 0.0853 0.0869 0.0885 0.0901 0.0918 0.0934 0.0951 0.0968
–1.2 0.0985 0.1003 0.1020 0.1038 0.1056 0.1075 0.1093 0.1112 0.1131 0.1151
–1.1 0.1170 0.1190 0.1210 0.1230 0.1251 0.1271 0.1292 0.1314 0.1335 0.1357
–1.0 0.1379 0.1401 0.1423 0.1446 0.1469 0.1492 0.1515 0.1539 0.1562 0.1587
–0.9 0.1611 0.1635 0.1660 0.1685 0.1711 0.1736 0.1762 0.1788 0.1814 0.1841
–0.8 0.1867 0.1894 0.1922 0.1949 0.1977 0.2005 0.2033 0.2061 0.2090 0.2119
–0.7 0.2148 0.2177 0.2206 0.2236 0.2266 0.2296 0.2327 0.2358 0.2389 0.2420
–0.6 0.2451 0.2483 0.2514 0.2546 0.2578 0.2611 0.2643 0.2676 0.2709 0.2743
–0.5 0.2776 0.2810 0.2843 0.2877 0.2912 0.2946 0.2981 0.3015 0.3050 0.3085
–0.4 0.3121 0.3156 0.3192 0.3228 0.3264 0.3300 0.3336 0.3372 0.3409 0.3446
–0.3 0.3483 0.3520 0.3557 0.3594 0.3632 0.3669 0.3707 0.3745 0.3783 0.3821
–0.2 0.3859 0.3897 0.3936 0.3974 0.4013 0.4052 0.4090 0.4129 0.4168 0.4207
–0.1 0.4247 0.4286 0.4325 0.4364 0.4404 0.4443 0.4483 0.4522 0.4562 0.4602
0.0 0.4641 0.4681 0.4721 0.4761 0.4801 0.4840 0.4880 0.4920 0.4960 0.5000
0 z
p(Z ≤ z)
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
Continued
Continued
H1518_Burke_appendix.indd 531 6/29/17 11:52 AM
532 Appendix E Areas under Standard Normal Distribution to the Left of Z-Values
Continued
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
H1518_Burke_appendix.indd 532 6/29/17 11:52 AM
533
appendix F F Distribution F0.10
H1518_Burke_appendix.indd 533 6/29/17 11:52 AM
534 Appendix F F Distribution F0.10
Numerator degrees of freedom
1 2 3 4 5 6 7 8 9 10 11
1 39.86 49.50 53.59 55.83 57.24 58.20 58.91 59.44 59.86 60.19 60.47
2 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38 9.39 9.40
3 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24 5.23 5.22
4 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94 3.92 3.91
5 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32 3.30 3.28
6 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96 2.94 2.92
7 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72 2.70 2.68
8 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56 2.54 2.52
9 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44 2.42 2.40
10 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35 2.32 2.30
11 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27 2.25 2.23
12 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21 2.19 2.17
13 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.20 2.16 2.14 2.12
14 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12 2.10 2.07
15 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09 2.06 2.04
16 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06 2.03 2.01
17 3.03 2.64 2.44 2.31 2.22 2.15 2.10 2.06 2.03 2.00 1.98
18 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00 1.98 1.95
19 2.99 2.61 2.40 2.27 2.18 2.11 2.06 2.02 1.98 1.96 1.93
20 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96 1.94 1.91
21 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95 1.92 1.90
22 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93 1.90 1.88
23 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92 1.89 1.87
24 2.93 2.54 2.33 2.19 2.10 2.04 1.98 1.94 1.91 1.88 1.85
25 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89 1.87 1.84
26 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88 1.86 1.83
27 2.90 2.51 2.30 2.17 2.07 2.00 1.95 1.91 1.87 1.85 1.82
28 2.89 2.50 2.29 2.16 2.06 2.00 1.94 1.90 1.87 1.84 1.81
29 2.89 2.50 2.28 2.15 2.06 1.99 1.93 1.89 1.86 1.83 1.80
30 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85 1.82 1.79
40 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79 1.76 1.74
60 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74 1.71 1.68
100 2.76 2.36 2.14 2.00 1.91 1.83 1.78 1.73 1.69 1.66 1.64
Continued
D e
n o
m in
a to
r d
e g
re e
s o
f fr
e e
d o
m
F distribution F0.10
H1518_Burke_appendix.indd 534 6/29/17 11:52 AM
Appendix F F Distribution F0.10 535
Numerator degrees of freedom
12 13 14 15 16 17 18 19 20 21 22
1 60.71 60.90 61.07 61.22 61.35 61.46 61.57 61.66 61.74 61.81 61.88
2 9.41 9.41 9.42 9.42 9.43 9.43 9.44 9.44 9.44 9.44 9.45
3 5.22 5.21 5.20 5.20 5.20 5.19 5.19 5.19 5.18 5.18 5.18
4 3.90 3.89 3.88 3.87 3.86 3.86 3.85 3.85 3.84 3.84 3.84
5 3.27 3.26 3.25 3.24 3.23 3.22 3.22 3.21 3.21 3.20 3.20
6 2.90 2.89 2.88 2.87 2.86 2.85 2.85 2.84 2.84 2.83 2.83
7 2.67 2.65 2.64 2.63 2.62 2.61 2.61 2.60 2.59 2.59 2.58
8 2.50 2.49 2.48 2.46 2.45 2.45 2.44 2.43 2.42 2.42 2.41
9 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.30 2.29 2.29
10 2.28 2.27 2.26 2.24 2.23 2.22 2.22 2.21 2.20 2.19 2.19
11 2.21 2.19 2.18 2.17 2.16 2.15 2.14 2.13 2.12 2.12 2.11
12 2.15 2.13 2.12 2.10 2.09 2.08 2.08 2.07 2.06 2.05 2.05
13 2.10 2.08 2.07 2.05 2.04 2.03 2.02 2.01 2.01 2.00 1.99
14 2.05 2.04 2.02 2.01 2.00 1.99 1.98 1.97 1.96 1.96 1.95
15 2.02 2.00 1.99 1.97 1.96 1.95 1.94 1.93 1.92 1.92 1.91
16 1.99 1.97 1.95 1.94 1.93 1.92 1.91 1.90 1.89 1.88 1.88
17 1.96 1.94 1.93 1.91 1.90 1.89 1.88 1.87 1.86 1.86 1.85
18 1.93 1.92 1.90 1.89 1.87 1.86 1.85 1.84 1.84 1.83 1.82
19 1.91 1.89 1.88 1.86 1.85 1.84 1.83 1.82 1.81 1.81 1.80
20 1.89 1.87 1.86 1.84 1.83 1.82 1.81 1.80 1.79 1.79 1.78
21 1.87 1.86 1.84 1.83 1.81 1.80 1.79 1.78 1.78 1.77 1.76
22 1.86 1.84 1.83 1.81 1.80 1.79 1.78 1.77 1.76 1.75 1.74
23 1.84 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.74 1.73
24 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.72 1.71
25 1.82 1.80 1.79 1.77 1.76 1.75 1.74 1.73 1.72 1.71 1.70
26 1.81 1.79 1.77 1.76 1.75 1.73 1.72 1.71 1.71 1.70 1.69
27 1.80 1.78 1.76 1.75 1.74 1.72 1.71 1.70 1.70 1.69 1.68
28 1.79 1.77 1.75 1.74 1.73 1.71 1.70 1.69 1.69 1.68 1.67
29 1.78 1.76 1.75 1.73 1.72 1.71 1.69 1.68 1.68 1.67 1.66
30 1.77 1.75 1.74 1.72 1.71 1.70 1.69 1.68 1.67 1.66 1.65
40 1.71 1.70 1.68 1.66 1.65 1.64 1.62 1.61 1.61 1.60 1.59
60 1.66 1.64 1.62 1.60 1.59 1.58 1.56 1.55 1.54 1.53 1.53
100 1.61 1.59 1.57 1.56 1.54 1.53 1.52 1.50 1.49 1.48 1.48
Continued
D e
n o
m in
a to
r d
e g
re e
s o
f fr
e e
d o
m
F distribution F0.10 (continued)
H1518_Burke_appendix.indd 535 6/29/17 11:52 AM
536 Appendix F F Distribution F0.10
Numerator degrees of freedom
23 24 25 26 27 28 29 30 40 60 100
1 61.94 62.00 62.05 62.10 62.15 62.19 62.23 62.26 62.53 62.79 63.01
2 9.45 9.45 9.45 9.45 9.45 9.46 9.46 9.46 9.47 9.47 9.48
3 5.18 5.18 5.17 5.17 5.17 5.17 5.17 5.17 5.16 5.15 5.14
4 3.83 3.83 3.83 3.83 3.82 3.82 3.82 3.82 3.80 3.79 3.78
5 3.19 3.19 3.19 3.18 3.18 3.18 3.18 3.17 3.16 3.14 3.13
6 2.82 2.82 2.81 2.81 2.81 2.81 2.80 2.80 2.78 2.76 2.75
7 2.58 2.58 2.57 2.57 2.56 2.56 2.56 2.56 2.54 2.51 2.50
8 2.41 2.40 2.40 2.40 2.39 2.39 2.39 2.38 2.36 2.34 2.32
9 2.28 2.28 2.27 2.27 2.26 2.26 2.26 2.25 2.23 2.21 2.19
10 2.18 2.18 2.17 2.17 2.17 2.16 2.16 2.16 2.13 2.11 2.09
11 2.11 2.10 2.10 2.09 2.09 2.08 2.08 2.08 2.05 2.03 2.01
12 2.04 2.04 2.03 2.03 2.02 2.02 2.01 2.01 1.99 1.96 1.94
13 1.99 1.98 1.98 1.97 1.97 1.96 1.96 1.96 1.93 1.90 1.88
14 1.94 1.94 1.93 1.93 1.92 1.92 1.92 1.91 1.89 1.86 1.83
15 1.90 1.90 1.89 1.89 1.88 1.88 1.88 1.87 1.85 1.82 1.79
16 1.87 1.87 1.86 1.86 1.85 1.85 1.84 1.84 1.81 1.78 1.76
17 1.84 1.84 1.83 1.83 1.82 1.82 1.81 1.81 1.78 1.75 1.73
18 1.82 1.81 1.80 1.80 1.80 1.79 1.79 1.78 1.75 1.72 1.70
19 1.79 1.79 1.78 1.78 1.77 1.77 1.76 1.76 1.73 1.70 1.67
20 1.77 1.77 1.76 1.76 1.75 1.75 1.74 1.74 1.71 1.68 1.65
21 1.75 1.75 1.74 1.74 1.73 1.73 1.72 1.72 1.69 1.66 1.63
22 1.74 1.73 1.73 1.72 1.72 1.71 1.71 1.70 1.67 1.64 1.61
23 1.72 1.72 1.71 1.70 1.70 1.69 1.69 1.69 1.66 1.62 1.59
24 1.71 1.70 1.70 1.69 1.69 1.68 1.68 1.67 1.64 1.61 1.58
25 1.70 1.69 1.68 1.68 1.67 1.67 1.66 1.66 1.63 1.59 1.56
26 1.68 1.68 1.67 1.67 1.66 1.66 1.65 1.65 1.61 1.58 1.55
27 1.67 1.67 1.66 1.65 1.65 1.64 1.64 1.64 1.60 1.57 1.54
28 1.66 1.66 1.65 1.64 1.64 1.63 1.63 1.63 1.59 1.56 1.53
29 1.65 1.65 1.64 1.63 1.63 1.62 1.62 1.62 1.58 1.55 1.52
30 1.64 1.64 1.63 1.63 1.62 1.62 1.61 1.61 1.57 1.54 1.51
40 1.58 1.57 1.57 1.56 1.56 1.55 1.55 1.54 1.51 1.47 1.43
60 1.52 1.51 1.50 1.50 1.49 1.49 1.48 1.48 1.44 1.40 1.36
100 1.47 1.46 1.45 1.45 1.44 1.43 1.43 1.42 1.38 1.34 1.29
D e
n o
m in
a to
r d
e g
re e
s o
f fr
e e
d o
m
F distribution F0.10 (continued)
H1518_Burke_appendix.indd 536 6/29/17 11:52 AM
537
appendix G F Distribution F0.05
H1518_Burke_appendix.indd 537 6/29/17 11:52 AM
538 Appendix G F Distribution F0.05
Numerator degrees of freedom
1 2 3 4 5 6 7 8 9 10 11
1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.0
2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.40
3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.76
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.94
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.70
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.03
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.60
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.31
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.10
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.94
11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.82
12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.72
13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.63
14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.57
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.51
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.46
17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.41
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.37
19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.34
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.31
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.28
22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.26
23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.24
24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.22
25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.20
26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.18
27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.17
28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.15
29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.14
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.13
40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.04
60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.95
100 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97 1.93 1.89
Continued
D e
n o
m in
a to
r d
e g
re e
s o
f fr
e e
d o
m
F distribution F0.05
H1518_Burke_appendix.indd 538 6/29/17 11:52 AM
Appendix G F Distribution F0.05 539
Numerator degrees of freedom
12 13 14 15 16 17 18 19 20 21 22
1 243.9 244.7 245.4 245.9 246.5 246.9 247.3 247.7 248.0 248.3 248.6
2 19.41 19.42 19.42 19.43 19.43 19.44 19.44 19.44 19.45 19.45 19.45
3 8.74 8.73 8.71 8.70 8.69 8.68 8.67 8.67 8.66 8.65 8.65
4 5.91 5.89 5.87 5.86 5.84 5.83 5.82 5.81 5.80 5.79 5.79
5 4.68 4.66 4.64 4.62 4.60 4.59 4.58 4.57 4.56 4.55 4.54
6 4.00 3.98 3.96 3.94 3.92 3.91 3.90 3.88 3.87 3.86 3.86
7 3.57 3.55 3.53 3.51 3.49 3.48 3.47 3.46 3.44 3.43 3.43
8 3.28 3.26 3.24 3.22 3.20 3.19 3.17 3.16 3.15 3.14 3.13
9 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94 2.93 2.92
10 2.91 2.89 2.86 2.85 2.83 2.81 2.80 2.79 2.77 2.76 2.75
11 2.79 2.76 2.74 2.72 2.70 2.69 2.67 2.66 2.65 2.64 2.63
12 2.69 2.66 2.64 2.62 2.60 2.58 2.57 2.56 2.54 2.53 2.52
13 2.60 2.58 2.55 2.53 2.51 2.50 2.48 2.47 2.46 2.45 2.44
14 2.53 2.51 2.48 2.46 2.44 2.43 2.41 2.40 2.39 2.38 2.37
15 2.48 2.45 2.42 2.40 2.38 2.37 2.35 2.34 2.33 2.32 2.31
16 2.42 2.40 2.37 2.35 2.33 2.32 2.30 2.29 2.28 2.26 2.25
17 2.38 2.35 2.33 2.31 2.29 2.27 2.26 2.24 2.23 2.22 2.21
18 2.34 2.31 2.29 2.27 2.25 2.23 2.22 2.20 2.19 2.18 2.17
19 2.31 2.28 2.26 2.23 2.21 2.20 2.18 2.17 2.16 2.14 2.13
20 2.28 2.25 2.22 2.20 2.18 2.17 2.15 2.14 2.12 2.11 2.10
21 2.25 2.22 2.20 2.18 2.16 2.14 2.12 2.11 2.10 2.08 2.07
22 2.23 2.20 2.17 2.15 2.13 2.11 2.10 2.08 2.07 2.06 2.05
23 2.20 2.18 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.04 2.02
24 2.18 2.15 2.13 2.11 2.09 2.07 2.05 2.04 2.03 2.01 2.00
25 2.16 2.14 2.11 2.09 2.07 2.05 2.04 2.02 2.01 2.00 1.98
26 2.15 2.12 2.09 2.07 2.05 2.03 2.02 2.00 1.99 1.98 1.97
27 2.13 2.10 2.08 2.06 2.04 2.02 2.00 1.99 1.97 1.96 1.95
28 2.12 2.09 2.06 2.04 2.02 2.00 1.99 1.97 1.96 1.95 1.93
29 2.10 2.08 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.93 1.92
30 2.09 2.06 2.04 2.01 1.99 1.98 1.96 1.95 1.93 1.92 1.91
40 2.00 1.97 1.95 1.92 1.90 1.89 1.87 1.85 1.84 1.83 1.81
60 1.92 1.89 1.86 1.84 1.82 1.80 1.78 1.76 1.75 1.73 1.72
100 1.85 1.82 1.79 1.77 1.75 1.73 1.71 1.69 1.68 1.66 1.65
Continued
D e
n o
m in
a to
r d
e g
re e
s o
f fr
e e
d o
m
F distribution F0.05 (continued)
H1518_Burke_appendix.indd 539 6/29/17 11:52 AM
540 Appendix G F Distribution F0.05
Numerator degrees of freedom
23 24 25 26 27 28 29 30 40 60 100
1 248.8 249.1 249.3 249.5 249.6 249.8 250.0 250.1 251.1 252.2 253.0
2 19.45 19.45 19.46 19.46 19.46 19.46 19.46 19.46 19.47 19.48 19.49
3 8.64 8.64 8.63 8.63 8.63 8.62 8.62 8.62 8.59 8.57 8.55
4 5.78 5.77 5.77 5.76 5.76 5.75 5.75 5.75 5.72 5.69 5.66
5 4.53 4.53 4.52 4.52 4.51 4.50 4.50 4.50 4.46 4.43 4.41
6 3.85 3.84 3.83 3.83 3.82 3.82 3.81 3.81 3.77 3.74 3.71
7 3.42 3.41 3.40 3.40 3.39 3.39 3.38 3.38 3.34 3.30 3.27
8 3.12 3.12 3.11 3.10 3.10 3.09 3.08 3.08 3.04 3.01 2.97
9 2.91 2.90 2.89 2.89 2.88 2.87 2.87 2.86 2.83 2.79 2.76
10 2.75 2.74 2.73 2.72 2.72 2.71 2.70 2.70 2.66 2.62 2.59
11 2.62 2.61 2.60 2.59 2.59 2.58 2.58 2.57 2.53 2.49 2.46
12 2.51 2.51 2.50 2.49 2.48 2.48 2.47 2.47 2.43 2.38 2.35
13 2.43 2.42 2.41 2.41 2.40 2.39 2.39 2.38 2.34 2.30 2.26
14 2.36 2.35 2.34 2.33 2.33 2.32 2.31 2.31 2.27 2.22 2.19
15 2.30 2.29 2.28 2.27 2.27 2.26 2.25 2.25 2.20 2.16 2.12
16 2.24 2.24 2.23 2.22 2.21 2.21 2.20 2.19 2.15 2.11 2.07
17 2.20 2.19 2.18 2.17 2.17 2.16 2.15 2.15 2.10 2.06 2.02
18 2.16 2.15 2.14 2.13 2.13 2.12 2.11 2.11 2.06 2.02 1.98
19 2.12 2.11 2.11 2.10 2.09 2.08 2.08 2.07 2.03 1.98 1.94
20 2.09 2.08 2.07 2.07 2.06 2.05 2.05 2.04 1.99 1.95 1.91
21 2.06 2.05 2.05 2.04 2.03 2.02 2.02 2.01 1.96 1.92 1.88
22 2.04 2.03 2.02 2.01 2.00 2.00 1.99 1.98 1.94 1.89 1.85
23 2.01 2.01 2.00 1.99 1.98 1.97 1.97 1.96 1.91 1.86 1.82
24 1.99 1.98 1.97 1.97 1.96 1.95 1.95 1.94 1.89 1.84 1.80
25 1.97 1.96 1.96 1.95 1.94 1.93 1.93 1.92 1.87 1.82 1.78
26 1.96 1.95 1.94 1.93 1.92 1.91 1.91 1.90 1.85 1.80 1.76
27 1.94 1.93 1.92 1.91 1.90 1.90 1.89 1.88 1.84 1.79 1.74
28 1.92 1.91 1.91 1.90 1.89 1.88 1.88 1.87 1.82 1.77 1.73
29 1.91 1.90 1.89 1.88 1.88 1.87 1.86 1.85 1.81 1.75 1.71
30 1.90 1.89 1.88 1.87 1.86 1.85 1.85 1.84 1.79 1.74 1.70
40 1.80 1.79 1.78 1.77 1.77 1.76 1.75 1.74 1.69 1.64 1.59
60 1.71 1.70 1.69 1.68 1.67 1.66 1.66 1.65 1.59 1.53 1.48
100 1.64 1.63 1.62 1.61 1.60 1.59 1.58 1.57 1.52 1.45 1.39
D e
n o
m in
a to
r d
e g
re e
s o
f fr
e e
d o
m
F distribution F0.05 (continued)
H1518_Burke_appendix.indd 540 6/29/17 11:52 AM
541
appendix h F Distribution F0.01
H1518_Burke_appendix.indd 541 6/29/17 11:52 AM
542 Appendix H F Distribution F0.01
Numerator degrees of freedom
1 2 3 4 5 6 7 8 9 10 11
1 4052 4999 5404 5624 5764 5859 5928 5981 6022 6056 6083
2 98.5 99 99.16 99.25 99.3 99.33 99.36 99.38 99.39 99.4 99.41
3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.34 27.23 27.13
4 21.2 18 16.69 15.98 15.52 15.21 14.98 14.8 14.66 14.55 14.45
5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.963
6 13.75 10.92 9.78 9.148 8.746 8.466 8.26 8.102 7.976 7.874 7.79
7 12.25 9.547 8.451 7.847 7.46 7.191 6.993 6.84 6.719 6.62 6.538
8 11.26 8.649 7.591 7.006 6.632 6.371 6.178 6.029 5.911 5.814 5.734
9 10.56 8.022 6.992 6.422 6.057 5.802 5.613 5.467 5.351 5.257 5.178
10 10.04 7.559 6.552 5.994 5.636 5.386 5.2 5.057 4.942 4.849 4.772
11 9.646 7.206 6.217 5.668 5.316 5.069 4.886 4.744 4.632 4.539 4.462
12 9.33 6.927 5.953 5.412 5.064 4.821 4.64 4.499 4.388 4.296 4.22
13 9.074 6.701 5.739 5.205 4.862 4.62 4.441 4.302 4.191 4.1 4.025
14 8.862 6.515 5.564 5.035 4.695 4.456 4.278 4.14 4.03 3.939 3.864
15 8.683 6.359 5.417 4.893 4.556 4.318 4.142 4.004 3.895 3.805 3.73
16 8.531 6.226 5.292 4.773 4.437 4.202 4.026 3.89 3.78 3.691 3.616
17 8.4 6.112 5.185 4.669 4.336 4.101 3.927 3.791 3.682 3.593 3.518
18 8.285 6.013 5.092 4.579 4.248 4.015 3.841 3.705 3.597 3.508 3.434
19 8.185 5.926 5.01 4.5 4.171 3.939 3.765 3.631 3.523 3.434 3.36
20 8.096 5.849 4.938 4.431 4.103 3.871 3.699 3.564 3.457 3.368 3.294
21 8.017 5.78 4.874 4.369 4.042 3.812 3.64 3.506 3.398 3.31 3.236
22 7.945 5.719 4.817 4.313 3.988 3.758 3.587 3.453 3.346 3.258 3.184
23 7.881 5.664 4.765 4.264 3.939 3.71 3.539 3.406 3.299 3.211 3.137
24 7.823 5.614 4.718 4.218 3.895 3.667 3.496 3.363 3.256 3.168 3.094
25 7.77 5.568 4.675 4.177 3.855 3.627 3.457 3.324 3.217 3.129 3.056
26 7.721 5.526 4.637 4.14 3.818 3.591 3.421 3.288 3.182 3.094 3.021
27 7.677 5.488 4.601 4.106 3.785 3.558 3.388 3.256 3.149 3.062 2.988
28 7.636 5.453 4.568 4.074 3.754 3.528 3.358 3.226 3.12 3.032 2.959
29 7.598 5.42 4.538 4.045 3.725 3.499 3.33 3.198 3.092 3.005 2.931
30 7.562 5.39 4.51 4.018 3.699 3.473 3.305 3.173 3.067 2.979 2.906
40 7.314 5.178 4.313 3.828 3.514 3.291 3.124 2.993 2.888 2.801 2.727
60 7.077 4.977 4.126 3.649 3.339 3.119 2.953 2.823 2.718 2.632 2.559
100 6.895 4.824 3.984 3.513 3.206 2.988 2.823 2.694 2.59 2.503 2.43
Continued
D e
n o
m in
a to
r d
e g
re e
s o
f fr
e e
d o
m
F distribution F0.01
H1518_Burke_appendix.indd 542 6/29/17 11:52 AM
Appendix H F Distribution F0.01 543
Numerator degrees of freedom
12 13 14 15 16 17 18 19 20 21 22
1 6107 6126 6143 6157 6170 6181 6191 6201 6208.7 6216.1 6223.1
2 99.42 99.42 99.43 99.43 99.44 99.44 99.44 99.45 99.448 99.451 99.455
3 27.05 26.98 26.92 26.87 26.83 26.79 26.75 26.72 26.69 26.664 26.639
4 14.37 14.31 14.25 14.2 14.15 14.11 14.08 14.05 14.019 13.994 13.97
5 9.888 9.825 9.77 9.722 9.68 9.643 9.609 9.58 9.5527 9.5281 9.5058
6 7.718 7.657 7.605 7.559 7.519 7.483 7.451 7.422 7.3958 7.3721 7.3506
7 6.469 6.41 6.359 6.314 6.275 6.24 6.209 6.181 6.1555 6.1324 6.1113
8 5.667 5.609 5.559 5.515 5.477 5.442 5.412 5.384 5.3591 5.3365 5.3157
9 5.111 5.055 5.005 4.962 4.924 4.89 4.86 4.833 4.808 4.7855 4.7651
10 4.706 4.65 4.601 4.558 4.52 4.487 4.457 4.43 4.4054 4.3831 4.3628
11 4.397 4.342 4.293 4.251 4.213 4.18 4.15 4.123 4.099 4.0769 4.0566
12 4.155 4.1 4.052 4.01 3.972 3.939 3.91 3.883 3.8584 3.8363 3.8161
13 3.96 3.905 3.857 3.815 3.778 3.745 3.716 3.689 3.6646 3.6425 3.6223
14 3.8 3.745 3.698 3.656 3.619 3.586 3.556 3.529 3.5052 3.4832 3.463
15 3.666 3.612 3.564 3.522 3.485 3.452 3.423 3.396 3.3719 3.3498 3.3297
16 3.553 3.498 3.451 3.409 3.372 3.339 3.31 3.283 3.2587 3.2367 3.2165
17 3.455 3.401 3.353 3.312 3.275 3.242 3.212 3.186 3.1615 3.1394 3.1192
18 3.371 3.316 3.269 3.227 3.19 3.158 3.128 3.101 3.0771 3.055 3.0348
19 3.297 3.242 3.195 3.153 3.116 3.084 3.054 3.027 3.0031 2.981 2.9607
20 3.231 3.177 3.13 3.088 3.051 3.018 2.989 2.962 2.9377 2.9156 2.8953
21 3.173 3.119 3.072 3.03 2.993 2.96 2.931 2.904 2.8795 2.8574 2.837
22 3.121 3.067 3.019 2.978 2.941 2.908 2.879 2.852 2.8274 2.8052 2.7849
23 3.074 3.02 2.973 2.931 2.894 2.861 2.832 2.805 2.7805 2.7582 2.7378
24 3.032 2.977 2.93 2.889 2.852 2.819 2.789 2.762 2.738 2.7157 2.6953
25 2.993 2.939 2.892 2.85 2.813 2.78 2.751 2.724 2.6993 2.677 2.6565
26 2.958 2.904 2.857 2.815 2.778 2.745 2.715 2.688 2.664 2.6416 2.6211
27 2.926 2.872 2.824 2.783 2.746 2.713 2.683 2.656 2.6316 2.609 2.5886
28 2.896 2.842 2.795 2.753 2.716 2.683 2.653 2.626 2.6018 2.5793 2.5587
29 2.868 2.814 2.767 2.726 2.689 2.656 2.626 2.599 2.5742 2.5517 2.5311
30 2.843 2.789 2.742 2.7 2.663 2.63 2.6 2.573 2.5487 2.5262 2.5055
40 2.665 2.611 2.563 2.522 2.484 2.451 2.421 2.394 2.3689 2.3461 2.3252
60 2.496 2.442 2.394 2.352 2.315 2.281 2.251 2.223 2.1978 2.1747 2.1533
10 2.368 2.313 2.265 2.223 2.185 2.151 2.12 2.092 2.0666 2.0431 2.0214
Continued
D e
n o
m in
a to
r d
e g
re e
s o
f fr
e e
d o
m
F distribution F0.01 (continued)
H1518_Burke_appendix.indd 543 6/29/17 11:52 AM
Numerator degrees of freedom
23 24 25 26 27 28 29 30 40 60 100
1 6228.7 6234.3 6239.9 6244.5 6249.2 6252.9 6257.1 6260.4 6286.4 6313 6333.9
2 99.455 99.455 99.459 99.462 99.462 99.462 99.462 99.466 99.477 99.484 99.491
3 26.617 26.597 26.579 26.562 26.546 26.531 26.517 26.504 26.411 26.316 26.241
4 13.949 13.929 13.911 13.894 13.878 13.864 13.85 13.838 13.745 13.652 13.577
5 9.4853 9.4665 9.4492 9.4331 9.4183 9.4044 9.3914 9.3794 9.2912 9.202 9.13
6 7.3309 7.3128 7.296 7.2805 7.2661 7.2528 7.2403 7.2286 7.1432 7.0568 6.9867
7 6.092 6.0743 6.0579 6.0428 6.0287 6.0156 6.0035 5.992 5.9084 5.8236 5.7546
8 5.2967 5.2793 5.2631 5.2482 5.2344 5.2214 5.2094 5.1981 5.1156 5.0316 4.9633
9 4.7463 4.729 4.713 4.6982 4.6845 4.6717 4.6598 4.6486 4.5667 4.4831 4.415
10 4.3441 4.3269 4.3111 4.2963 4.2827 4.27 4.2582 4.2469 4.1653 4.0819 4.0137
11 4.038 4.0209 4.0051 3.9904 3.9768 3.9641 3.9522 3.9411 3.8596 3.7761 3.7077
12 3.7976 3.7805 3.7647 3.7501 3.7364 3.7238 3.7119 3.7008 3.6192 3.5355 3.4668
13 3.6038 3.5868 3.571 3.5563 3.5427 3.53 3.5182 3.507 3.4253 3.3413 3.2723
14 3.4445 3.4274 3.4116 3.3969 3.3833 3.3706 3.3587 3.3476 3.2657 3.1813 3.1118
15 3.3111 3.294 3.2782 3.2636 3.2499 3.2372 3.2253 3.2141 3.1319 3.0471 2.9772
16 3.1979 3.1808 3.165 3.1503 3.1366 3.1238 3.1119 3.1007 3.0182 2.933 2.8627
17 3.1006 3.0835 3.0676 3.0529 3.0392 3.0264 3.0145 3.0032 2.9204 2.8348 2.7639
18 3.0161 2.999 2.9831 2.9683 2.9546 2.9418 2.9298 2.9185 2.8354 2.7493 2.6779
19 2.9421 2.9249 2.9089 2.8942 2.8804 2.8675 2.8555 2.8442 2.7608 2.6742 2.6023
20 2.8766 2.8594 2.8434 2.8286 2.8148 2.8019 2.7898 2.7785 2.6947 2.6077 2.5353
21 2.8183 2.801 2.785 2.7702 2.7563 2.7434 2.7313 2.72 2.6359 2.5484 2.4755
22 2.7661 2.7488 2.7328 2.7179 2.704 2.691 2.6789 2.6675 2.5831 2.4951 2.4218
23 2.7191 2.7017 2.6857 2.6707 2.6568 2.6438 2.6316 2.6202 2.5355 2.4471 2.3732
24 2.6764 2.6591 2.643 2.628 2.614 2.601 2.5888 2.5773 2.4923 2.4035 2.3291
25 2.6377 2.6203 2.6041 2.5891 2.5751 2.562 2.5498 2.5383 2.453 2.3637 2.2888
26 2.6022 2.5848 2.5686 2.5535 2.5395 2.5264 2.5142 2.5026 2.417 2.3273 2.2519
27 2.5697 2.5522 2.536 2.5209 2.5069 2.4937 2.4814 2.4699 2.384 2.2938 2.218
28 2.5398 2.5223 2.506 2.4909 2.4768 2.4636 2.4513 2.4397 2.3535 2.2629 2.1867
29 2.5121 2.4946 2.4783 2.4631 2.449 2.4358 2.4234 2.4118 2.3253 2.2344 2.1577
30 2.4865 2.4689 2.4526 2.4374 2.4233 2.41 2.3976 2.386 2.2992 2.2079 2.1307
40 2.3059 2.288 2.2714 2.2559 2.2415 2.228 2.2153 2.2034 2.1142 2.0194 1.9383
60 2.1336 2.1154 2.0984 2.0825 2.0677 2.0538 2.0408 2.0285 1.936 1.8363 1.7493
100 2.0012 1.9826 1.9651 1.9489 1.9337 1.9194 1.9059 1.8933 1.7972 1.6918 1.5977
D e
n o
m in
a to
r d
e g
re e
s o
f fr
e e
d o
m
F distribution F0.01 (continued)
544 Appendix H F Distribution F0.01
H1518_Burke_appendix.indd 544 6/29/17 11:52 AM
545
p
n x 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
2 0 0.980 0.960 0.941 0.922 0.903 0.884 0.865 0.846 0.828 0.810 0.723 0.640 0.563 0.490 0.423 0.360 0.303 0.250
2 1 1.000 1.000 0.999 0.998 0.998 0.996 0.995 0.994 0.992 0.990 0.978 0.960 0.938 0.910 0.878 0.840 0.798 0.750
3 0 0.970 0.941 0.913 0.885 0.857 0.831 0.804 0.779 0.754 0.729 0.614 0.512 0.422 0.343 0.275 0.216 0.166 0.125
3 1 1.000 0.999 0.997 0.995 0.993 0.990 0.986 0.982 0.977 0.972 0.939 0.896 0.844 0.784 0.718 0.648 0.575 0.500
3 2 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.999 0.997 0.992 0.984 0.973 0.957 0.936 0.909 0.875
4 0 0.961 0.922 0.885 0.849 0.815 0.781 0.748 0.716 0.686 0.656 0.522 0.410 0.316 0.240 0.179 0.130 0.092 0.063
4 1 0.999 0.998 0.995 0.991 0.986 0.980 0.973 0.966 0.957 0.948 0.890 0.819 0.738 0.652 0.563 0.475 0.391 0.313
4 2 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.998 0.997 0.996 0.988 0.973 0.949 0.916 0.874 0.821 0.759 0.688
4 3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.998 0.996 0.992 0.985 0.974 0.959 0.938
5 0 0.951 0.904 0.859 0.815 0.774 0.734 0.696 0.659 0.624 0.590 0.444 0.328 0.237 0.168 0.116 0.078 0.050 0.031
5 1 0.999 0.996 0.992 0.985 0.977 0.968 0.958 0.946 0.933 0.919 0.835 0.737 0.633 0.528 0.428 0.337 0.256 0.188
5 2 1.000 1.000 1.000 0.999 0.999 0.998 0.997 0.995 0.994 0.991 0.973 0.942 0.896 0.837 0.765 0.683 0.593 0.500
5 3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.993 0.984 0.969 0.946 0.913 0.869 0.813
5 4 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.998 0.995 0.990 0.982 0.969
6 0 0.941 0.886 0.833 0.783 0.735 0.690 0.647 0.606 0.568 0.531 0.377 0.262 0.178 0.118 0.075 0.047 0.028 0.016
6 1 0.999 0.994 0.988 0.978 0.967 0.954 0.939 0.923 0.905 0.886 0.776 0.655 0.534 0.420 0.319 0.233 0.164 0.109
6 2 1.000 1.000 0.999 0.999 0.998 0.996 0.994 0.991 0.988 0.984 0.953 0.901 0.831 0.744 0.647 0.544 0.442 0.344
6 3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.999 0.994 0.983 0.962 0.930 0.883 0.821 0.745 0.656
6 4 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.995 0.989 0.978 0.959 0.931 0.891
6 5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.998 0.996 0.992 0.984
7 0 0.932 0.868 0.808 0.751 0.698 0.648 0.602 0.558 0.517 0.478 0.321 0.210 0.133 0.082 0.049 0.028 0.015 0.008
7 1 0.998 0.992 0.983 0.971 0.956 0.938 0.919 0.897 0.875 0.850 0.717 0.577 0.445 0.329 0.234 0.159 0.102 0.063
7 2 1.000 1.000 0.999 0.998 0.996 0.994 0.990 0.986 0.981 0.974 0.926 0.852 0.756 0.647 0.532 0.420 0.316 0.227
7 3 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.998 0.997 0.988 0.967 0.929 0.874 0.800 0.710 0.608 0.500
7 4 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.995 0.987 0.971 0.944 0.904 0.847 0.773
7 5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.996 0.991 0.981 0.964 0.938
7 6 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.998 0.996 0.992
Continued
Binomial distribution
appendix i Binomial Distribution
Probability of x or fewer occurrences in a sample of size n
H1518_Burke_appendix.indd 545 6/29/17 11:52 AM
546 Appendix I Binomial Distribution
p
n x 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
8 0 0.923 0.851 0.784 0.721 0.663 0.610 0.560 0.513 0.470 0.430 0.272 0.168 0.100 0.058 0.032 0.017 0.008 0.004
8 1 0.997 0.990 0.978 0.962 0.943 0.921 0.897 0.870 0.842 0.813 0.657 0.503 0.367 0.255 0.169 0.106 0.063 0.035
8 2 1.000 1.000 0.999 0.997 0.994 0.990 0.985 0.979 0.971 0.962 0.895 0.797 0.679 0.552 0.428 0.315 0.220 0.145
8 3 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.998 0.997 0.995 0.979 0.944 0.886 0.806 0.706 0.594 0.477 0.363
8 4 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.997 0.990 0.973 0.942 0.894 0.826 0.740 0.637
8 5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.996 0.989 0.975 0.950 0.912 0.855
8 6 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.996 0.991 0.982 0.965
8 7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.998 0.996
9 0 0.914 0.834 0.760 0.693 0.630 0.573 0.520 0.472 0.428 0.387 0.232 0.134 0.075 0.040 0.021 0.010 0.005 0.002
9 1 0.997 0.987 0.972 0.952 0.929 0.902 0.873 0.842 0.809 0.775 0.599 0.436 0.300 0.196 0.121 0.071 0.039 0.020
9 2 1.000 0.999 0.998 0.996 0.992 0.986 0.979 0.970 0.960 0.947 0.859 0.738 0.601 0.463 0.337 0.232 0.150 0.090
9 3 1.000 1.000 1.000 1.000 0.999 0.999 0.998 0.996 0.994 0.992 0.966 0.914 0.834 0.730 0.609 0.483 0.361 0.254
9 4 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.994 0.980 0.951 0.901 0.828 0.733 0.621 0.500
9 5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.997 0.990 0.975 0.946 0.901 0.834 0.746
9 6 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.996 0.989 0.975 0.950 0.910
9 7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.996 0.991 0.980
9 8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.998
10 0 0.904 0.817 0.737 0.665 0.599 0.539 0.484 0.434 0.389 0.349 0.197 0.107 0.056 0.028 0.013 0.006 0.003 0.001
10 1 0.996 0.984 0.965 0.942 0.914 0.882 0.848 0.812 0.775 0.736 0.544 0.376 0.244 0.149 0.086 0.046 0.023 0.011
10 2 1.000 0.999 0.997 0.994 0.988 0.981 0.972 0.960 0.946 0.930 0.820 0.678 0.526 0.383 0.262 0.167 0.100 0.055
10 3 1.000 1.000 1.000 1.000 0.999 0.998 0.996 0.994 0.991 0.987 0.950 0.879 0.776 0.650 0.514 0.382 0.266 0.172
10 4 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.998 0.990 0.967 0.922 0.850 0.751 0.633 0.504 0.377
10 5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.994 0.980 0.953 0.905 0.834 0.738 0.623
Binomial distribution (continued)
H1518_Burke_appendix.indd 546 6/29/17 11:52 AM
547
df χ20.995 χ20.99 χ20.975 χ20.95 χ20.90 χ20.10 χ20.05 χ20.025 χ20.01 χ20.005 1 0.000 0.000 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.086 16.750
6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
8 1.344 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 3.565 4.107 5.009 5.892 7.041 19.812 22.362 24.736 27.688 29.819
14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.558
25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
27 11.808 12.878 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.994
Continued
Chi-square distribution
appendix J Chi-Square Distribution
H1518_Burke_appendix.indd 547 6/29/17 11:52 AM
548 Appendix J Chi-Square Distribution
df χ20.995 χ20.99 χ20.975 χ20.95 χ20.90 χ20.10 χ20.05 χ20.025 χ20.01 χ20.005 29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.335
30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
31 14.458 15.655 17.539 19.281 21.434 41.422 44.985 48.232 52.191 55.002
32 15.134 16.362 18.291 20.072 22.271 42.585 46.194 49.480 53.486 56.328
33 15.815 17.073 19.047 20.867 23.110 43.745 47.400 50.725 54.775 57.648
34 16.501 17.789 19.806 21.664 23.952 44.903 48.602 51.966 56.061 58.964
35 17.192 18.509 20.569 22.465 24.797 46.059 49.802 53.203 57.342 60.275
40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
45 24.311 25.901 28.366 30.612 33.350 57.505 61.656 65.410 69.957 73.166
50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
55 31.735 33.571 36.398 38.958 42.060 68.796 73.311 77.380 82.292 85.749
60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
65 39.383 41.444 44.603 47.450 50.883 79.973 84.821 89.177 94.422 98.105
70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
75 47.206 49.475 52.942 56.054 59.795 91.061 96.217 100.839 106.393 110.285
80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
85 55.170 57.634 61.389 64.749 68.777 102.079 107.522 112.393 118.236 122.324
90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
95 63.250 65.898 69.925 73.520 77.818 113.038 118.752 123.858 129.973 134.247
100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.170
Chi-square distribution (continued)
H1518_Burke_appendix.indd 548 6/29/17 11:52 AM
549
X
00000.100000.00
48409.061590.01.0
37818.072181.02.0
28047.081952.03.0
23076.086923.04.0
35606.074393.05.0
18845.091154.06.0
95694.014305.07.0
33944.076055.08.0
75604.034395.09.0
88763.021236.01
78233.031766.01.1
91103.018896.02.1
35272.074727.03.1
06642.004357.04.1
31322.078677.05.1
09102.001897.06.1
86281.023718.07.1
03561.007438.08.1
75941.034058.09.1
43531.066468.02
64221.045778.01.2
08011.002988.02.2
62001.047998.03.2
27090.082909.04.2
80280.029719.05.2
72470.037529.06.2
Continued
Exponential distribution
Area to left of X
Area to right of X
appendix K Exponential Distribution
H1518_Burke_appendix.indd 549 6/29/17 11:52 AM
550 Appendix K Exponential Distribution
X
12760.097239.07.2
18060.091939.08.2
20550.089449.09.2
97940.012059.03
50540.059459.01.3
67040.042959.02.3
88630.021369.03.3
73330.036669.04.3
02030.008969.05.3
23720.086279.06.3
27420.082579.07.3
73220.036779.08.3
42020.067979.09.3
23810.086189.04
75610.034389.01.4
00510.000589.02.4
75310.034689.03.4
82210.027789.04.4
11110.098889.05.4
50010.059989.06.4
01900.009099.07.4
32800.077199.08.4
54700.055299.09.4
47600.062399.05
01600.009399.01.5
25500.084499.02.5
99400.010599.03.5
25400.084599.04.5
90400.019599.05.5
07300.003699.06.5
53300.056699.07.5
30300.079699.08.5
47200.062799.09.5
84200.025799.06
Exponential distribution (continued)
Area to left of X
Area to right of X
H1518_Burke_appendix.indd 550 6/29/17 11:52 AM
551
x↓ n→ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0.005 0.995 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.01 0.990 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.02 0.980 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.03 0.970 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.04 0.961 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.05 0.951 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.06 0.942 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.07 0.932 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.08 0.923 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.09 0.914 0.996 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.1 0.905 0.995 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.15 0.861 0.990 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.2 0.819 0.982 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.25 0.779 0.974 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.3 0.741 0.963 0.996 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.35 0.705 0.951 0.994 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.4 0.670 0.938 0.992 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.5 0.607 0.910 0.986 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.6 0.549 0.878 0.977 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.7 0.497 0.844 0.966 0.994 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.8 0.449 0.809 0.953 0.991 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.9 0.407 0.772 0.937 0.987 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1 0.368 0.736 0.920 0.981 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.2 0.301 0.663 0.879 0.966 0.992 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.4 0.247 0.592 0.833 0.946 0.986 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.6 0.202 0.525 0.783 0.921 0.976 0.994 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.8 0.165 0.463 0.731 0.891 0.964 0.990 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2 0.135 0.406 0.677 0.857 0.947 0.983 0.995 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
Continued
Poisson distribution
appendix l Poisson Distribution
Probability of x or fewer occurrences of an event
H1518_Burke_appendix.indd 551 6/29/17 11:52 AM
552 Appendix L Poisson Distribution
x↓ n→ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
2.2 0.111 0.355 0.623 0.819 0.928 0.975 0.993 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2.4 0.091 0.308 0.570 0.779 0.904 0.964 0.988 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2.6 0.074 0.267 0.518 0.736 0.877 0.951 0.983 0.995 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2.8 0.061 0.231 0.469 0.692 0.848 0.935 0.976 0.992 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3 0.050 0.199 0.423 0.647 0.815 0.916 0.966 0.988 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3.2 0.041 0.171 0.380 0.603 0.781 0.895 0.955 0.983 0.994 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3.4 0.033 0.147 0.340 0.558 0.744 0.871 0.942 0.977 0.992 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3.6 0.027 0.126 0.303 0.515 0.706 0.844 0.927 0.969 0.988 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3.8 0.022 0.107 0.269 0.473 0.668 0.816 0.909 0.960 0.984 0.994 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000
4 0.018 0.092 0.238 0.433 0.629 0.785 0.889 0.949 0.979 0.992 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000
4.5 0.011 0.061 0.174 0.342 0.532 0.703 0.831 0.913 0.960 0.983 0.993 0.998 0.999 1.000 1.000 1.000 1.000 1.000
5 0.007 0.040 0.125 0.265 0.440 0.616 0.762 0.867 0.932 0.968 0.986 0.995 0.998 0.999 1.000 1.000 1.000 1.000
5.5 0.004 0.027 0.088 0.202 0.358 0.529 0.686 0.809 0.894 0.946 0.975 0.989 0.996 0.998 0.999 1.000 1.000 1.000
6 0.002 0.017 0.062 0.151 0.285 0.446 0.606 0.744 0.847 0.916 0.957 0.980 0.991 0.996 0.999 0.999 1.000 1.000
6.5 0.002 0.011 0.043 0.112 0.224 0.369 0.527 0.673 0.792 0.877 0.933 0.966 0.984 0.993 0.997 0.999 1.000 1.000
7 0.001 0.007 0.030 0.082 0.173 0.301 0.450 0.599 0.729 0.830 0.901 0.947 0.973 0.987 0.994 0.998 0.999 1.000
7.5 0.001 0.005 0.020 0.059 0.132 0.241 0.378 0.525 0.662 0.776 0.862 0.921 0.957 0.978 0.990 0.995 0.998 0.999
8 0.000 0.003 0.014 0.042 0.100 0.191 0.313 0.453 0.593 0.717 0.816 0.888 0.936 0.966 0.983 0.992 0.996 0.998
8.5 0.000 0.002 0.009 0.030 0.074 0.150 0.256 0.386 0.523 0.653 0.763 0.849 0.909 0.949 0.973 0.986 0.993 0.997
9 0.000 0.001 0.006 0.021 0.055 0.116 0.207 0.324 0.456 0.587 0.706 0.803 0.876 0.926 0.959 0.978 0.989 0.995
9.5 0.000 0.001 0.004 0.015 0.040 0.089 0.165 0.269 0.392 0.522 0.645 0.752 0.836 0.898 0.940 0.967 0.982 0.991
10 0.000 0.000 0.003 0.010 0.029 0.067 0.130 0.220 0.333 0.458 0.583 0.697 0.792 0.864 0.917 0.951 0.973 0.986
10.5 0.000 0.000 0.002 0.007 0.021 0.050 0.102 0.179 0.279 0.397 0.521 0.639 0.742 0.825 0.888 0.932 0.960 0.978
Poisson distribution (continued)
H1518_Burke_appendix.indd 552 6/29/17 11:52 AM
553
n 1 2 3 4 5 6 7 8 9 10 11 12
1 0.500 0.292 0.206 0.159 0.130 0.109 0.095 0.083 0.074 0.067 0.061 0.056
2 0.708 0.500 0.386 0.315 0.266 0.230 0.202 0.181 0.163 0.149 0.137
3 0.794 0.614 0.500 0.422 0.365 0.321 0.287 0.260 0.237 0.218
4 0.841 0.685 0.578 0.500 0.440 0.394 0.356 0.325 0.298
5 0.870 0.734 0.635 0.560 0.500 0.452 0.412 0.379
6 0.891 0.770 0.679 0.606 0.548 0.500 0.460
7 0.905 0.798 0.713 0.644 0.588 0.540
8 0.917 0.819 0.740 0.675 0.621
9 0.926 0.837 0.763 0.702
10 0.933 0.851 0.782
11 0.939 0.863
12 0.944
n 13 14 15 16 17 18 19 20 21 22 23 24
1 0.052 0.049 0.045 0.043 0.040 0.038 0.036 0.034 0.033 0.031 0.030 0.029
2 0.127 0.118 0.110 0.104 0.098 0.092 0.088 0.083 0.079 0.076 0.073 0.070
3 0.201 0.188 0.175 0.165 0.155 0.147 0.139 0.132 0.126 0.121 0.115 0.111
4 0.276 0.257 0.240 0.226 0.213 0.201 0.191 0.181 0.173 0.165 0.158 0.152
5 0.351 0.326 0.305 0.287 0.270 0.255 0.242 0.230 0.220 0.210 0.201 0.193
6 0.425 0.396 0.370 0.348 0.328 0.310 0.294 0.279 0.266 0.254 0.244 0.234
7 0.500 0.465 0.435 0.409 0.385 0.364 0.345 0.328 0.313 0.299 0.286 0.275
8 0.575 0.535 0.500 0.470 0.443 0.418 0.397 0.377 0.360 0.344 0.329 0.316
9 0.649 0.604 0.565 0.530 0.500 0.473 0.448 0.426 0.407 0.388 0.372 0.357
10 0.724 0.674 0.630 0.591 0.557 0.527 0.500 0.475 0.453 0.433 0.415 0.398
11 0.799 0.743 0.695 0.652 0.615 0.582 0.552 0.525 0.500 0.478 0.457 0.439
12 0.873 0.813 0.760 0.713 0.672 0.636 0.603 0.574 0.547 0.522 0.500 0.480
Continued
Median ranks
appendix M Median Ranks
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554 Appendix M Median Ranks
n 13 14 15 16 17 18 19 20 21 22 23 24
13 0.948 0.882 0.825 0.774 0.730 0.690 0.655 0.623 0.593 0.567 0.543 0.520
14 0.951 0.890 0.835 0.787 0.745 0.706 0.672 0.640 0.612 0.585 0.561
15 0.955 0.896 0.845 0.799 0.758 0.721 0.687 0.656 0.628 0.602
16 0.957 0.902 0.853 0.809 0.770 0.734 0.701 0.671 0.643
17 0.960 0.908 0.861 0.819 0.780 0.746 0.714 0.684
18 0.962 0.912 0.868 0.827 0.790 0.756 0.725
19 0.964 0.917 0.874 0.835 0.799 0.766
20 0.966 0.921 0.879 0.842 0.807
21 0.967 0.924 0.885 0.848
22 0.969 0.927 0.889
23 0.970 0.930
24 0.971
Median ranks (continued)
H1518_Burke_appendix.indd 554 6/29/17 11:52 AM
555
n = 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 -1.05 -1.18 -1.28 -1.36 -1.43 -1.50 -1.55 -1.59 -1.64 -1.68 -1.71 -1.74 -1.77 -1.80
2 -0.30 -0.50 -0.64 -0.76 -0.85 -0.93 -1.00 -1.06 -1.11 -1.16 -1.20 -1.24 -1.28 -1.32
3 0.30 0.00 -0.20 -0.35 -0.47 -0.57 -0.65 -0.73 -0.79 -0.85 -0.90 -0.94 -0.99 -1.03
4 1.05 0.50 0.20 0.00 -0.15 -0.27 -0.37 -0.46 -0.53 -0.60 -0.66 -0.71 -0.76 -0.80
5 1.18 0.64 0.35 0.15 0.00 -0.12 -0.22 -0.31 -0.39 -0.45 -0.51 -0.57 -0.62
6 1.28 0.76 0.47 0.27 0.12 0.00 -0.10 -0.19 -0.27 -0.33 -0.39 -0.45
7 1.36 0.85 0.57 0.37 0.22 0.10 0.00 -0.09 -0.16 -0.23 -0.29
8 1.43 0.93 0.65 0.46 0.31 0.19 0.09 0.00 -0.08 -0.15
9 1.50 1.00 0.73 0.53 0.39 0.27 0.16 0.08 0.00
10 1.55 1.06 0.79 0.60 0.45 0.33 0.23 0.15
11 1.59 1.11 0.85 0.66 0.51 0.39 0.29
12 1.64 1.16 0.90 0.71 0.57 0.45
13 1.68 1.20 0.94 0.76 0.62
14 1.71 1.24 0.99 0.80
15 1.74 1.28 1.03
16 1.77 1.32
17 1.80
18
19
20
21
22
23
24
25
26
27
28
29
30
Continued
Normal scores
appendix N Normal Scores
H1518_Burke_appendix.indd 555 6/29/17 11:52 AM
556 Appendix N Normal Scores
n = 18 19 20 21 22 23 24 25 26 27 28 29 30
1 -1.82 -1.85 -1.87 -1.89 -1.91 -1.93 -1.95 -1.97 -1.98 -2.00 -2.01 -2.03 -2.04
2 -1.35 -1.38 -1.40 -1.43 -1.45 -1.48 -1.50 -1.52 -1.54 -1.56 -1.58 -1.59 -1.61
3 -1.06 -1.10 -1.13 -1.16 -1.18 -1.21 -1.24 -1.26 -1.28 -1.30 -1.32 -1.34 -1.36
4 -0.84 -0.88 -0.92 -0.95 -0.98 -1.01 -1.04 -1.06 -1.09 -1.11 -1.13 -1.15 -1.17
5 -0.66 -0.70 -0.74 -0.78 -0.81 -0.84 -0.87 -0.90 -0.93 -0.95 -0.98 -1.00 -1.02
6 -0.50 -0.54 -0.59 -0.63 -0.66 -0.70 -0.73 -0.76 -0.79 -0.82 -0.84 -0.87 -0.89
7 -0.35 -0.40 -0.45 -0.49 -0.53 -0.57 -0.60 -0.63 -0.66 -0.69 -0.72 -0.75 -0.77
8 -0.21 -0.26 -0.31 -0.36 -0.40 -0.44 -0.48 -0.52 -0.55 -0.58 -0.61 -0.64 -0.67
9 -0.07 -0.13 -0.19 -0.24 -0.28 -0.33 -0.37 -0.41 -0.44 -0.48 -0.51 -0.54 -0.57
10 0.07 0.00 -0.06 -0.12 -0.17 -0.22 -0.26 -0.30 -0.34 -0.38 -0.41 -0.44 -0.47
11 0.21 0.13 0.06 0.00 -0.06 -0.11 -0.15 -0.20 -0.24 -0.28 -0.31 -0.35 -0.38
12 0.35 0.26 0.19 0.12 0.06 0.00 -0.05 -0.10 -0.14 -0.18 -0.22 -0.26 -0.29
13 0.50 0.40 0.31 0.24 0.17 0.11 0.05 0.00 -0.05 -0.09 -0.13 -0.17 -0.21
14 0.66 0.54 0.45 0.36 0.28 0.22 0.15 0.10 0.05 0.00 -0.04 -0.09 -0.12
15 0.84 0.70 0.59 0.49 0.40 0.33 0.26 0.20 0.14 0.09 0.04 0.00 -0.04
16 1.06 0.88 0.74 0.63 0.53 0.44 0.37 0.30 0.24 0.18 0.13 0.09 0.04
17 1.35 1.10 0.92 0.78 0.66 0.57 0.48 0.41 0.34 0.28 0.22 0.17 0.12
18 1.82 1.38 1.13 0.95 0.81 0.70 0.60 0.52 0.44 0.38 0.31 0.26 0.21
19 1.85 1.40 1.16 0.98 0.84 0.73 0.63 0.55 0.48 0.41 0.35 0.29
20 1.87 1.43 1.18 1.01 0.87 0.76 0.66 0.58 0.51 0.44 0.38
21 1.89 1.45 1.21 1.04 0.90 0.79 0.69 0.61 0.54 0.47
22 1.91 1.48 1.24 1.06 0.93 0.82 0.72 0.64 0.57
23 1.93 1.50 1.26 1.09 0.95 0.84 0.75 0.67
24 1.95 1.52 1.28 1.11 0.98 0.87 0.77
25 1.97 1.54 1.30 1.13 1.00 0.89
26 1.98 1.56 1.32 1.15 1.02
27 2.00 1.58 1.34 1.17
28 2.01 1.59 1.36
29 2.03 1.61
30 2.04
Normal scores (continued)
H1518_Burke_appendix.indd 556 6/29/17 11:52 AM
557
df t0.10 t0.05 t0.025 t0.01 t0.005 df
1 3.078 6.314 12.706 31.821 63.656 1
2 1.886 2.920 4.303 6.965 9.925 2
3 1.638 2.353 3.182 4.541 5.841 3
4 1.533 2.132 2.776 3.747 4.604 4
5 1.476 2.015 2.571 3.365 4.032 5
6 1.440 1.943 2.447 3.143 3.707 6
7 1.415 1.895 2.365 2.998 3.499 7
8 1.397 1.860 2.306 2.896 3.355 8
9 1.383 1.833 2.262 2.821 3.250 9
10 1.372 1.812 2.228 2.764 3.169 10
11 1.363 1.796 2.201 2.718 3.106 11
12 1.356 1.782 2.179 2.681 3.055 12
13 1.350 1.771 2.160 2.650 3.012 13
14 1.345 1.761 2.145 2.624 2.977 14
15 1.341 1.753 2.131 2.602 2.947 15
16 1.337 1.746 2.120 2.583 2.921 16
17 1.333 1.740 2.110 2.567 2.898 17
18 1.330 1.734 2.101 2.552 2.878 18
19 1.328 1.729 2.093 2.539 2.861 19
20 1.325 1.725 2.086 2.528 2.845 20
21 1.323 1.721 2.080 2.518 2.831 21
22 1.321 1.717 2.074 2.508 2.819 22
23 1.319 1.714 2.069 2.500 2.807 23
24 1.318 1.711 2.064 2.492 2.797 24
25 1.316 1.708 2.060 2.485 2.787 25
26 1.315 1.706 2.056 2.479 2.779 26
27 1.314 1.703 2.052 2.473 2.771 27
28 1.313 1.701 2.048 2.467 2.763 28
Continued
Values of t distribution
appendix O Values of t Distribution
H1518_Burke_appendix.indd 557 6/29/17 11:52 AM
558 Appendix O Values of t Distribution
df df
29 1.311 1.699 2.045 2.462 2.756 29
30 1.310 1.697 2.042 2.457 2.750 30
31 1.309 1.696 2.040 2.453 2.744 31
32 1.309 1.694 2.037 2.449 2.738 32
33 1.308 1.692 2.035 2.445 2.733 33
34 1.307 1.691 2.032 2.441 2.728 34
35 1.306 1.690 2.030 2.438 2.724 35
40 1.303 1.684 2.021 2.423 2.704 40
45 1.301 1.679 2.014 2.412 2.690 45
50 1.299 1.676 2.009 2.403 2.678 50
55 1.297 1.673 2.004 2.396 2.668 55
60 1.296 1.671 2.000 2.390 2.660 60
70 1.294 1.667 1.994 2.381 2.648 70
80 1.292 1.664 1.990 2.374 2.639 80
90 1.291 1.662 1.987 2.368 2.632 90
100 1.290 1.660 1.984 2.364 2.626 100
200 1.286 1.653 1.972 2.345 2.601 200
400 1.284 1.649 1.966 2.336 2.588 400
600 1.283 1.647 1.964 2.333 2.584 600
800 1.283 1.647 1.963 2.331 2.582 800
999 1.282 1.646 1.962 2.330 2.581 999
Values of t distribution (continued)
t0.10 t0.05 t0.025 t0.01 t0.005
H1518_Burke_appendix.indd 558 6/29/17 11:52 AM
559
appendix P Selected National and International
Quality System Standards
american National standards institute* 1430 Broadway New York, NY 10018
ANSI/ASQ Z1.4-2003 (R2013) Sampling Procedures and Tables for Inspection by Attributes
ANSI/ASQ Z1.9-2003 (R2013) Sampling Procedures and Tables for Inspection by Variables for Percent Nonconforming
ANSI/ASQC C1-1996 (ANSI Z1.8-1971) Specifications of General Requirements for a Quality Program
ANSI/ASQC D1160-1995 Formal Design Review
ANSI/ISO/ASQ Q10005-2005 Quality management—Guidelines for quality plans
ANSI/ISO/ASQ Q10006-2003 Quality management—Guidelines for quality management in projects
ANSI/ISO/ASQ Q9004-2009 Quality management systems—Guidelines for performance improvements
ANSI/ISO/ASQ QE19011S-2008 Guidelines for management systems auditing— U.S. version with supplemental guidance added
ASQ/ANSI/ISO 14001:2015 Environmental management systems—Requirements with guidance for use
ASQ/ANSI/ISO 9000:2015 Quality management systems—Fundamentals and vocabulary
ASQ/ANSI/ISO 9001:2015 Quality management systems—Requirements
* Copies of these standards can be ordered in print or electronic format from American Society for Quality (ASQ), PO Box 3005, Milwaukee, WI 53201-3005, www.asq.org/quality-press.
H1518_Burke_appendix.indd 559 6/29/17 11:52 AM
560 Appendix P Selected National and International Quality System Standards
ISO 10002:2014 Quality management—Customer satisfaction—Guidelines for complaints handling in organizations
ISO 10007:2017 Quality management—Guidelines for configuration management
ISO/TS 9002:2016 Quality management systems—Guidelines for the application of ISO 9001:2015
North atlantic treaty Organization Autoroute De Zaventem 1110 NATO (Brussels), Belgium
AQAP-1: NATO Requirements for an Industrial Quality Control System
AQAP-2: Guide for the Evaluation of a Contractor’s Quality Control System for Compliance with AQAP-1
AQAP-4: NATO Inspection Systems Requirements for Industry
AQAP-5: Guide for the Evaluation of a Contractor’s Inspection System for Compliance with AQAP-4
AQAP-7: Guide for the Evaluation of a Contractor’s Measurement and Calibration System for Compliance with AQAP-6
IEC Guide 102 (1996-03): Specifications Structure for Quality
british standards institution 101 Pentonville Road London N19ND, England
BSI HDBK 22-1981: Quality Assurance (Contains 15 Publications)
Canadian standards association (Csa) 178 Rexdale Boulevard Rexdale, Ontario Canada M9W IR3
CAN3 Z299-1—CSA: Quality Assurance Program—Category 1
CAN3 Z299-2—CSA: Quality Assurance Program—Category 2
CAN3 Z299-3—CSA: Quality Assurance Program—Category 3
H1518_Burke_appendix.indd 560 6/29/17 11:52 AM
Appendix P Selected National and International Quality System Standards 561
international Organization for standardization (isO)* 1, rue de Varembé, Case postale 56 CH-1211 Geneva 20, Switzerland
ISO 9000:2015: Quality management systems—Fundamentals and vocabulary
ISO 9001:2015: Quality management systems—Requirements
ISO 9004:2009: Quality management systems—Guidelines for performance improvements
ISO 10019:2005: Guidelines for the selection of quality management system con- sultants and use of their services
ISO 13485:2016: Medical devices: Quality management systems—Requirements for regulatory purposes
ISO/TR 10013:2001: Guidelines for quality management system documentation
the Department of Defense (DOD) The Pentagon Washington, DC 20301-1155
MIL-HDBK-50: Evaluation of a Contractor’s Quality Program
MIL-Q-9858A: Quality Program Requirements
MIL-STD-1521B: Technical Reviews and Audits of System, Equipment, and Computer Software
MIL-STD-1535A: Supplier Quality Assurance Quality Requirements
MIL-STD-2164: Failure Reporting, Analysis, and Corrective Action Systems
MIL-T-50301: Quality Control System Requirements for Technical Data
* Copies of these standards can be ordered in print or electronic format from American Society for Quality (ASQ), PO Box 3005, Milwaukee, WI 53201-3005, www.asq.org/quality-press.
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563
For many of the terms in this glossary, we provide a chapter (X) and section (Y) refer- ence in the form [X.Y]. Please note that some of the definitions in this glossary are based on the references in the chapter and section mentioned.
#
2k designs—A factorial design with k factors, each with two levels. These designs are referred to as “two-to-the-k” designs. See also factorial design. [6.H]
5S—A lean methodology of visual control based on five Japanese words, each begin- ning with the letter s: seiri (sort), seiton (straighten), seiso (shine), seiketsu (standard- ize), and shitsuke (sustain). [5.D]
a
acceptable quality limit (AQL)—The maximum percentage or proportion of variant units in a lot or batch that, for purposes of acceptance sampling, can be considered satisfactory as a process average. [4.C]
acceptance sampling—Sampling inspection in which decisions are made to accept or not accept a product or service; also, the methodology that deals with procedures by which decisions to accept or not accept are based on the results of the inspection of samples. [4.C]
accuracy—A qualitative term that describes the closeness of alignment between an observed value and an accepted reference value.
action plan—The detailed plan to implement the actions needed to achieve strategic goals and objectives. [1.B]
activity network diagram (AND) (arrow diagram)—A management and planning tool used to develop the best possible schedule and appropriate controls to accomplish the schedule; the critical path method (CPM) and the program evaluation review technique (PERT) make use of arrow diagrams. [5.B]
advanced product quality planning (APQP) and control plan—APQP is a compre- hensive quality planning and control system specifying protocols for product and process design and development, validation, assessment, and corrective action. [7.B]
glossary
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564 Glossary
advanced quality planning (AQP)—A comprehensive system of applying quality dis- ciplines during a product or process development effort. [4.A]
affinity diagram—A quality management tool used to help collect, organize, summa- rize, and communicate facts, opinions, and ideas based on natural relationships. [5.B]
aliasing—See confounding. [6.H]
alternative hypothesis (Ha)—In statistical hypothesis testing, this is the hypothesis that the null hypothesis is tested against. The hypothesis test is conducted under the assumption that the null hypothesis is true. If evidence is found against the null hypothesis, the null hypothesis is rejected. [6.D]
American National Standards Institute (ANSI)—An organization that creates and pro- motes guidelines and standards across many industries.
American Society for Quality (ASQ)—An organization that provides the community with training, professional certifications, and knowledge related to quality.
analysis of variance (ANOVA)—A partitioning of total variability into components due to factors or other sources of variation. The sources of variation as well as their cor- responding sums of squares and degrees of freedom are usually given in an analysis of variance table. [6.D]
appraisal costs—The costs associated with measuring, evaluating, or auditing products or services to ensure conformance to quality standards and performance require- ments. [2.E]
assignable causes—See special causes. [6.F]
assumptions—Conditions that must be true in order for a statistical procedure to be valid.
attributes data—Data that are categorized for analysis or evaluation. Attributes data may involve measurements as long as the measurements are used only to place a given piece of data in a category for further analysis or evaluation. Contrast with variables data. [6.A]
audit—A systematic and independent evaluation of the quality system and its execu- tion. [2.D]
auditee—The individual or organization being audited. [2.D]
autocorrelation—A measure of the linear relationship between sequential observations, typically associated with time series. [6.E]
availability—The probability that a system is properly operating at time t. [3.E]
average outgoing quality (AOQ)—The expected average quality of outgoing product following the use of an acceptance sampling plan for a given value of incoming product quality. [4.C]
average outgoing quality limit (AOQL)—For a given acceptance sampling plan, the maximum AOQ over all possible levels of incoming quality. [4.C]
average run length (ARL)—In process monitoring and statistical process control, the average number of time periods or samples that elapse until the process signals out- of-control or produces an out- of-control signal. [6.F]
average sample number (ASN)—The average number of sample units per lot used for making decisions (acceptance or nonacceptance). [4.C]
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Glossary 565
B
bathtub curve—A general failure rate model for the life cycle of a system with the fail- ure rate plotted against time. Named because of its shape, the curve has a decreasing failure rate (see infant mortality phase), followed by constant failure rate, followed by an increasing failure rate (see wear-out phase). [3.E]
benchmark—An organization, part of an organization, or measurement that serves as a reference point or point of comparison. [1.B]
benefit–cost analysis—A collection of the dollar value of benefits derived from an ini- tiative, divided by the associated costs incurred.
Bernoulli trial—A trial that results in one of two possible outcomes, usually defined as “success” and “failure.” [6.C]
bias—A quantitative term representing the systematic difference between results or measurements obtained and the true quantity of interest. In measurement system analysis, bias describes the difference between the average of measurements made on the same unit and its reference or master value. [4.F]
binomial distribution—A discrete distribution describing the number of successes in a set or series of n independent Bernoulli trials where the probability of a success is constant from trial to trial. [6.C]
bivariate distribution—A joint distribution of two random variables. The joint distribu- tion of two normally distributed random variables is the bivariate normal distribu- tion. [6.C]
block diagram—A diagram that describes the operation, interrelationships, and inter- dependencies of components in a system. Boxes, or blocks (hence the name), repre- sent the components; connecting lines between the blocks represent interfaces. See also functional block diagram and reliability block diagram.
blocking—A principle of experimental design used to group experiments into relatively homogenous experimental conditions in order to reduce the variability transmitted from nuisance factors. [6.H]
box plot—A graphical method for displaying characteristics of a set of data. The box represents the interquartile range (middle 50% of the data). The whiskers extend from each end of the box to some specified bounds, usually the minimum and maxi- mum. Also known as the box-and-whisker plot. [6.A]
brainstorming—A problem- solving tool that teams use to generate as many ideas as possible related to a particular subject. Team members begin by offering all their ideas; the ideas are not discussed or reviewed until after the brainstorming session. [1.E]
C
c chart—A control chart that monitors the number of nonconformities in a process. Compare with u chart. [6.F]
calibrate—To determine whether an instrument is functioning within prescribed accu- racy objectives. [4.E]
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566 Glossary
calibration—The disciplines necessary to compare and/or control a measurement instrument or system of unverified accuracy with a measurement instrument or sys- tem of known accuracy. Calibration can also be used to detect any variation from the true value. [4.E]
categorical variable—A variable whose possible outcomes are categories that have no numerical significance. [6.A]
causation—The principle that a change in a factor or regressor variable causes a change in the response. Can only be determined by a designed experiment. Contrast with correlation. [6.E]
cause-and-effect diagram—A graphical aid used to organize and identify possible causes of a problem or effect. The causal factors are variables that when changed or manipulated may result in an effect. Also known as a fishbone diagram or Ishikawa diagram. [5.A]
central limit theorem—The principle that the distribution of the sum of independent, identically distributed random variables approaches a normal distribution as the sample size increases toward infinity. Often used to approximate the distribution of sample means. [6.C]
certified quality engineer (CQE)—A professional who understands the application of the quality principles of product and service quality evaluation and control, which are outlined by the American Society for Quality.
chance cause variation—Variation due to an inherent part of the process. Also known as common cause variation. [6.F]
change agent—The person who takes the lead in transforming a company into a quality organization by providing guidance during the planning phase, facilitating imple- mentation, and supporting those who pioneer the changes.
check sheet—One of the seven quality control tools; used to count event occurrences when collecting data. Check sheets are often used to summarize types of defects. [5.A]
chi-square distribution—A continuous probability distribution that results from the sum of k squared independent normal random variables. [6.C]
coefficient of determination (R2)—The proportion of the total variability in the response that can be explained by the regression line. Provides a measure (between 0 and 1 inclusive) of how adequate the current regression model is for a particular set of data. [6.E]
complement—In probability, an event that contains all the outcomes in the sample space that are not in the event itself. The complement of an event A is denoted A’ or AC. [6.B]
confidence interval—An interval of values (L, U) that is believed to contain the true parameter value of interest. The probability level refers only to the interval con- structed and its properties and not the unknown parameter being estimated. [6.D]
confidence level—Is equal to 100 multiplied by (1 – significance level). The interpre- tation of a 99% confidence interval would be that if the estimation procedure is repeated over and over again, 99% of all the constructed intervals would contain the true parameter of interest. [6.D]
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Glossary 567
conflict resolution—A process for resolving disagreements in a manner acceptable to all parties. [1.E]
confounding—A design of experiments term that refers to indistinguishable effects. Two factors that are confounded means that their effects cannot be distinguished from one another. Mathematically, this means that the effect calculations are com- puted by the same linear combination. [6.D]
consumer’s risk (β)—For a sampling plan, refers to the probability of acceptance of a lot, the quality of which has a designated numerical value representing a level that is seldom desirable. Usually the value will be the lot tolerance percent defective (LTPD). Also known as the beta risk or probability of a type II error. [4.C]
contingency table—A table for grouping data from two or more categorical variables. Categories can be represented as rows and columns. The values in the table cells represent counts. [6.D]
continuous variable—A numerical variable that can take on any possible value inside a provided interval. [6.A]
contour plot—A graph that displays the predicted response over a range of the vari- ables in a regression equation. Can be used to determine optimal or robust settings of factors. [6.H]
control chart—A chart used to monitor a critical- to-quality characteristic of interest. A control chart generally consists of three horizontal lines: one representing the mean or target level, one representing an upper limit, and one representing a lower limit (although there are many instances when only an upper limit or only a lower limit is of interest). The limits are statistically determined. [6.F]
control factor—In design of experiments, a factor or process input that can be manipu- lated by the experimenter and is presumed to affect the output of a process. [6.H]
control plan—A document used to communicate the procedures used to monitor and control a process. [4.A]
coordinate measuring machine (CMM)—A machine used to calculate a physical repre- sentation of a three- dimensional rectilinear coordinate system. CMMs are used for defining the geometry of different- shaped workpieces. [4.D]
corrective action—Action taken to eliminate the root cause(s) and symptom(s) of an existing deviation or nonconformity to prevent recurrence. [5.E]
correlation—A general term describing the degree of interdependence between two or more variables. [6.E]
correlation coefficient—A measure of the linear relationship between two random variables. The correlation coefficient is dimensionless and can take on any value between –1 and 1 inclusive. [6.E]
cost of quality—A methodology that allows an organization to determine the extent to which its resources are used for activities that prevent poor quality, that appraise the quality of the organization’s products or services, and that result from internal and external failures. [2.E]
Crawford slip method—A method of gathering and presenting anonymous data from a group by using various voting schemes. [1.E]
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568 Glossary
critical defect—A defect that may lead to severe injury or catastrophic loss. [4.B]
critical path—The sequence of tasks that take the longest time and determine a project’s completion date. [1.B]
critical path method (CPM)—An activity- oriented project management technique that uses arrow- diagramming techniques to demonstrate both the time and the cost required to complete a project. It provides one time estimate—normal (most likely) time. [1.B]
critical region—In hypothesis testing, it is the value of the test statistic that will lead to rejection of the null hypothesis. [6.D]
critical value—In hypothesis testing, the value (or values) with which the value of the test statistic is compared to determine whether the null hypothesis can be rejected. The critical value is determined from the significance level of the test. [6.D]
criticality—An indication of the consequences that are expected to result from a fail- ure. [7.B]
critical-to-quality characteristic—The most important or key features of a product, pro- cess, or service. [5.C]
cumulative distribution function (cdf)—Used to describe a probability distribution for a random variable. For a random variable X, the cumulative distribution function would be given by P(X ≤ x), where x is some numeric value. [6.C]
cumulative sum (CUSUM)—The sum of the current and past observations of a sequence of data. [6.F]
cycle time—The time that it takes to complete a process from beginning to end. [5.D]
d
dashboard—A visual display that shows at- a-glance key business indicators. [1.B]
decision tree—A planning tool used to help estimate the expected value of gain or loss in a project. The tree lists potential outcomes and the financial payout for each out- come, with a probability assigned to each branch. [1.B]
defect—A nonconformity severe enough to cause the product to not satisfy normal usage requirements. [4.B]
defining relation—In design of experiments, an expression for a fractional factorial design that contains all possible combinations of columns in the design matrix that do not change (are equal to the identity column). [6.H]
density function—See probability density function. [6.C]
dependent events—Two events A and B are dependent if the probability of one event occurring is affected by the occurrence of the other event. Contrast with independent events. [6.B]
descriptive statistics—Techniques for displaying and summarizing data. Examples include histograms, run charts, and summary statistics such as the mean and stan- dard deviation. [6.A]
design of experiments (DOE)—A formal method including pre- experimental planning, setting up and running an experiment, analyzing the data, and drawing objective conclusions. [6.H]
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Glossary 569
design review—A documented, comprehensive, and systematic examination of a design to evaluate its capability to fulfill the requirements for quality. [3.B]
designed experiment—A formal statistical process for determining the test or series of runs in which the experimenter manipulates the factor levels to determine potential causal effects on the response. [6.H]
detection—A criterion for risk used in the risk priority number and is defined as the likelihood of detecting a failure once it has occurred. Detection is evaluated based on a 10-point scale. At the lowest end of the scale (1) it is assumed that a design control will detect a failure with certainty. At the highest end of the scale (10) it is assumed that a design control will not detect a failure if a failure occurs. [7.B]
dial indicator—A measurement tool that magnifies the dimension deviation from a standard to which a gage is set. [4.D]
discrete variable—A numeric variable whose possible values form a finite or at most countably infinite set. [6.A]
disjoint events—See mutually exclusive events. [6.B]
distribution function—See cumulative distribution function. [6.C]
DMAIC—An acronym denoting a sequence used in the methodology most often associ- ated with Six Sigma: define, measure, analyze, improve, control. [5.C]
E
effect estimate—In design of experiments, the difference in the average response at the high and low levels of a factor or combination of factors. [6.H]
environmental stress screening—A process designed to trigger burgeoning defects into detectable failures by applying environmental stresses, such as temperature or vibration, to hardware. [7.B]
error—1. The difference between the estimated value and the true value of a measured quantity. 2. A fault resulting from defective judgment, deficient knowledge, or care- lessness. It is not to be confused with measurement error, which is the difference between a computed or measured value and the true or theoretical value.
error-proofing—See foolproofing. [5.F]
estimate—A numerical value for a population parameter based on information col- lected from a sample. Also known as a point estimate. [6.D]
estimator—A statistic used to compute estimates of a parameter. [6.D]
event—In probability, a subset of the sample space. [6.B]
expected frequency—In the chi- squared goodness- of-fit test, the number of noncon- forming units that would be expected in each category if the sample exactly fol- lowed the historical percentages. [6.D]
expected value—The mean of a random variable. [6.C]
experiment—See designed experiment. [6.H]
experimental error—See random error.
exponential distribution—A continuous probability distribution often used to model problems in reliability that have a constant failure rate. [6.C]
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570 Glossary
exponentially weighted moving average (EWMA)—A statistic that takes a sequential weighted average of previous observations in a series of data. [6.F]
external failure costs—Costs of quality associated with defects found during or after delivery of the product or service. [2.E]
extrapolation—In regression analysis, the process of predicting new observations with predictor values outside the range of the original data. Contrast with interpolation. [6.E]
F
F distribution—A continuous probability distribution defined by the ratio of two inde- pendent chi- square random variables divided by their respective degrees of free- dom. [6.C]
facilitator—An individual who is responsible for creating favorable conditions that will enable a team to reach its purpose or achieve its goals by bringing together the nec- essary tools, information, and resources to get the job done. Provides support to the team while allowing the team to maintain ownership of its decisions. [1.E]
factor—In design of experiments, an independent variable chosen by the experimenter to determine what effect, if any, it has on the response in an experiment. [6.H]
factorial design—In design of experiments, a type of design where all possible combi- nations of factor levels are examined. [6.H]
fail-safe device—A method of preventive action to ensure problems or abnormalities in a process are discovered in a manner that maintains a safe working environment and ensures that quality is not compromised. [5.F]
failure—The termination, due to one or more defects, of the ability of an item, product, or service to perform its required function when called on to do so. A failure may be partial, complete, or intermittent.
failure density function—A probability density function that represents the distribu- tion of failure time. [3.E]
failure mode effects and criticality analysis (FMECA)—A failure modes and effects analysis (FMEA) that includes a criticality metric in the evaluation of potential fail- ure modes of a system, subsystem, product, or process. [7.B]
failure modes and effects analysis (FMEA)—A team- based problem- solving procedure for helping users identify and eliminate or reduce the negative effects of potential failures by evaluating a risk priority number of potential failure modes of a system, subsystem, product, or process. [7.B]
fault tree—A top- down technique for analyzing complex systems to determine poten- tial failure modes and the probabilities of their occurrence. Some of the symbols in the fault tree include AND gate, OR gate, basic fault events, and priority AND gate. [7.B]
filters—Relative to human- to-human communication, those perceptions (based on cul- ture, language, demographics, experience, etc.) that affect how a message is trans- mitted by the sender and how a message is interpreted by the receiver.
fishbone diagram—See cause-and-effect diagram. [5.A]
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Glossary 571
fixed effect—In analysis of variance, an effect is fixed if the factor levels included in the test or experiment are the only levels of interest. That is, the levels of the factor included in the experiment are the only ones to which the results of testing will apply.
flowchart—One of the seven tools of quality; a graphical representation of the elements, components, or tasks associated with a process. [5.A]
foolproofing—A process in preventive action to make a product or process immune to errors on the part of the user or operator. Synonymous with error-proofing. [5.F]
force field analysis—A quality management method for organizing ideas based on driving and opposing forces associated with a desired change in an organization. Compare with affinity diagram. [5.B]
fraction nonconforming—In quality control, the proportion of the total number of units under study that do not meet specifications. Also known as fraction defective.
fractional factorial design—In design of experiments, a design consisting of only a sub- set or fraction of all possible combinations of a factorial design. [6.H]
functional block diagram—A block diagram that shows a system’s subsystems and lower- level products, their interrelationships, and interfaces with other systems. [6.B]
g
gage block—A system for producing precision lengths; may be used as a reference when calibrating dial indicators. [4.D]
gage repeatability and reproducibility (gage R&R)—Measures the capability of a gage to determine whether it is suitable for use in its intended application. Repeatability represents the gage variability when the gage is used to measure the same unit with the same setup or operator. Reproducibility refers to the variability arising from dif- ferent setups or operators.
Gantt chart—A type of bar chart used in process or project planning and control to dis- play planned and finished work in relation to time. Also called a milestone chart. [1.B]
gauging—A procedure that determines product conformance with specifications with the aid of measuring instruments such as calipers, micrometers, templates, and other mechanical, optical, and electronic devices. [4.D]
Gaussian distribution—See normal distribution. [6.C]
go/no-go gage—A tool to measure inspection by attributes, made to sizes identical to the design specification limits of the dimension to be inspected. The “go” end checks the characteristic at the maximum material condition while the “no-go” end detects conditions of excessive clearance. Also called a limit gage. [4.D]
H
hazard and operability analysis (HAZOP)—A technique used to identify operability issues and potential hazards that may lead to unacceptable products, processes, ser- vices, or risk to personnel. [7.B]
hazard rate function—A function defined by the limit of the failure rate as the time interval approaches zero; provides an instantaneous failure rate at time t. [3.E]
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572 Glossary
heredity—In design of experiments, the principle that if an interaction between two main effects is significant, then both main effects are also significant. [6.H]
hierarchical relationship—A set of relationships that can be ordered or arranged from general to specific. [1.B]
hierarchy—In design of experiments, the principle that if a higher- order term is signifi- cant, then a lower- order term also containing that factor should be included in the model. [6.H]
histogram—A graphical display of observations from a sample where the class frequen- cies are represented by areas of rectangles over the interval for each class. [6.A]
hold point—A point, defined in an appropriate document, beyond which an activity must not proceed without the approval of a designated organization or authority.
hypothesis testing—A formal statistical procedure for testing a statement about a pop- ulation using sample data. The statement to be tested may also concern a distribu- tional form of a quality characteristic of interest. [6.D]
i
I and MR chart—A pair of control charts used to monitor a variable in which the sample size is one (also called individual measurements). The MR values are typically cal- culated by computing the difference between sequential pairs of individual observa- tions (see moving range [MR]) and are representative of sample variability. [6.F]
independent events—Two events A and B are said to be independent if the occurrence of one event does not depend on the occurrence or lack of occurrence of another (or preceding) event. If two events are independent, then the probability that they both occur is the product of the probabilities of their individual occurrence. [6.B]
Industry 4.0—A concept that refers to a fourth industrial revolution in manufacturing, often characterized by big data, advanced analytics, and human- machine interfaces. [1.B]
infant mortality phase—In reliability, represents the first phase in a bathtub curve. It is often characterized by a decreasing failure rate where failures are typically attrib- uted to defects in the manufacturing processes, assemblies, and shipping of the product. [3.E]
inferential statistics—Techniques for reaching conclusions about a population based on analysis of data from a sample. [6.B]
information system—Technology-based system used to support operations, aid day- to-day decision making, and support strategic analysis (other names often used include management information system, decision system, information technology [IT], and data processing). [1.B]
inspection—The process of measuring, examining, testing, gauging, or otherwise com- paring a unit with the applicable requirements. [1.H]
interaction—A term used to describe that the relationship between a response variable and an input variable may change in the presence of one or more other variables. [6.D]
interaction plot—A graph in which the average response (y-axis) is plotted against a factor (x-axis) over various levels of a second factor, used to determine the presence of an interaction between two factors. [6.H]
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Glossary 573
internal failure costs—Costs associated with defects found before the product or ser- vice is delivered; including scrap, rework, and material review. [2.E]
International Organization for Standardization (ISO)—The organization responsible for the ISO 9000 and other management standard series.
interpolation—In regression analysis, the process of predicting new observations with predictor values inside the range of the original data. Contrast with extrapolation. [6.E]
interquartile range—The difference between the 75th and the 25th quantiles. [6.B]
interrelationship digraph—A tool used to help discover, visualize, and communicate a high- level sequential and/or cause- and-effect relationship. [5.B]
intersection—In probability, the event consisting of all outcomes that are contained in both A and B. [6.B]
Ishikawa diagram—See cause-and-effect diagram. [5.A]
J
joint distribution—A probability distribution representing two or more variables that are involved in a random experiment. Also known as joint probability distribution.
just-in-time—A lean principle that refers to the delivery of material, components, or parts just prior to their use in order to minimize inventory costs. [5.D]
K
kaizen—A Japanese word for the philosophy that defines management’s role in con- tinuously encouraging small improvements involving everyone in an organization. [5.C]
kaizen blitz—An intense team approach to employing the concepts and techniques of continuous improvement in a short time frame (e.g., to reduce cycle time or increase throughput). [5.D]
kanban—A system to simplify and improve inventory resupply procedures. [5.D]
k-out-of-n system—A system with n components or subcomponents where k of the components must be functioning for the system to operate properly. [3.E]
L
lean—A process for eliminating waste from a system, such that only value- added activi- ties remain. [5.D]
least squares estimation—In regression analysis, a method for estimating parameters by minimizing the sum of the squared differences between the actual or observed responses and the values predicted by the fitted model. [6.E]
level of significance (α)—See significance level. [6.D] levels—In experimental design, the chosen values of a factor of interest to be varied in
an experiment. [6.H]
limit gage—See go/no-go gage. [4.D]
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574 Glossary
linear regression—See regression analysis. [6.E]
linearity—A measure of how changes in the size of the part being measured will affect measurement system bias over the expected process range. [4.F]
lognormal distribution—A continuous probability distribution often used to model product life spans. It is the distribution of a random variable whose logarithm fol- lows the normal distribution. [6.C]
lot tolerance percent defective (LTPD)—Expressed in percent defective, the poorest quality in an individual lot that should be accepted. [4.C]
M
main effect—The effect on a response due to a change in a factor or variable indepen- dent of all other factors or variables in the system. [6.D]
main effect plot—A plot that represents the average change in the response (plotted on the y-axis) over the values of a particular factor (plotted on the x-axis). [6.H]
maintainability—The measure of the ability of an item to be retained or restored to a specified condition when maintenance is performed by personnel having specified skill levels and using prescribed procedures and resources at each prescribed level of maintenance and repair. [3.E]
major defect—A defect that interferes with normal or reasonable foreseeable use but does not cause a risk of damage or injury. [4.B]
material control—A broad collection of tools for managing the items and lots in a pro- duction process. [4.B]
materials review board—A quality control committee or team, usually employed in manufacturing or other materials- processing installations, that has the responsibility and authority to deal with items or materials that do not conform to fitness- for-use specifications. [4.B]
matrix diagram—A tool used to help people discover, visualize, and communicate rela- tionships within a single set of factors or between two or more sets of factors. [5.B]
mean—A measure of central tendency. For random variables, it is also the expected value. For a sample of data of size n, it is the sum of the observations divided by n. [6.A]
mean squares—In analysis of variance, mean squares are estimates of variances. In gen- eral, they are found by dividing the sum of squares by the appropriate degrees of freedom. [6.D]
mean time between failures (MTBF)—The expected time between two successive fail- ures when the system is repairable. [3.E]
mean time to failure (MTTF)—The expected time to failure between two successive failures when the system is nonrepairable. [3.E]
mean time to repair (MTTR)—The expected time to repair a failure, not including wait- ing time for parts or tools to start the repair. [3.E]
measurement—1. The process of evaluating a property or characteristic of an object and describing it with a numerical or nominal value. 2. A series of manipulations of physi- cal objects or systems according to a defined protocol that results in a number. [4.D]
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Glossary 575
measurement process—Repeated application of a test method using a measuring sys- tem. [4.D]
measurement system—The entire process for obtaining measurement on some quality characteristic of interest, including standards, personnel, and methods of measure- ment. [4.F]
measurement system analysis (MSA)—Qualifying the measurement process, deter- mining the adequacy of the measurement system for use, and identifying and esti- mating the process error. [4.F]
measuring system—In general, the elements of a measuring system include the instru- mentation, calibration standards, environmental influences, human operator limita- tions, and features of the workpiece or object being measured. [4.D]
median—A measure of central tendency that divides an ordered data set in half; 50% of the data are at or below this value and 50% of the data are above this value. [6.A]
metrology—The science of precision measurement. [4.E]
milestone—A specific time when a critical event is to occur; a symbol placed on a mile- stone chart to locate the point when a critical event is to occur.
milestone chart—See Gantt chart. [1.B]
minor defect—A defect that may cause difficulty in assembly or use of a product but does not prevent the product from being properly used and does not pose any haz- ard to users. [4.B]
mode—The value in a data set that occurs most often. There can be more than one mode for a sample. [6.A]
moving average—An unweighted average of observations in a series of data over a specified span. Contrast with exponentially weighted moving average. [6.F]
moving range (MR)—The difference between two successive observations. [6.F]
muda—The seven classes of waste: overproduction, delay, transportation, processing, inventory, wasted motion, and defective parts. [5.D]
multivoting—A decision- making tool that enables a group to sort through a long list of ideas to identify priorities. [1.E]
mutually exclusive events—Events that do not have outcomes in common or that do not occur jointly. [6.B]
Myers-Briggs Type Indicator—A method and instrument for assessing personality that can be used for building a team with complementary skills. Based on Carl Jung’s theory of personality preferences. [1.B]
n
National Institute of Standards and Technology (NIST)—An organization for the stan- dards of measurement, established by an act of Congress. NIST maintains the base units of measurements that are used in calibration. [4.D]
natural tolerance limits—Limits based on the natural variation of the process (mea- sured by the process standard deviation). [6.G]
noise factor—See nuisance factor. [6.H]
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576 Glossary
nominal group technique—A technique similar to brainstorming, used by teams to generate and make a selection from ideas on a particular subject based on group prioritization. [1.E]
nominal variable—A categorical variable with no order implied in the values. [6.A]
nonconformity—A failure of a quality characteristic to meet its intended level or state, occurring with severity sufficient to cause the product to not meet a specification. Sometimes the specification is based on fitness- for-use requirements. [4.B]
non-value-added activity—Activities that the customer is not willing to pay for and/or do not change the form or function of the product or service. [5.D]
normal distribution—A symmetric, bell- shaped continuous probability distribution. Another name for the Gaussian distribution, often attributed to Karl Gauss. [6.C]
normal probability plot—See probability plot. [6.A]
np chart—A control chart used to monitor the number of nonconforming units in a sample. Compare with p chart. [6.F]
nuisance factor—A factor that may influence the response in an experiment (see designed experiment) but is not of direct interest to the experimenter. [6.H]
null hypothesis—In hypothesis testing, a statement about a population parameter or distributional form of a quality characteristic that is to be tested. It is often the state- ment of no difference.
o
observation—The process of determining the presence or absence of attributes or mak- ing measurements of a variable. Also, the result of the process of determining the presence or absence of attributes or making a measurement of a variable.
observational study—Analysis of data collected from a process without imposing changes on the process.
occurrence—A criterion for risk used in the risk priority number and is defined as the likelihood of a failure occurring. Occurrence is evaluated based on a 10-point scale. At the lowest end of the scale (1) it is assumed that the probability of a failure is unlikely. At the highest end of the scale (10) it is assumed that the probability of a failure is nearly inevitable. [7.B]
one-way ANOVA—See analysis of variance. [6.D]
operating characteristic (OC) curve—For a sampling plan, a plot that indicates the probability of accepting a lot based on the sample size to be taken and the fraction defective in the batch. [4.C]
ordinal variable—A categorical variable for which the levels that the variable takes on have an inherent or natural order (e.g., shirt size of small, medium, or large). [6.A]
outlier—One or more observations that deviate significantly from the majority of the sample from which they came.
P
p chart—A control chart that monitors the fraction nonconforming in a process. Com- pare with np chart. [6.F]
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Glossary 577
paired data—Data in which observations from two samples are dependent, where an observation from one sample is paired with an observation from the other sample. [6.D]
parallel system—A system where the components are connected in such a way that failure of one or more units still allows the remaining units to perform properly. The system fails when all units fail. [3.E]
parameter—A constant or coefficient that describes some characteristic of a population.
Pareto diagram—One of the seven quality control tools; used to rank causes of prob- lems from most significant to least significant. [5.A]
percentile—See quantile. [6.A]
Phase 1 analysis—In statistical process control, the development of trial control limits based on preliminary samples of data. [6.F]
Phase 2 analysis—In statistical process control, the development of reliable control chart limits that can be used for monitoring future production. [6.F]
plan-do-check-act (PDCA)—A continuous improvement methodology developed by Shewhart that is made up of a four- stage improvement process. Also called plan-do- study-act (PDSA). [5.C]
point estimate—See estimate. [6.A]
Poisson distribution—A discrete probability distribution where values take on integer values, often used to model count data such as the number of nonconformities. [6.C]
poka-yoke—A term that means to mistake- proof a process by building safeguards into the system that avoid or immediately find errors. The term comes from the Japanese terms poka, which means “error,” and yokeru, which means “to avoid.” [5.D]
pooled variance—An estimator for the variance of the difference between two popula- tion means; used when the population variances are unknown but assumed roughly equal. [6.D]
population—All possible outcomes or objects of interest. [6.A]
power—In statistical inference, the probability of rejecting a false null hypothesis.
practical significance—In statistical inference, identifying a meaningful difference associated with a parameter of interest. Compare with statistical significance. [6.D]
precision—The closeness of agreement between randomly selected individual mea- surements or test results.
prediction—Estimation of new or future observations using a statistical model.
predictor variable—See regressor. [6.E]
prevention costs—Costs of quality related to activities specifically designed to prevent poor quality in products or services. [2.E]
probability—A numerical measure assigned to events in a sample space that represents the likelihood that a particular outcome will occur. It takes on values between 0 and 1 inclusive. [6.B]
probability density function (pdf)—A function that describes the probability distribu- tion of a continuous random variable. [6.C]
probability mass function (pmf)—A function that describes the probability distribu- tion of a discrete random variable. [6.C]
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578 Glossary
probability plot—A graphical display that shows the actual data on the x-axis plotted against percentiles based on the hypothesized or assumed distribution of interest on the y-axis; used to assess whether a data set follows a specified distribution, for example, with a normal probability plot. [6.C]
process capability—The ability of a process to meet its intended purpose. It is a mea- sure of how well the process produces outcomes that meet specifications. [6.G]
process decision program chart (PDPC)—A planning tool to help organize and evalu- ate process- related events and contingencies with respect to implementation and/or early operations. [5.B]
process map—A flowchart of a work process in detail, including key measurements. [5.B]
process stability—See stable process. [6.F]
process value chain diagram—A planning tool that depicts a sequence of cause- to-effect and effect- to-cause relationships between business results and outcomes and basic physical, economic, and social variables. [5.B]
producer’s risk (α)—For a sampling plan, refers to the probability of not accepting a lot, the quality of which has a designated numerical value representing a level that is generally desirable. Usually the designated value will be the acceptable quality level. Also called alpha risk or probability of a type I error.
product identification—A means of marking parts with a label, etching, engraving, ink, or other means so that different part numbers and other key attributes can be identified.
program evaluation and review technique (PERT)—An event- oriented project man- agement planning and measurement technique that utilizes an arrow diagram or road map to identify all major project events and demonstrates the amount of time (critical path) needed to complete a project. It provides three time estimates: opti- mistic, most likely, and pessimistic. [1.B]
programmable logic controller (PLC)—A computer system used for the control of a manufacturing process. [1.B]
pull system—A lean concept for continuous flow manufacturing. A process where each activity moves a component through the value stream so that it arrives at the next activity at the time it is needed. See also just-in-time. [5.D]
push system—A system in manufacturing where production is driven by forecasts of demand for a product. Contrast with pull system. [5.D]
p-value—The probability of getting a value of the test statistic as extreme as or more extreme than that observed if the null hypothesis is true. The p-value is the actual or observed significance level for a test.
Q
qualitative variable—A variable whose possible outcomes are nonnumeric or categori- cal. See also categorical variable. [6.A]
quality—A term with many interpretations and definitions, including fitness for use and conformance to specifications.
quality assurance—All the planned or systematic actions necessary to provide adequate confidence that a product or service will satisfy given needs.
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Glossary 579
quality audit—A systematic, independent examination and review to determine whether quality activities and related results comply with planned arrangements and whether these arrangements are implemented effectively and are suitable to achieve the objectives. [1.H]
quality control—The operational techniques and the activities that sustain a quality of a product or service that will satisfy given needs; also, the use of such techniques and activities. [6.F]
quality control tools—Seven common tools used in statistical process control: flow- charts, cause- and-effect diagrams, check sheets, histograms, Pareto charts, control charts, and scatter diagrams. [5.B]
quality cost—See cost of quality. [2.E]
quality council—The group driving the quality improvement effort and usually hav- ing oversight responsibility for the implementation and maintenance of the quality management system; it is operated in parallel with the normal operation of the busi- ness. Sometimes referred to as a quality steering committee. [5.E]
quality function deployment (QFD)—A structured method in which customer require- ments are translated into appropriate technical requirements for each stage of prod- uct development and production. The QFD process is often referred to as listening to the voice of the customer. [1.G]
quality information system—A collection of data, rules, and equipment that creates information about quality in a systematic way. The system will collect, store, ana- lyze, and manage quality- related data from customers, suppliers, and internal pro- cesses. [1.B]
quality manual—A document stating the quality policy and describing the quality sys- tem of an organization. [2.B]
quality policy—Top management’s formally stated intentions and direction for the organization pertaining to quality. [2.B]
quality surveillance—Continual monitoring and verification of the status of an entity and analysis of records to ensure that specified requirements are being fulfilled. [6.F]
quality system—The organizational structure, procedures, processes, and resources needed to implement quality management.
quantile—A value x such that 100q% of the sample is less than x. [6.A]
quantitative variable—A variable whose outcomes are numeric, continuous or dis- crete. [6.A]
quartile—A boundary point of a sorted data set divided into four approximately equal subsets. The second quartile is also called the median. [6.A]
r
random effect—In analysis of variance, an effect is random if the factor levels included in the test or experiment are randomly selected from a larger population of possible levels. The results of the test conducted would then apply to the entire population of factor levels and not just those included in the experiment.
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580 Glossary
random error—Error that occurs as a result of natural variation in a process or system. It is variation that occurs when taking repeated measurements on the same unit or item under identical conditions. Also referred to as experimental error.
random experiment—An experiment that has more than one possible outcome.
random sampling—The process of selecting units for a sample such that all combina- tions of units under consideration have an equal or ascertainable chance of being selected as the sample. [6.A]
random variable—A function that associates a real number to each outcome in an experiment.
randomized block design—An experimental design where there is a nuisance factor that may have some influence on the results. [6.H]
range—The difference between the largest and smallest value or observation in a data set. It provides a measure of dispersion in a set of data. [6.A]
rational subgrouping—A method for collecting data that will allow for minimizing the chance of variability due to assignable causes while maximizing the chance of vari- ability due to chance or natural causes. A fundamental and nontrivial concept in statistical process control. [6.F]
readability—The ease of reading the instrument scale when a dimension is being mea- sured. [4.E]
regression analysis—Statistical techniques for determining and modeling the relation- ship between a dependent variable and one or more independent variables. The response variable is also referred to as a response, and the independent variables are also referred to as regressors or predictors. [6.E]
regression coefficient—The parameters in a linear regression model that define the mathematical relationship between the response and regressors. [6.E]
regressor—In regression analysis, it is the independent variable. Also known as the pre- dictor variable. [6.H]
rejection region—In significance testing, the values of the test statistic that will lead to rejection of the null hypothesis. Sometimes referred to as the critical region. [6.D]
reliability—The probability that an item can perform its intended function for a speci- fied interval under stated conditions. [3.E]
reliability block diagram—A block diagram that is similar to the functional block diagram except that it is modified to emphasize those aspects influencing reliabil- ity. [3.E]
repeatability—Variability due to the gage or test instrument used to measure the same part under identical measuring conditions. [4.F]
replication—The repetition of the set of all the treatment combinations to be compared in an experiment. Each of the repetitions is called a replicate. [6.H]
reproducibility—Variability due to different operators or setup measuring the same parts using the same measuring device. Thus, reproducibility represents the vari- ability due to the measurement system. [4.F]
residual—The difference between the actual or observed response and the predicted response for the variable of interest. [6.E]
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Glossary 581
residual analysis—An examination of the residuals used to determine the adequacy of a fitted model and to check the validity of assumptions made in model fitting.
resolution—A design is of resolution R if no p-factor effect is aliased with another effect containing less than R – p factors. For example, a resolution III design is one in which the main effects are aliased with two- factor interactions. [6.H]
resource requirements matrix—A tool to relate the resources required to the project tasks requiring them (used to indicate types of individuals needed, material needed, subcontractors, etc.).
response surface plot—A three- dimensional plot of the response variable versus two of the regressors, generated by a regression model.
response variable—The variable that shows the observed results of an experimental treatment. It is the dependent variable in regression analysis.
return on investment (ROI)—An umbrella term for a variety of ratios measuring an organization’s business performance, calculated by dividing some measure of return by a measure of investment and then multiplying by 100 to provide a percentage. In its most basic form, ROI indicates what remains from all money taken in after all expenses are paid.
risk priority number (RPN)—The multiplication of the three scores for severity (S), occurrence (O), and detection (D) to assess risk. Because each scale (S, O, and D) ranges from 1 to 10, the minimum RPN is 1 and the maximum is 1000. [7.B]
robust designs—Products or processes that continue to perform as intended in spite of manufacturing variation and extreme environmental conditions during use.
robustness—The condition of a product or process design that remains relatively stable with a minimum of variation even though factors that influence operations or usage, such as environment and wear, are constantly changing.
run chart—See time series plot.
S
sample—A subset of units or observations selected from a population of interest. A sample can provide information that may be used as a basis for making a decision concerning the larger quantity. [6.B]
sample integrity—Procedures implemented so that samples are maintained in a unique manner in order to avoid corruption or confusion with others. [4.C]
sample space—The set of all possible outcomes of a random process or random experi- ment. [6.B]
sample standard deviation—A measure of dispersion for a set of observations in the same unit of measure as the original data. It is the positive square root of the sample variance. [6.A]
sample variance—See variance. [6.A]
sampling distribution—The probability distribution of a statistic calculated from a ran- dom sample of a given size. [6.C]
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582 Glossary
sampling plan—A plan used for acceptance sampling defined by the sample size and acceptance number. The plan may contain multiple sample sizes and acceptance numbers if double or multiple sampling is used. [4.C]
scatter diagram—A two- dimensional plot of data resulting from two random variables (bivariate data). The scatter diagram is a tool that can reveal associations between two variables. Also known as a scatter plot. [6.A]
sensitivity—The least perceptible change in dimension detected by the measuring instrument and shown by the indicator. [4.E]
sequential sampling plan—A type of sampling plan used when tests are either destruc- tive or expensive. Units are sampled and compared against two equations repre- senting a reject zone line and an accept zone line. [4.C]
series system—A system composed of n components or subsystems connected end- to-end such that a failure of any component results in the failure of the entire system. [3.E]
serious defect—A defect that may lead to injury or significant economic loss. [4.B]
severity—A criterion for risk used in the risk priority number and is an indicator of the severity of a failure should a failure occur. Severity can be evaluated based on a 10-point scale. At the lowest end of the scale (1) it is assumed that a failure will have no noticeable effect. At the highest end of the scale (10) it is assumed that a failure will impact safe operation or violate compliance with a regulatory mandate. [7.B]
significance level—A stated or fixed probability of wrongly rejecting a true null hypoth- esis that the practitioner is willing to accept. It is the probability of committing a type I error. [6.D]
simple random sampling—See random sampling. [6.A]
single minute exchange of dies (SMED)—A system used to reduce changeover time and improve timely response to demand. [5.D]
SIPOC diagram—A high- level process map used to identify the important aspects of the current process (suppliers, inputs, process, outputs, and customers); often used in quality planning activities. [5.B]
Six Sigma—A continuous improvement methodology; a collection of techniques and tools for use in reducing variation; a program of improvement that focuses on strong leadership tools and emphasizes bottom- line financial results. [5.C]
sparsity-of-effects principle—The belief that the system under investigation is domi- nated by the main effects and low- order interactions. The assumption made in typi- cal designed experiments is that some higher- order interactions (orders higher than two- factor interactions) are negligible. [6.H]
special causes—Causes of variation that arise because of special circumstances or unusual events. They are not an inherent part of a process. Special causes are also referred to as assignable causes. [6.F]
specification limits—Limits of a quality characteristic determined externally, for exam- ple, by the customer.
stable process—A process in which no special causes of variation are present. [6.F]
stakeholders—People, departments, and/or parties that have an investment or interest in the success of or actions taken by the organization.
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Glossary 583
standard—A statement, specification, or quantity of material against which measured outputs from a process may be judged as acceptable or unacceptable.
standard deviation—A measure of dispersion or spread in the same units as the unit of measure. It is equal to the positive square root of the variance. [6.A]
standard error—The standard deviation of the sampling distribution of a statistic. In general, it is the standard deviation of any estimator of a parameter and provides a measure of precision of the estimate. [6.D]
standardized work—A lean tool that states that each activity should be performed the same way every time. [5.D]
standby system—A form of a redundant system where standby components function only upon the failure of the main component. [3.E]
statistic—A quantity calculated from a sample of observations, most often to form an estimate of some population parameter. [6.D]
statistical control—See stable process. [6.F]
statistical process control (SPC)—The application of formal statistical methods that seek to improve process performance and reduce/understand variability in critical- to-quality metrics. A control chart is one of the primary SPC techniques. [6.F]
statistical quality control (SQC)—The use of statistical and engineering technology for quality improvement within an organization. Three primary areas of statistical quality control are statistical process control, design of experiments, and acceptance sampling.
statistical significance—The rejection of the null hypothesis at a prespecified level. A statistically significant result (one in favor of the alternative hypothesis) may not have practical significance in some cases (e.g., if two mean values are different with statistical significance, but their difference is smaller than 0.001 of their value, this might not be of practical importance). [6.D]
stem-and-leaf plot—A graphical display using the actual observations of a sample. The “stem” consists of the leading digit(s) of the observation, and the “leaf” consists of the next digit of the observation. [6.A]
stratified sampling—A method of sampling used when the population is divided into groups. Items are randomly selected within each group and each group makes up a proportional part of the stratified sample. [6.A]
sum of squares—The sum of squared observations between two values. [6.D]
supervisory control and data acquisition (SCADA) system—A large- scale networked control system that uses programmable logic controllers for the management of a manufacturing process. [1.B]
supply chain—The series of processes and/or organizations that are involved in pro- ducing and delivering a product to the final user.
surface metrology—The measurement of the difference between what a surface actu- ally is and what it is intended to be. It may involve other terms such as surface rough- ness and surface finish.
survival function—A reliability function used to model the probability that an object of interest (or component) will survive beyond a specified time. [3.E]
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584 Glossary
SWOT analysis—An assessment of an organization’s key strengths, weaknesses, opportunities, and threats. It considers factors such as the organization’s industry, competitive position, functional areas, and management.
system—A composite of equipment, skills, and techniques capable of performing or supporting an operational role, or both. A complete system includes all equipment, related facilities, material, software, services, and personnel required for its opera- tion and support to the degree that it can be considered self- sufficient in its intended operating environment.
systematic error—Error that remains the same over repeated measurements taken under assumed identical conditions.
T
t distribution—The distribution of the ratio of two independent random variables. The random variable in the numerator is a standard normal random variable. The ran- dom variable in the denominator is the square root of a chi- square random variable divided by its degrees of freedom. Also known as Student’s t distribution. [6.C]
takt time—The rate needed to complete a product in order to meet customer demand. [5.D]
tally sheet—See check sheet. [5.A]
test for significance of regression—An analysis of variance procedure used to deter- mine whether any of the regression coefficients in the linear regression model (except for the intercept) is different from a value of zero. [6.E]
test statistic—A quantity calculated from a sample of data, which is based on the null hypothesis in a statistical hypothesis test and used to make a statistical decision (reject or fail to reject the null hypothesis). [6.D]
testing—A means of determining the capability of an item to meet specified require- ments by subjecting the item to a set of physical, chemical, environmental, or operat- ing actions and conditions.
theory of constraints (TOC)—A problem- solving methodology that employs a systems approach to identify and focus on the weakest link in a chain of processes in order to improve the entire system. [5.C]
time series analysis—The use of mathematical modeling and statistical inference to summarize and predict observations dependent on time and/or each other. [6.E]
time series plot—A graphical depiction of data (observations) over time.
tolerance interval—A statistical interval that contains a stated percentage of a popula- tion with a specified level of confidence. [6.F]
total quality management (TQM)—A structured approach to managing quality improve- ment methods within an organization, with a focus on the customer, employee empowerment, and leadership. [5.C]
total sum of squares—The sum of the squared differences between each observation in a data set and the overall mean of the data set. [6.D]
traceability—The ability to trace the history, application, or location of an item or activ- ity and like items or activities by means of recorded identification. [4.B]
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Glossary 585
treatment—Levels of a factor in an experiment. [6.D]
tree diagram—A planning tool to visualize hierarchical relationships between critical events. See fault tree. [5.B]
type I error—The failure to reject the null hypothesis given that the null is true. The probability of type I error is called the significance level (α).
type II error—The rejection of the null hypothesis given that the null is false. The prob- ability of type II error is referred to as β.
u
u chart—A control chart used to measure the average number of nonconformities. Com- pare with c chart. [6.F]
unbiased estimator—An estimator whose expected value is equal to the parameter for which it is an estimator. [6.D]
uniform distribution—A distribution whose values are equally distributed over an interval. Each possible outcome is assigned equal probability. The uniform distribu- tion is defined for both continuous and discrete random variables. [6.C]
union—The union of two events A and B is that event consisting of all outcomes con- tained in A, in B, or in both. [6.B]
utility function—A mathematical equation in which alternatives are ranked based on their utility to an individual. [7.B]
v
validation—Confirmation, through the provision of objective evidence, that the require- ments for a specific intended use or application have been fulfilled. [3.D]
value added—Work activities in a process that change the form and/or function of the product or service. [5.D]
value stream mapping—A tool based on the principles of lean, used to identify oppor- tunities for improvement of a process and track performance. [5.D]
variables data—Data resulting from the measurement of a parameter or a variable. The resulting measurements may be recorded on a continuous scale. Contrast with attri- butes data.
variables sampling plan—The use of the actual measurements of sample products for decision making rather than classifying products as conforming or nonconforming Compare with sampling plans. [4.C]
variance—A measure of dispersion. For a set of data, it is the sum of the squared differ- ences between the individual observations and the mean of the observations divided by the degrees of freedom. For population variance, the degrees of freedom is the total population size; for sample variance, the degrees of freedom is the sample size minus one. [6.A]
verification—Confirmation, through the provision of objective evidence, that specified requirements have been fulfilled. [3.D]
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586 Glossary
visual control—A lean tool used to create a visual factory, a facility in which locations for tools, inventory, safety equipment, etc. are clearly marked and identified. [5.D]
voice of the customer—The customer’s expectations and preferences, commonly cap- tured using quality function deployment and SIPOC diagrams.
W
wear-out phase—In reliability, represents the final phase in a bathtub curve. It is often characterized by an increasing failure rate due to examples such as fatigue loading and friction between mating surfaces. [3.E]
Weibull distribution—A continuous probability distribution, typically used to model failure rates, including non- constant failure rates. [6.C]
work breakdown structure (WBS)—A project management technique by which a proj- ect is divided into tasks, subtasks, and units of work to be performed. [1.B]
x
x– and R charts—A pair of control charts for variables (continuous) subgroup data, where the x– chart (pronounced “x-bar”) is used to monitor the process mean and the R chart is used to monitor the process variability. R is computed as the subgroup range. x– and R charts are typically used when the subgroup size is less than or equal to 10. [6.F]
x– and s charts—A pair of control charts for variables (continuous) subgroup data, where the x– chart (pronounced “x-bar”) is used to monitor the process mean and the s chart is used to monitor the process variability. s is computed as the subgroup sample standard deviation. x– and s charts are typically used when the subgroup size is greater than 10 or of variable size. [6.F]
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587
Agresti, A. 1988. “A Model for Agreement between Ratings on an Ordinal Scale.” Biometrics 44: 539–48.
———. 1992. “Modeling Patterns of Agreement and Disagreement.” Statistical Methods in Medical Research 1: 201–18.
Agresti, A., and B. Coull. 1998. “Approximate Is Better Than Exact for Interval Estimation of Binomial Proportions.” American Statistician 52: 119–26.
Agresti, A., and J. B. Lang. 1993. “Quasi-symmetric Latent Class Models, with Application to Rater Agreement.” Biometrics 49: 131–39.
Akao, Y., ed. 1990. Quality Function Deployment. Portland, OR: Productivity Press. AlMaian, R. Y., K. L. Needy, K. D. Walsh, T. D. C. Alves, and N. M. Scala. 2016. “Analyzing
Supplier Quality Management Practices in the Construction Industry.” Quality Engineer- ing 28 (2): 175–83.
Anderson, N. C., and J. V. Kovach. 2014. “Reducing Welding Defects in Turnaround Proj- ects: A Lean Six Sigma Case Study.” Quality Engineering 26 (2): 168–81.
Anderson-Cook, C. M. 2017. “Optimizing in a Complex World: A Statistician’s Role in Deci- sion Making.” Quality Engineering 29 (1): 27–41.
Anderson-Cook, C. M., and L. Lu. 2015. “Much-Needed Structure: A New 5-Step Decision- Making Process Helps You Evaluate, Balance Competing Objectives.” ASQ Quality Prog- ress 48 (10): 42–50.
ANSI/ASQ Z1.4-2003 (R2013). 2013. Sampling Procedures and Tables for Inspection by Attri- butes. Milwaukee, WI: ASQ Quality Press.
Automotive Industry Action Group (AIAG). 2006. Production Part Approval Process Manual. Detroit, MI: AIAG.
———. 2010. Measurement System Analysis Reference Manual. Detroit, MI: AIAG. Badiru, A. B., and P. S. Pulat. 1995. Comprehensive Project Management. Englewood Cliffs, NJ:
Prentice Hall. Banerjee, M., M. Capozzoli, L. McSweeney, and D. Sinha. 1999. “Beyond Kappa: A Review
of Interrater Agreement Measures.” Canadian Journal of Statistics 27: 3–23. Barlow, R. E., J. B. Fussell, and N. D. Singpurwalla. 1975. Reliability and Fault Tree Analysis.
Philadelphia: SIAM. Barnes, R. M. 1980. Motion and Time Study Design and Measurement of Work. 7th ed. New York:
John Wiley & Sons. Barrentine, L. 2003. Concepts for R&R Studies. Milwaukee, WI: ASQ Quality Press. Bartlett, L. M. 2007. “Fault Trees.” In Encyclopedia of Statistics in Quality and Reliability.
doi:10.1002/9780470061572.eqr353. Baur, C., and D. Wee. 2015. “Manufacturing’s Next Act.” McKinsey Quarterly, June.
references
H1518_Burke.indd 587 6/29/17 11:49 AM
588 References
Belanger, B. C. 1980. Measurement of Quality Control and the Use of NBS Measurement Assistance Program. NBS Special Publication 620-A. Washington, DC: US Department of Commerce.
Besterfield, D. H. 1999. Total Quality Management. 2nd ed. Englewood Cliffs, NJ: Prentice Hall. ———. 2001. Quality Control. 6th ed. Englewood Cliffs, NJ: Prentice Hall. Bisgaard, S. 2008. “Must a Process Be in Statistical Control before Conducting Designed
Experiments?” Quality Engineering 20 (2): 143–50. doi: 10.1080/08982110701826721. Bisgaard, S., and M. Kulahci. 2011. Time Series Analysis and Forecasting by Example. Hobo-
ken, NJ: Wiley & Sons. Bloch, D. A., and H. C. Kraemer. 1989. “2 × 2 Kappa Coefficients: Measures of Agreement or
Association.” Biometrics 45: 269–87. Bond, T. P. 1983. “Basics of an MRB.” Quality (November): 48. Borror, C. M., D. C. Montgomery, and G. C. Runger. 1997. “Confidence Intervals for Vari-
ance Components from Gauge Capability Studies.” Quality and Reliability Engineering International 13: 361–69.
Bosch, J. A. 1984. 66 Centuries of Measurement. Dayton, OH: Sheffield Measurement Division. Bothe, D. R. 1997. Measuring Process Capability: Techniques and Calculations for Quality and
Manufacturing Engineers. New York: McGraw- Hill Companies. ———. 2001. “Back to Basics: Use Check Sheets to Identify the Causes of Downtime.” ASQ
Quality Progress 34 (4): 136. Boulanger, M., M. E. Johnson, and S. N. Luko. 2012. “Reviews of Standards and Related
Material: Statistical Standards and ISO, Part 1.” Quality Engineering 24 (1): 94–101. Box, G. E. 1954. “Some Theorems on Quadratic Forms Applied in the Study of Analysis of
Variance Problems, I. Effect of Inequality of Variance in the One-Way Classification.” The Annals of Mathematical Statistics 25 (2): 290–302.
Box, G. E. P., W. Hunter, and J. S. Hunter. 2005. Statistics for Experimenters: Design, Innovation, and Discovery. 2nd ed. Hoboken, NJ: John Wiley & Sons.
Box, G. E. P., and W. H. Woodall. 2012. “Innovation, Quality Engineering, and Statistics.” Quality Engineering 24 (1): 20–29.
Boyles, R. A. 2001. “Gauge Capability for Pass- Fail Inspection.” Technometrics 43: 223–29. Brassard, M. 1989. The Memory Jogger Plus+. Methuen, MA: Goal/QPC Press. Brettel, M., N. Friederichsen, M. Keller, and M. Rosenberg. 2014. “How Virtualization,
Decentralization and Network Building Change the Manufacturing Landscape: An Industry 4.0 Perspective.” International Journal of Mechanical, Industrial Science and Engi- neering 8 (1) : 37–44.
Breyfogle, F. W. 1999. Implementing Six Sigma: Smarter Solutions Using Six Sigma. New York: John Wiley & Sons.
———. 2003. “Control Charting at the 30,000-Foot-Level.” ASQ Quality Progress (Novem- ber): 66–70.
Breyfogle, F. W., J. Cupello, and B. Meadows. 2000. Managing Six Sigma: A Practical Guide to Understanding, Assessing, and Implementing the Strategy That Yields Bottom- Line Success. New York: John Wiley & Sons.
Britz, G. C., and D. W. Emerling. 2000. Improving Performance through Statistical Thinking. Milwaukee, WI: ASQ Quality Press.
Burdick, R. K., E. Allen, and G. Larsen. 2002. “Comparing Variability of Two Measurement Processes Using R&R Studies.” Journal of Quality Technology 34: 97–105.
Burdick, R. K., C. M. Borror, and D. C. Montgomery. 2003. “A Review of Methods for Mea- surement Systems Capability Analysis.” Journal of Quality Technology 35: 342–54.
H1518_Burke.indd 588 6/29/17 11:49 AM
References 589
———. 2005. Design and Analysis of Gauge R&R Studies: Making Decisions with Confidence Intervals in Random and Mixed ANOVA Models. Philadelphia: ASA- SIAM Series on Statis- tics and Applied Probability.
Burdick, R. K., and G. Larsen. 1997. “Confidence Intervals on Measures of Variability in Gauge R&R Studies.” Journal of Quality Technology 29: 261–73.
Burnett, R. E. 2005. Technical Communication. 6th ed. Boston: Wadsworth/ITP. Camgoz-Akdag, H., I. H. Pinar, and E. K. Nazli. 2016. “Internal Customer Satisfaction
Improvement with QFD Technique.” Business Process Management Journal 22 (5): 957–68. Camp, R. C. 1989. Benchmarking: The Search for Industry Best Practices That Lead to Superior
Performance. Milwaukee, WI: ASQC Quality Press. ———. 1995. Business Process Benchmarking. Milwaukee, WI: ASQC Quality Press. Capizzi, G. 2015. “Recent Advances in Process Monitoring: Nonparametric and Variable-
Selection Methods for Phase I and Phase II.” Quality Engineering 27 (1): 44–67. Chakraborti, S., S. W. Human, and M. A. Graham. 2008. “Phase I Statistical Process Control
Charts: An Overview and Some Results.” Quality Engineering 21 (1): 52–62. Champ, C. W., and W. H. Woodall. 1987. “Exact Results for Shewhart Control Charts with
Supplementary Runs Rules.” Technometrics 29 (4): 393–99. Cianfrani, C. A., and J. E. West. 2015. ISO 9001:2015 Explained. 4th ed. Milwaukee, WI: ASQ
Quality Press. Cicchetti, D. V., and A. R. Feinstein. 1990. “High Agreement but Low Kappa: II. Resolving
the Paradoxes.” Journal of Clinical Epidemiology 43: 551–58. Cohen, J. 1960. “Coefficient of Agreement for Nominal Scales.” Educational and Psychological
Measurement 20: 37–46. Collins, J. C., and J. I. Porras. 1997. Built to Last: Successful Habits of Visionary Companies. New
York: Harper Business. Conger, A. J. 1980. “Integration and Generalization of Kappas for Multiple Raters.” Psycho-
logical Bulletin 88: 322–28. Cox, L. A. T. 2008. “What’s Wrong with Risk Matrices?” Risk Analysis 28 (2): 497–512. Creasy, T., and S. Ramey. 2013. “Don’t Lose Patients.” ASQ Quality Progress 46 (2): 42–49. Crosby, P. B. 1979. Quality Is Free. New York: McGraw- Hill. Crowder, S. V. 1989. “Design of Exponentially Weighted Moving Average Schemes.” Journal
of Quality Technology 21: 155–62. Daepp, M. I., M. J. Hamilton, G. B. West, and L. M. Bettencourt. 2015. “The Mortality of
Companies.” Journal of The Royal Society Interface 12 (106): 20150120. Dalkey, N. C. 1967. Analysis of the Future: The Delphi Method. Santa Monica, CA: RAND Cor-
poration. http://www.rand.org/pubs/papers/P3558.html. Daniel, C. 1959. “Use of Half- Normal Plots in Interpreting Factorial Two Level Experi-
ments.” Technometrics 1: 311–42. Darmody, W. J. 1967. “Elements of a Generalized Measuring System.” In Handbook of Indus-
trial Metrology. Englewood Cliffs, NJ: Prentice- Hall (ASTME). Day, R. G. 1993. Quality Function Deployment: Linking a Company with Its Customers. Milwau-
kee, WI: ASQC Quality Press. De Mast, J., and W. N. van Wieringen. 2004. “Measurement System Analysis for Bounded
Ordinal Data.” Quality and Reliability Engineering International 20: 383–95. ———. 2007. “Measurement System Analysis for Categorical Measurements: Agreement
and Kappa- Type Indices.” Journal of Quality Technology 39: 191–202. DeBono, E. 1992. Serious Creativity: Using the Power of Lateral Thinking to Create New Ideas.
New York: HarperCollins.
H1518_Burke.indd 589 6/29/17 11:49 AM
590 References
Defense Acquisition University (DAU). 2017. “Introduction to Systems Engineering.” In Defense Acquisition Guidebook. https://www.dau.mil/tools/dag.
Defeo, J. A. 2016. Juran’s Quality Handbook: The Complete Guide to Performance Excellence. 7th ed. New York: McGraw Hill Professional.
Deming, W. E. 1982. Quality, Productivity, and Competitive Position. Cambridge, MA: M.I.T. Center for Advanced Engineering Study.
———. 1986. Out of the Crisis. Cambridge, MA: M.I.T. Center for Advanced Engineering Study.
Devore, J. 2016. Probability and Statistics for Engineering and the Sciences. 9th ed. Pacific Grove, CA: Duxbury Press.
Dhillon, B. S., and C. Singh. 1981. Engineering Reliability: New Techniques and Applications. New York: John Wiley & Sons.
Dodson, B., and D. Nolan. 1999. Reliability Engineering Handbook. Tucson, AZ: QA Publish- ing, LLC.
Doganaksoy, N., and G. J. Hahn. 2012. “Getting the Right Data Up Front: A Key Challenge.” Quality Engineering 24 (4): 446–59.
Doiron, T. 2007. “20 °C—A Short History of the Standard Reference Temperature for Indus- trial Dimensional Measurements.” Journal of Research of the National Institute of Standards and Technology 112 (1): 1–23.
Dolezal, K. K., R. K. Burdick, and N. J. Birch. 1998. “Analysis of a Two- Factor R&R Study with Fixed Operators.” Journal of Quality Technology 30: 163–70.
Drews, W. E. 1978. “How to Measure Roundness.” Tooling and Production (June): 156–60. Duffy, G. L. 2014. Modular Kaizen: Continuous and Breakthrough Improvement. Milwaukee, WI:
ASQ Quality Press. Duncan, A. J. 1986. Quality Control and Industrial Statistics. 5th ed. Homewood, IL: Rich-
ard D. Irwin. Durivage, M. A. 2016. Practical Process Validation. Milwaukee, WI: ASQ Quality Press. ———. 2017. “Work Smarter, Not Harder.” ASQ Quality Progress 50 (3): 41–43. Ebeling, C. E. 2009. Introduction to Reliability and Maintainability Engineering. Long Grove, IL:
Waveland Press. Elsayed, E. A. 1996. Reliability Engineering. Reading, PA: Addison Wesley. ———. 2000. “Perspectives and Challenges for Research in Quality and Reliability Engi-
neering.” International Journal of Production Research 38 (9): 1953–76. Engel, J., and B. deVries. 1997. “Evaluating a Well- Known Criterion for Measurement Preci-
sion.” Journal of Quality Technology 29: 469–76. Erdmann, T. P., R. J. M. M. Does, and S. Bisgaard. 2009. “Quality Quandaries: A Gage R&R
Study in a Hospital.” Quality Engineering 22 (1): 46–53. Feigenbaum, A. V. 2004. Total Quality Control. 4th ed. New York: McGraw- Hill. Feinstein, A. R., and D. V. Cicchetti. 1990. “High Agreement but Low Kappa: I. The Prob-
lems of Two Paradoxes.” Journal of Clinical Epidemiology 43: 543–49. Filho, M. G., A. Boschi, A. F. Rentes, M. Thurer, and T. M. Bertani. 2015. “Improving Hos-
pital Performance by Use of Lean Techniques: An Action Research Project in Brazil.” Quality Engineering 27 (2): 196–211.
Fisher, R. A. 1925. Statistical Methods for Research Workers. London: Oliver and Boyd. Fleiss, J. L. 1971. “Measuring Nominal Scale Agreement among Many Raters.” Psychological
Bulletin 76: 378–82. Foster, S. T. 1998. “The Ups and Downs of Customer- Driven Quality.” ASQ Quality Progress
(October): 67–72.
H1518_Burke.indd 590 6/29/17 11:49 AM
References 591
Freedman, D., and P. Diaconis. 1981. “On the Histogram as a Density Estimator: L2 Theory.” Zeit. Wahr. ver. Geb. 57: 453–76.
Gale, B. T., with R. C. Wood. 1994. Managing Customer Value: Creating Quality and Service That Customers Can See. New York: The Free Press.
Gan, F. F. 1991. “An Optimal Design of CUSUM Quality Control Charts.” Journal of Quality Technology 23 (4): 279–86.
Garrett, D. F., and J. Lee. 2011. “Lean Construction Submittal Process—A Case Study.” Qual- ity Engineering 23 (1): 83–93.
Garvey, P. R. 2009. Analytical Methods for Risk Management. Boca Raton, FL: Taylor & Fran- cis Group.
Garvin, D. A. 1987. “Competing in the Eight Dimensions of Quality.” Harvard Business Review 87 (6): 101–9.
Gee, G., P. McGrath, and M. Izadi. 1996. “A Team Approach to Kaizen.” Journal of Industrial Technology (Fall): 45–48.
Gilbreth, F., and L. M. Gilbreth. 1921. Process Charts. New York: American Society of Mechanical Engineers.
Gillette, B., R. Johnson, E. Polashek, J. Thornburg, and C. White. 1993. The Art of Working Together: A Guide to Effective Collaboration. Ames, IA: C. I. White and Associates.
Givens, G. H., and J. A. Hoeting. 2012. Computational Statistics. Vol. 710. Hoboken, NJ: John Wiley & Sons.
Godfrey, A. J. R., G. K. G. Russell, and B. D. Betz- Stablein. 2016. “Monitoring Acute and Chronic Kidney Failure Using Statistical Process Control Techniques.” Quality Engineer- ing 28 (2): 184–92.
Goldratt, E. M. 1997. Critical Chain. Great Barrington, MA: The North River Press. Gosavi, A., and E. Cudney. 2012. “Form Errors in Precision Metrology: A Survey of Mea-
surement Techniques.” Quality Engineering 24 (3): 369–80. Gryna, F. M. 1988a. “Manufacturing Planning.” In Juran’s Quality Control Handbook, 1–59.
4th ed. New York: McGraw- Hill. ———. 1988b. “Training for Quality.” In Juran’s Quality Control Handbook. 4th ed. New York:
McGraw- Hill. Gryna, F. M., R. C. H. Chua, and J. A. Defeo. 2007. Juran’s Quality Planning and Analysis for
Enterprise Quality. 5th ed. New York: McGraw- Hill. Hahn, G., N. Doganaksoy, and C. Stanard. 2001. “Statistical Tools for Six Sigma.” ASQ Qual-
ity Progress (September): 78–82. Hallock, M. L., S. J. Alper, and B. Karsh. 2006. “A Macroergonomic Work System Analy-
sis of the Diagnostic Testing Process in an Outpatient Health Care Facility for Process Improvement and Patient Safety.” Ergonomics 49 (5–6): 544–66.
Hawkins, D. M. 1981. “A CUSUM for a Scale Parameter.” Journal of Quality Technology 13 (4): 228–35.
———. 1993. “Cumulative Sum Control Charting: An Underutilized SPC Tool.” Quality Engineering 5 (3): 463–77.
Hawkins, D., and D. H. Olwell. 1998. Cumulative Sum Charts and Charting for Quality Improve- ment. New York: Springer- Verlag.
Hawkins, D. M., and Q. Wu. 2014. “The CUSUM and the EWMA Head- to-Head.” Quality Engineering 26 (2): 215–22.
Hayes, B. E. 2008. Measuring Customer Satisfaction: Survey Design, Use, and Statistical Analysis Methods. 3rd ed. Milwaukee, WI: ASQ Quality Press.
Hellier, C. 2012. Handbook of Nondestructive Evaluation. 2nd ed. New York: McGraw Hill.
H1518_Burke.indd 591 6/29/17 11:49 AM
592 References
Hill, H. M., and D. J. McClaskey. 1980. “Developing Awareness of Quality Responsibilities.” In ASQC Technical Conference Transactions. Milwaukee, WI: ASQC Quality Press.
Hoerl, R. 2001. “Six Sigma Black Belts: What Do They Need to Know?” (with discussion). Journal of Quality Technology 33: 391–435.
Hoerl, R., and R. D. Snee. 2012. Statistical Thinking: Improving Business Performance. Hobo- ken, NJ: John Wiley & Sons.
Hogg, R. V., E. A. Tanis, and D. Zimmerman. 2014. Probability and Statistical Inference. Engle- wood Cliffs, NJ: Pearson Higher Ed.
Houf, R., and D. Berman. 1988. “Statistical Analysis of Power Module Thermal Test Equip- ment Performance.” IEEE Transactions on Components, Hybrids, and Manufacturing Tech- nology 22: 516–20.
Hughes, T. A. 1995. Measurement and Control Basics. 2nd ed. Research Triangle Park, NC: Instrument Society of America.
Imai, M. 1986. Kaizen. New York: McGraw- Hill. Ishikawa, K. 1985. What Is Total Quality Control? The Japanese Way. Englewood Cliffs, NJ:
Prentice Hall. ISO. 2015. Moving from ISO 9001:2008 to ISO 9001:2015. Geneva: International Organization
for Standardization. ISO 9001:2015. 2015. Quality Management Systems—Requirements. Geneva: International
Organization for Standardization. ISO 9004:2009. 2009. Managing for the Sustained Success of an Organisation: A Quality Manage-
ment Approach. Geneva: International Organization for Standardization. ISO 31000:2009. 2009. Risk Management—Principles and Guidelines. Geneva: International
Organization for Standardization. Jardine, A. K. S., and J. A. Buzacott. 1983. “Equipment Reliability and Maintenance.” Euro-
pean Journal of Operational Research 19: 285–96. Jensen, C. R. 2002. “Variance Component Calculations: Common Methods and Misapplica-
tions in the Semiconductor Industry.” Quality Engineering 14: 645–57. Johnson, R. H., and R. T. Webber. 1985. Buying Quality: How Purchasing, Quality Control, and
Suppliers Work Together. New York: Franklin Watts. Juran, J. M. 1988. Juran’s Quality Handbook. 4th ed. New York: McGraw- Hill. ———. 1989. Juran on Leadership for Quality. New York: Free Press. Juran, J. M., and A. Godfrey. 1999. Juran’s Quality Handbook. 5th ed. New York: McGraw Hill. Juran, J. M., and F. N. Gryna Jr. 1980. Quality Planning and Analysis. New York: McGraw- Hill. Kaplan, R. S., and D. Norton. 1992. “The Balanced Scorecard: Measures That Drive Perfor-
mance.” Harvard Business Review 70 (1): 71–79. Kerns, D. T., and D. T. Nadler. 1992. Prophets in the Dark: How Xerox Reinvented Itself and Beat
Back the Japanese. New York: Harper Business. Kilmann, R. H., and K. W. Thomas. 1977. “Developing a Forced Choice Measure of
Conflict- Handling Behavior: The ‘MODE’ Instrument.” Educational and Psychological Measurement 37 (2): 309–25.
King, B. 1987. Better Designs in Half the Time. Methuen, MA: Goal/QPC Press. Kirkpatrick, D. L. 2006. Evaluating Training Programs: The Four Levels. 3rd ed. San Francisco:
Berrett- Koehler. Knowles, M. S. 1996. “Adult Learning.” In The ASTD Training and Development Handbook.
4th ed. New York: McGraw- Hill. Kolarik, W. J. 1995. Creating Quality: Concepts, Systems, Strategies, and Tools. New York:
McGraw- Hill. ———. 1999. Creating Quality: Process Design for Results. New York: McGraw- Hill.
H1518_Burke.indd 592 6/29/17 11:49 AM
References 593
Kotz, S., and C. Lovelace. 1998. Process Capability Indices in Theory and Practice. London: Arnold Press.
Kubiak, T. M. 2009. “Perusing Process Performance Metrics.” ASQ Quality Progress (August): 42.
Kutner, M. H., C. J. Nachtsheim, J. Neter, and W. Li. 2004. Applied Linear Statistical Models. 5th ed. Boston: McGraw- Hill Irwin.
Laford, R. J. 1986. Ship-to-Stock: An Alternative to Incoming Inspection. Milwaukee, WI: ASQC Quality Press.
Langdon, D. J. 1994. “A New Language of Work.” Quality Digest (October): 44–48. Larsen, G. A. 2002. “Measurement System Analysis—The Usual Metrics Can Be Noninfor-
mative.” Quality Engineering 15: 293–98. Ledolter, J., and C. Burrill. 1999. Statistical Quality Control: Strategies and Tools for Continual
Improvement. New York: John Wiley & Sons. Ledolter, J., and A. Swersey. 1997. “An Evaluation of Pre- Control.” Journal of Quality Technol-
ogy 29: 163–71. ———. 2007. Testing 1-2-3: Experimental Design with Applications in Marketing and Service
Operations. Los Angeles: Stanford University Press. Lee, H., and H. Awbi. 2004. “Effect of Internal Partitioning on Room Air Quality with Mix-
ing Ventilation—Statistical Analysis.” Renewable Energy 29: 1721–32. Long, C. S., and F. M. Gryna. 1999. Preferred Practices in Developing a Quality Information Sys-
tem. Report No. 907. Tampa, FL: College of Business. Lucas, J., and M. Sacucci. 1990. “Exponentially Weighted Moving Average Control Schemes:
Properties and Enhancements.” Technometrics 32: 1–29. Luceño, A. 1996. “A Process Capability Ratio with Reliable Confidence Intervals.” Commu-
nication in Statistics—Simulation and Computation 25: 235–46. Luko, S. N. 2013. “Risk Management Terminology.” Quality Engineering 25 (3): 292–97. Machinability Data Center. 1980. Machining Data Handbook. Cincinnati, OH: TechSolve. Mader, D. P., J. Prins, and R. E. Lampe. 1999. “The Economic Impact of Measurement Error.”
Quality Engineering 15: 293–98. Majeske, K. D., and R. W. Andrews. 2002. “Evaluating Measurement Systems and Manu-
facturing Processes Using Three Quality Measures.” Quality Engineering 15 (2): 243–51. Makino, T. 1984. “Mean Hazard Rate and Its Application to the Normal Approximation of
the Weibull Distribution.” Naval Research Logistics Quarterly 31: 1–8. Mallette, P. 1993. “Improving Through Creativity.” Quality Digest (May): 81–85. Malshe, A., K. Rajurkar, A. Samant, H. N. Hansen, S. Bapat, and W. Jiang. 2013. “Bio-inspired
Functional Surfaces for Advanced Applications.” CIRP Annals- Manufacturing Technology 62 (2): 607–28.
Manos, T. 2006. “Value Stream Mapping—An Introduction.” ASQ Quality Progress (June): 64–69.
McCaslin, J. A., and G. F. Gruska. 1976. “Analysis of Attribute Gage Systems.” ASQC Techni- cal Conference Transactions 30: 392–99.
McNish, A. 1967. “The Nature of Measurement.” In Handbook of Industrial Metrology. Engle- wood Cliffs, NJ: Prentice Hall.
Menesatti, P., C. Beni, G. Paglia, S. Marcelli, and S. D’Andrea. 1999. “Predictive Statistical Model for the Analysis of Drop Impact Damage on Peach.” Journal of Agricultural Engi- neering Research 73 (3): 275–82.
MIL-HDBK-61A(SE). 2001. Military Handbook: Configuration Management Guidance. Wash- ington, DC: Department of Defense.
MIL-STD-1629A. 1980. Procedures for Performing a Failure Mode, Effects, and Criticality Analy- sis. Washington, DC: Department of Defense.
H1518_Burke.indd 593 6/29/17 11:49 AM
594 References
Mizuno, S., ed. 1988. Management for Quality Improvement. Portland, OR: Productivity Press. Montgomery, D. C. 2013. Introduction to Statistical Quality Control. 7th ed. Hoboken, NJ: John
Wiley & Sons. ———. 2017. Design and Analysis of Experiments. 9th ed. New York: John Wiley & Sons. Montgomery, D. C., C. Jennings, and M. Kulahci. 2015. Introduction to Time Series Analysis
and Forecasting. 2nd ed. New York: John Wiley & Sons. Montgomery, D. C., and G. C. Runger. 2013. Applied Statistics and Probability for Engineers.
6th ed. Hoboken, NJ: John Wiley & Sons. Montgomery, D. C., G. C. Runger, and N. F. Hubele. 2010. Engineering Statistics. 5th ed.
Hoboken, NJ: John Wiley & Sons. Myers, R. H., D. C. Montgomery, and C. M. Anderson- Cook. 2016. Response Surface Meth-
odology. 4th ed. New York: John Wiley & Sons. Nadler, G., and S. Hibino. 1994. Breakthrough Thinking. 2nd ed. Rocklin, CA: Prima
Publishing. Nepal, B., S. Mohanty, and L. Kay. 2013. “Quality Improvement of Medical Wire Manufac-
turing Process.” Quality Engineering 25 (2): 151–63. Neubauer, D. V., and S. N. Luko. 2013a. “Comparing Acceptance Sampling Standards,
Part 1.” Quality Engineering 25 (1): 73–77. ———. 2013b. “Comparing Acceptance Sampling Standards, Part 2.” Quality Engineering
25 (2): 181–87. NIST (National Institute for Standards and Technology). 1981. A Brief History of Measurement
Systems. Special Publication 304A. Washington, DC: US Department of Commerce. Page, E. S. 1961. “Cumulative Sum Control Charts.” Technometrics 3: 1–9. Palady, P. 1997. Failure Modes and Effects Analysis: Practical Applications. Ann Arbor, MI:
Library of Congress. Park, C. S. 2007. Contemporary Engineering Economics. 4th ed. New Jersey: Prentice Hall. Parsowith, B. S. 1995. Fundamentals of Quality Auditing. Milwaukee, WI: ASQC Quality Press. Pearlson, K. E., and C. S. Saunders. 2004. Managing and Using Information Systems: A Strategic
Approach. New York: John Wiley & Sons. Perez-Wilson, M. 1997. “Process Capability: Minding Your Cpk’s.” Quality Digest. Accessed
December 20, 2016. http://www.qualitydigest.com/magazine/1997/dec/article/process- capability-minding-your-cpks.html.
Perry, B. 1998. “Seeing Your Customers in a Whole New Light.” Journal for Quality and Par- ticipation 21 (6): 38–43.
Phillips, J. J. 2003. Return on Investment in Training and Performance Improvement Programs. New York: Butterworth Heinemann.
Pilot, S. 2002. “What Is a Fault Tree Analysis?” ASQ Quality Progress (March): 120. Pinto, C. A., and P. R. Garvey. 2012. Advanced Risk Analysis in Engineering Enterprise Systems.
Boca Raton, FL: CRC Press. Reason, R. E. 1960. The Measurement of Surface Texture. London: CleaverHume Press. ReVelle, J. B. 2004. Quality Essentials: A Reference Guide from A to Z. Milwaukee, WI: ASQ
Quality Press. Rice, G. O. 1986. “Metrology.” In Quality Management Handbook, edited by L. Walsh,
R. Wurster, and R. J. Kimber. Milwaukee, WI: ASQC Quality Press; New York: Marcel Dekker.
Robbins, S. P., and T. Judge. 2012. Essentials of Organizational Behavior. Upper Saddle River, NJ: Pearson.
Robert, C., and G. Casella. 2009. Introducing Monte Carlo Methods with R. New York: Springer Science & Business Media.
H1518_Burke.indd 594 6/29/17 11:49 AM
References 595
Roberts, S. W. 1959. “Control Chart Tests Based on Geometric Moving Averages.” Techno- metrics 1: 97–102.
Rodriguez-Perez, J. 2012. Quality Risk Management in the FDA- Regulated Industry. Milwau- kee, WI: ASQ Quality Press.
Rother, M., and J. Shook. 1999. Learning to See. Brookline, MA: The Lean Enterprise Institute. Ruggeri, F., R. S. Kenett, and F. W. Faltin, eds. 2007. Encyclopedia of Statistics in Quality and
Reliability. Chichester, England: Wiley. Russell, J. P., ed. 2013. The ASQ Auditing Handbook. 4th ed. Milwaukee, WI: ASQ Quality
Press. Saaty, T. 1982. Decision Making for Leaders. Belmont, CA: Lifetime Learning Publications. Salegna, G., and F. Fazel. 2000. “Obstacles to Implementing Quality.” ASQ Quality Progress
( July): 53–57. Schall, S. O. 2012. “Variability Reduction: A Statistical Engineering Approach to Engage
Operations Teams in Process Improvement.” Quality Engineering 24 (2): 264–79. Schoonhoven, M., C. Lubbers, and R. J. M. M. Does. 2013. “Quality Quandaries: Shortening
the Throughput Time at a Hospital’s Billing Process.” Quality Engineering 25 (2): 188–93. Scott, D. 1979. “On Optimal and Data- Based Histograms.” Biometrika 66: 605–10. Shewhart, W. A. 1980. Economic Control of Quality Manufactured Product. Milwaukee, WI:
ASQC Quality Press. Shingo, S. 1986. Zero Quality Control: Source Inspection and the Poka- Yoke System. Portland,
OR: Productivity Press. Simpson, J. A. 1981. “Foundations of Metrology.” Journal of Research of the National Bureau of
Standards 86 (3): 36–42. Simpson, J. R., C. M. Listak, and G. T. Hutto. 2013. “Guidelines for Planning and Evidence
for Assessing a Well- Designed Experiment.” Quality Engineering 25 (4): 333–55. Snee, R. D., and R. W. Hoerl. 2003. Leading Six Sigma: A Step- by-Step Guide Based on Experi-
ence with GE and Other Six Sigma Companies. Upper Saddle River, NJ: Financial Times Prentice Hall.
———. 2012. “Leadership—Essential for Developing the Discipline of Statistical Engineer- ing.” Quality Engineering 24: 162–70.
Society for Automotive Engineers. 2014. Aerospace First Article Inspection Requirement. Standard AS9102b.
Somerville, S. E., and D. C. Montgomery. 1996. “Process Capability Indices and Nonnormal Distributions.” Quality Engineering 9 (2): 305–16.
Spragg, R. C. 1976. “Advanced System for the Measurement of Errors of Form.” SME Paper No. IQ 76-807.
Stamatis, D. 2003. Failure Mode and Effect Analysis: FMEA Theory to Execution. 2nd ed. Mil- waukee, WI: ASQ Quality Press.
Stephens, K. S. 2016. “Practitioner Advice: Dodge and Romig Sampling Tables: Revisited, Refined, and Extended- Practitioner Advice.” Quality Engineering 28 (2): 238–44.
Stevenson, W. 2000. “Supercharging Your Pareto Analysis.” ASQ Quality Progress (October): 51–55.
Stolovitch, H. D., and E. J. Keeps, eds. 1992. The Handbook of Human Performance Technology. San Francisco: Jossey- Bass.
Sturges, H. A. 1926. “The Choice of a Class Interval.” Journal of the American Statistical Asso- ciation 21: 65–66.
Sullivan, L. P. 1986. “Quality Function Deployment.” ASQC Quality Progress (June): 39–50. Sumithra, B., and S. Bhattacharya. 2008. “Toasting of Corn Flakes: Product Characteristics
as a Function of Processing Conditions.” Journal of Food Engineering 88: 419–28.
H1518_Burke.indd 595 6/29/17 11:49 AM
596 References
Sweet, A. L., S. Tjokrodjojo, and P. Wijaya. 2005. “An Investigation of the Measurements Systems Analysis ‘Analytic Method’ for Attribute Gages.” Quality Engineering 17: 219–26.
Taguchi, G. 1986. Introduction to Quality Engineering: Designing Quality into Products and Pro- cesses. White Plains, NY: Kraus International; UNIPUB (Asian Productivity Organization).
Tague, N. 2005. The Quality Toolbox. 2nd ed. Milwaukee, WI: ASQ Quality Press. Tan, R. Y. C., M. Met- Domestici, K. Zhou, A. B. Guzman, S. T. Lim, K. C. Soo, T. W. Feeley, and
J. Ngeow. 2016. “Using Quality Improvement Methods and Time- Driven Activity- Based Costing to Improve Value- Based Cancer Care Delivery at a Cancer Genetics Clinic.” Journal of Oncology Practice 12 (3): e320–31. JOPR007765.
Taylor, B. N., and A. Thompson, eds. 2008. The International System of Units (SI). NIST Special Publication SP 330-2008. Washington, DC: Government Printing Office.
Tobias, P. A., and D. C. Trindade. 2011. Applied Reliability. 3rd ed. Boca Raton, FL: CRC Press. Townsend, A., N. Senin, L. Blunt, R. K. Leach, and J. S. Taylor. 2016. “Surface Texture Metrol-
ogy for Metal Additive Manufacturing: A Review.” Precision Engineering 46: 34–47. Trip, A., and R. J. Does. 2010. “Quality Quandaries: Interpretation of Signals from Runs
Rules in Shewhart Control Charts.” Quality Engineering 22 (4): 351–57. Tuckman, B. W. 1965. “Developmental Sequence in Small Groups.” Psychological Bulletin 63
(6): 384–99. Tuckman, B. W., and M. A. C. Jensen. 1977. “Stages of Small- Group Development Revis-
ited.” Group & Organizational Studies 2 (4): 419–27. Uebersax, J. S., and W. M. Grove. 1990. “Latent Class Analysis of Diagnostic Agreement.”
Statistical Methods 9: 559–72. Van Patten, J. 2006. “A Second Look at 5S.” ASQ Quality Progress 39 (10): 55–59. Van Wieringen, W. N., and E. R. van Heuvel. 2005. “A Comparison of Methods for the
Evaluation of Binary Measurement Systems.” Quality Engineering 17: 495–507. Vardeman, S. B., and E. S. VanValkenburg. 1999. “Two-Way Random- Effects Analyses and
Gauge R&R Sudies.” Technometrics 41: 202–11. Vendor-Vendee Technical Committee. 1977. How to Conduct a Supplier Survey. Milwaukee,
WI: ASQC Quality Press. Vining, G. 2009. “Technical Advice: Phase I and Phase II Control Charts.” Quality Engineer-
ing 21: 478–79. ———. 2011. “Technical Advice: Essential Elements for Quality Improvement Programs,”
Quality Engineering 23: 395–97. ———. 2013. “Technical Advice: Scientific Method and Approaches for Collecting Data.”
Quality Engineering 25 (2): 194–201. Vining, G. G., and S. Kowalski. 2011. Statistical Methods for Engineers. 3rd ed. Pacific Grove,
CA: Brooks- Cole. Vogt, T. L. 1980. Optimizing Calibration Recall Intervals and Algorithms. NIST Publication
NBS- GCR-80-283. Wald, A. 1973. Sequential Analysis. New York: Dover. Wang, J. X., and M. L. Roush. 2000. What Every Engineer Should Know about Risk Engineering
and Management. Boca Raton, FL: CRC Press. Watson, G. H. 1993. Strategic Benchmarking. New York: John Wiley & Sons. Weaver, B. P., M. S. Hamada, S. B. Vardeman, and A. G. Wilson. 2012. “A Bayesian Approach
to the Analysis of Gauge R&R Data.” Quality Engineering 24 (4): 486–500. Weber, E. U. 1997. “The Utility of Measuring and Modeling Perceived Risk.” In Choice, Deci-
sion, and Measurement: Essays in Honor of R. Duncan Luce, edited by A. A. J. Marley, 45–57. Mahwah, NJ: Erlbaum.
H1518_Burke.indd 596 6/29/17 11:49 AM
References 597
———. 1998. “Who’s Afraid of a Little Risk? New Evidence for General Risk Aversion.” In Decision Research from Bayesian Approaches to Normative Systems: Reflections on the Con- tributions of Ward Edwards, edited by J. Shanteau, B. A. Mellers, and D. Schum. Norwell, MA: Kluwer Academic Press.
Weber, E. U., A.-R. Blais, and N. E. Betz. 2002. “A Domain- Specific RiskAttitude Scale: Mea- suring Risk Perceptions and Risk Behaviors.” Journal of Behavioral Decision Making 15 (4): 263–90.
Westcott, R. T. 2003. Stepping Up to ISO 9004:2000. Chico, CA: Paton Press. Western Electric Company. 1956. Statistical Quality Control Handbook. Indianapolis, IN: West-
ern Electric Co. Wheeler, D. J. 2004. The Six Sigma Practitioner’s Guide to Data Analysis. Knoxville, TN:
SPC Press. Wheeler, D. J., and R. W. Lyday. 1989. Evaluating the Measurement Process. 2nd ed. Knox-
ville, TN: SPC Press. Whitehouse, D. 2002. Surfaces and Their Measurements. London: Kogan Page Science. ———. 2010. Handbook of Surface and Nanometrology. 2nd ed. Boca Raton, FL: Taylor and
Francis Group. Wijma, J., A. Trip, R. J. M. M. Does, and S. Bisgaard. 2009. “Quality Quandaries: Efficiency
Improvement in a Nursing Department,” Quality Engineering 21 (2): 222–28. Windsor, S. E. 2003. “Attribute Gage R&R.” Six Sigma Forum Magazine 2 (4): 23–28. Woodall, W. 2017. “Bridging the Gap between Theory and Practice in Basic Statistical Pro-
cess Monitoring.” Quality Engineering 21 (1): 2–15. Woodall, W. H., and M. Adams. 1993. “The Statistical Design of CUSUM Charts.” Quality
Engineering 5: 559–70. Woodall, W. H., and C. M. Borror. 2008. “Some Relationships between Gage R&R Criteria.”
Quality and Reliability Engineering International 24: 99–104. Woodall, W., and D. C. Montgomery. 1999. “Research Issues and Ideas in Statistical Process
Control.” Journal of Quality Technology 31 (4): 376–86. ———. 2014. “Some Current Directions in the Theory and Application of Statistical Pro-
cess Monitoring.” Journal of Quality Technology 46 (1): 78–94. Zwetsloot, I. M., and R. J. M. M. Does. 2015. “Quality Quandaries: Improving Revenue by
Attracting More Clients Online.” Quality Engineering 27 (1): 130–38.
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599
index
Note: Page numbers followed by f or t refer to figures or tables, respectively.
a absolute instruments, 180 acceptable quality level (AQL), 150, 154,
163, 169 acceptable quality limit (AQL), 150, 154,
163, 169 acceptance number, 154, 156f, 156t acceptance sampling, 148–78
acceptance number in, 154, 156f, 156t by attributes, 150, 158–63, 168 basic concepts in, 150–58 categories of plans for, 150 continuous sampling plan for, 173–76, 174t definition of, 66, 148 double sampling plan for, 158–63, 159t,
160f, 160t, 161f, 163f, 166 lot size in, 154–58, 157f, 157t multiple sampling plan for, 161, 166 risks of, 150, 152 sample integrity in, 176–78 sample size in, 154–58, 155f, 158f sequential sampling plan for, 172–73, 172f single sampling plan for, 151–54, 152f,
153t, 155f, 155t, 166 standards for, 158, 163–67, 169–72 uses for, 148–49 by variables, 150, 168–72, 168t
accuracy of confidence intervals, 352 in data collection, 301–3 of measurements, 199, 203–4
action plans definition of, 16 in deployment of quality management
system, 15–16 implementation schedule for, 30, 32f in project planning, 30, 31f, 32f
activities coupled, 244
definition of, 244 non-value-added, 256, 272–76 parallel, 244 sequential, 244
activity network diagrams (AND), 27, 244–48, 245t, 246f, 247t
actual cost, 93, 95 ad hoc groups, 46 ADDIE model of training, 103, 104 addition rule, general, 316, 321 aerospace industry, supplier management
in, 65 affective conflict, 54–55 affinity diagrams, 231, 232f, 236 affinity principle, 231 Agresti, A., 358 AHP. See analytical hierarchy process AIAG. See Automotive Industry Action Group alarm value, 131 AlMaian, R. Y., 65 Alper, S. J., 219 alternative hypothesis, 359 American National Standards Institute. See
ANSI American Society for Quality (ASQ)
The ASQ Auditing Handbook, 88 awards of, 4 code of ethics of, 40, 41f, 43 establishment of, 4 standards of (See ANSI/ASQ)
American Society for Quality Control, 4. See also American Society for Quality
American Society for Testing and Materials (ASTM), 175
American Society of Mechanical Engineers, 82 American Supplier Institute, 59 analysis of experiments. See also design of
experiments steps in process of, 461–63, 461t terminology in, 457–60
H1518_Burke.indd 599 6/29/17 11:49 AM
600 Index
analysis of variance (ANOVA), 383–92 in gage R&R studies, 207–16, 208t, 211t,
212t vs. model fitting, 481 in one- factor experiments, 464–65, 467–70,
467t, 469t one-way, 383–89, 385t, 387t, 388t for testing significance of regression,
408–9, 408t, 409t in two- factor experiments, 471, 471t in two- level factorial experiments, 478–81,
480t, 481t two-way, 389–92, 391t
analytical hierarchy process (AHP), 241 analytical studies, 313 AND. See activity network diagrams AND gate, 497–98, 498f Anderson-Cook, C. M., 270 angle measurements, 180–82, 181t, 196 ANOVA. See analysis of variance ANSI (American National Standards
Institute), 456, 559–60 ANSI/ASQ QE 19011S standard, 89 ANSI/ASQ Z1.4-2003 standard, 152, 154,
158, 161, 163–67 ANSI/ASQ Z1.9-2003 standard, 169–72,
170f ANSI/ASQC Z1.4-1993 standard, 154,
163–64 ANSI/EIA-649 standard, 143, 178 AOQ. See average outgoing quality AOQL. See average outgoing quality limit appraisal costs, 93, 94–95, 96t–97t approximations, rules for, 338, 338f AQL. See acceptable quality level; acceptable
quality limit ARIMA. See autoregressive integrated
moving average ARL. See average run length arrow diagrams, 244 art vs. science, quality as, 229 artifacts, physical, 197 ASN. See average sample number ASQ. See American Society for Quality ASQ Auditing Handbook, The, 88 assembly instructions, 142 assessment phase of risk management, 494.
See also risk assessment assignable causes, 415 assumptions
in control charts, 413, 416, 442 in gage R&R studies, 214–15
in linear regression, 410 in process capability studies, 451 in statistical tests, 313
ASTM. See American Society for Testing and Materials
ASTM-E2819-11 standard, 173, 175–76 AT&T, 5t attribute data. See discrete data attribute sampling plans, 150, 158–63, 168 attributes control charts, 425–35, 525 audits, 88–92
classification of, 89 definition of, 65 vs. inspections, 66, 88 planning and implementation of, 90–91 process, 66, 89 product, 66, 89 quality system, 88–92 reporting and follow- up on, 91–92 risk, 523 roles and responsibilities in, 89–90 scope of, 89 in supplier management, 65–66 system, 66, 70
autocorrelation, 413–14 automatic gauging, 300–301 automotive industry
FMEA in, 503 industry-specific standards in, 87 rules for statistical process control in,
443–44 supplier management in, 65
Automotive Industry Action Group (AIAG), 65, 179, 443–44, 503
autoregressive integrated moving average (ARIMA) models, 414
availability in reliability, 129–31 steady state, 130 system, 129
average. See mean average moving range, 423 average outgoing quality (AOQ), 150–51
curve of, 151, 151f, 154, 155f definition of, 150 for double sampling plans, 161, 161f for single sampling plans, 154, 155f, 155t
average outgoing quality limit (AOQL), 151, 161
average run length (ARL), 444 average sample number (ASN), 161–63, 163f Awbi, H., 474
H1518_Burke.indd 600 6/29/17 11:49 AM
Index 601
B balanced scorecard, 20 balancing point. See mean Baldrige, Malcolm, 87–88 base units, standards for definitions of,
195–97, 195t, 196t batches, sample integrity with, 177 bathtub curve, 131–32, 131f behavioral theory of leadership, 43 Bell System/Bell Telephone Laboratories, 4,
5t–6t, 8, 497 benchmarking, 16–17, 252–55
and breakthrough thinking, 254–55, 254f collaborative, 17 competitive, 16, 253 definition of, 16, 253 evolution of, 253 external, 17 functional, 253 generic, 253 internal, 16–17, 253 steps in, 17, 253–54
best practices in benchmarking, 16–17, 252–54 in communication, 56
Betz, N. E., 494 Bhattacharya, S., 345 bias
in accuracy of measurement, 203–4, 203t in point estimators, 347
bilateral tolerance, 112 bimodal distribution, 309 binomial distribution, 333–35, 338t
in acceptance sampling, 151–54 definition of, 333 in k-out-of-n systems, 126–27 negative, 337 probabilities for, 545–46 rules for, 338
biomedical industry, industry- specific standards in, 87
Bisgaard, Søren, 492 bivariate normal distribution, 331–32, 398 Blais, A.-R., 494 blocking
definition of, 464 one-factor experiments without, 464–65,
466t, 467t blocking factors, 465–70 Bond, T. P., 148 Bothe, D. R., 450
Box, G. E. P., 6t–7t box plots, 303–4, 303f–5f, 388, 388f box-and-whisker diagrams, 303–4, 303f brainstorming
in corrective action, 291 as facilitation tool, 52–53 structured vs. unstructured, 52–53
breakdown value, 131 breakthrough thinking, 254–55, 254f Breyfogle, F. W., 268 bridge-type configuration, 188–89 British Standards Institution, 560 Brumbaugh, Martin, 4 Brumbaugh Award, 4 budgets
project, 27 quality cost program in, 100–101
Building on Baldrige (Council on Competitiveness), 87–88
Burdick, R. K., 205 business processes, in balanced scorecard, 20 business risks, management of, 494
C c charts, 426, 432–33, 433f
short-run, 449, 449f CADDIEM model of training, 103 calibration, 200–202 calipers, 181 Camgoz-Akdag, H., 59 Camp, R. C., 253 “can,” 108 Canadian Standards Association (CSA),
275–76, 560 cantilever configuration, 188, 190f capability, process and performance, 449–56 capability studies, 312–13, 450–51 categorical frequency distribution, 311, 311t Cauchy distribution, 342 causation
in linear regression, 412 of variation, common and special, 414–15
cause-and-effect (Ishikawa) diagrams, 219–21 examples of, 220–21, 220f, 221f origins of, 10 in root cause analysis, 290, 291 steps in creation of, 219–20 uses for, 219
cdfs. See cumulative distribution functions center, of data, 304–7 central limit theorem, 339–40
H1518_Burke.indd 601 6/29/17 11:49 AM
602 Index
certification operator, in corrective action, 294 process, in corrective action, 293–94, 294t
change(s) planning for, in quality system, 77–78 sustaining and communicating, in Six
Sigma, 259, 270–71 change agents, 74 change control, 177–78 change management, 74 charter, project, 26 check sheets, 221–23, 223f, 300, 300f checklists
in audits, 90 in supplier surveys, 67–68
chi-square distribution, 341, 341f, 547–48 chi-square goodness- of-fit tests, 392–95 Cianfrani, C. A., 83 closed-loop failure reporting, 135 closing conference, 69 CM. See configuration management CMMs. See coordinate measuring machines coding, data, 301 coefficient of determination, 409 cold standby redundancy, 128 collaborative benchmarking, 17 Collins, J. C., 58 column-type configuration, 189, 190f common cause variation, 414–15 communication, 56–57
as barrier to quality improvement, 72 best practices in, 56 of change, in Six Sigma, 259, 270–71 with customers, 58 definition of, 56 feedback in, 57 forms of, 56–57 with suppliers, 65
community, as stakeholders, 18 comparators (comparative instruments),
180, 183 competitive benchmarking, 16, 253 complement, of events, 314 completely randomized design, 463 computer-assisted coordinate measuring
machines, 189 concept, vs. term, definitions of, 84 concept fans, 238–39 concept phase of design, 109 concurrent engineering, 110 condition(s)
given, 318 marginal, 101
maximum material, 112, 113f, 114–15 in statistical tests, 313
condition parameters, 131 conditional probability, 318–19, 321 conferences
closing, 69 opening, 68–69
confidence coefficient, 348 confidence intervals, 348–58
and hypothesis tests, 359 for paired data, 377–79 on population variance and standard
deviation, 355–56 practical interpretation of, 349 probabilistic interpretation of, 349 for single population mean, 350–55 for single population proportion, 356–58 for single population standard deviation,
355–56 for single population variance, 355–56 for two population means, 366–72 for two population proportions, 374–77 for two population variances, 372–74
configuration of coordinate measuring machines,
188–89, 190f definition of, 177
configuration control, 177–78 configuration management (CM), 143 conflict
interpersonal, 54–55 procedural, 54–55 substantive, 54–55
conflict resolution, 54–55 confounding, 463 consistency, of measurements, 200 constant(s), for control charts, 416, 526 constant failure rate, 124, 129, 132, 133f constant variance, 384, 389 constraints, theory of. See theory of constraints consumers. See customer(s) consumer’s risk, in acceptance sampling, 150 containment
in corrective action, 289–90 in reaction plans, 138
contingency tables, 316–17, 316t, 395–98, 395t, 396t
continuous data, definition of, 299 continuous distributions, 321–47
common types of, 324–33 definition of, 321 formulas for expected value and variance
for, 324
H1518_Burke.indd 602 6/29/17 11:49 AM
Index 603
summary of types of, 333, 333t theoretical probability functions for,
321–24 continuous flow manufacturing, 271–72 continuous improvement, 217–97
case studies in, 296–97 corrective action in, 286–94 lean thinking in, 271–86 methodologies of, 10–11, 255–71 preventive action in, 294–96 quality control tools in, 217–29 quality management and planning tools
in, 229–55 continuous sample spaces, 313 continuous sampling plans (CSPs),
173–76, 174t continuous uniform distribution, 330–31,
331f, 333t contour plots, 484–85, 484f control
change, 177–78 configuration, 177–78 document, 81 inventory, 278 material, 142–48 positive (positrol), 293, 293t product and process (See product and
process control) quality (See quality control) recurrence, 293–94 risk, 522–23 statistical (See statistical control) visual, 257, 276–78, 277f
control charts, 416–49 analysis of, 443–45 assumptions in, 413, 416, 442 attributes, 425–35, 525 constants for, 416, 526 control limits in, 416, 525 in corrective action, 292 cumulative sum, 435–40 exponentially weighted moving average,
439–41 functions of, 78, 416 in gage R&R studies, 213–14, 213f how to choose, 442, 442f as main tool in statistical process
control, 11 moving average, 441–42 origins of, 4, 8, 416 vs. pre- control charts, 445–47 rational subgrouping in, 415–16 vs. run charts, 226, 413
selection of variables for, 415 short-run, 448–49 in time series analysis, 413 variables, 417–25, 525
control factors, in experimental design, 458 control limits
formulas for, 416, 525 vs. specification and natural tolerance
limits, 452 control phase of risk management, 494 control plans, 137–39, 140f, 141f coordinate measuring machines (CMMs),
188–90 COQ. See cost of quality corporate university, 102 corrective action, 286–94
in continuous improvement, 286–94 eight disciplines (8D) model of, 286–87 failure and root cause analysis in, 289–91 phases of, 286, 286f problem correction in, 291–93 problem identification in, 288–89 in quality improvement, 101 in quality system, 79 recurrence control in, 293–94, 293t, 294t and reliability, 135 in risk management, 521 verification of effectiveness in, 294
corrective maintenance, 130 correlation(s)
autocorrelation, 413–14 definition of, 398 negative, 227, 228f, 399 no, 227, 228f positive, 227, 228f, 399 in scatter diagrams, 227, 228f simple linear, 398–401
correlation coefficient, population, 398–400 cost of quality (COQ), 92–102
categories of, 93, 94–95, 96t–98t collection of, 95–99 definition of, 92–93 implementation of, 95 management of, 94 principles and lessons from, 101–2 reporting on, 100 summary and analysis of, 99–100 using, 100–101
cost system, quality, goal of, 93–94 Coull, B., 358 Council on Competitiveness, Building on
Baldrige, 87–88 countdown, 295
H1518_Burke.indd 603 6/29/17 11:49 AM
604 Index
coupled activities, 244 Cp process capability index, 453–54, 455 Cpk process capability index, 454–55 CPM. See critical path method Cpm process capability index, 455 Cr process capability index, 453–54, 455 Crawford slip, 53 Creasy, T., 259 creativity
in breakthrough thinking, 254 components of, 52 in corrective action, 291–92 lost, 275 types of, 291–92
critical control points, 139–42 critical defects, 147, 164 critical design review, 112 critical failures, 135 critical path, in activity network diagrams,
244–48, 246f critical path method (CPM)
in activity network diagrams, 244–48, 246f in project planning, 27, 28, 30f, 32 in strategic planning, 14
critical values, in hypothesis testing, 359–60, 380
criticality, definition of, 514 criticality analysis, 514–17 Crosby, Philip, 3, 10 Crosby Medal, 4 Crosby Quality College, 10 Crowtha, Samuel, 5t CSA. See Canadian Standards Association C-shaped manufacturing cells, 274–75, 274f CSP. See continuous sampling plans cube plots, 475, 475f Cudney, E., 197 cumulative distribution functions (cdfs),
322–24 for continuous uniform distribution, 331 definition of, 323 and probability mass functions, 323–24 in reliability measures, 119, 121t, 122f, 124 for Weibull distribution, 330
cumulative frequency distribution, 310–11, 310t, 311t
cumulative sum (CUSUM) control charts, 435–40, 438f, 442
curriculum development, 103. See also training
customer(s), 58–64 in balanced scorecard, 20 needs and wants of, 58, 60, 79 in quality function deployment, 59–63
in quality planning, 79 reliability in perceptions of, 118 as stakeholders, 18
customer focus characteristics of companies with, 64 in implementation of Six Sigma, 261–62 lack of, as barrier to quality
improvement, 73 customer value analysis, 61–63, 63t customer-driven quality, 63–64 CUSUM. See cumulative sum cycle times, 284–86
d Daniel, C., 485 dashboards, 21–22, 21f data, 299–311
center of, 304–7 in descriptive statistics, 304–11 graphical summaries of, 303–4, 309–11 in information systems strategy matrix,
36f, 37 shape of, 309–11 spread of, 308–9 types of, 299
data coding, 301 data collection
accuracy and integrity in, 301–3 errors in, 301–2 measurement scales in, 299–300 methods of, 300–301 sampling in, 302–3
DeBono, E., 254 decision criteria, 239–41 decision making. See statistical decision
making decision trees, 22, 23–24 defects
classification of, 147, 148, 164 correction of, as waste, 275 critical, 147, 164 definition of, 147 major, 147, 164 in material control, 146–48 minor, 147, 164 as nonconformities, 78 vs. nonconformities, 147 reporting system for, 100 serious, 147 zero, 10
defects per million opportunities (DPMO), 453
defensive approach to risk, 495
H1518_Burke.indd 604 6/29/17 11:49 AM
Index 605
defining relations, in fractional factorial experiments, 488
definition phase of design, 108 degrees of freedom (df), in ANOVA, 386 Dell Corporation, 17 Delphi method, 518 Deming, W. Edwards
on analytical studies, 313 career and contributions of, 1, 4, 5t–6t,
8–9 on control charts, 443 14 points of, 9, 9f on inspections, 150 in origins of total quality management,
11, 256 work in Japan, 4, 8
Deming cycle. See plan- do-study-act cycle Deming Medal, 4 Deming Prize, 8 Department of Defense (DoD), 111, 143,
178, 561 dependent variables, in experimental design,
457–59 deployment of quality management system,
15–34 benchmarking in, 16–17 performance measures in, 18–22 project management in, 22–34 stakeholder analysis in, 17–18
depth rulers, 181 descriptive statistics, 304–11 design(s), 107–35
definition of, 108 FMEA applied to, 502–3, 506–14, 507f,
511t–12t, 515f inputs for, 110 maintainability in, 117–18 meaning of quality in, 75–76 phases of, 108–9 of quality system, 79 reliability in (See reliability) review of, 111–12 technical drawings and specifications in,
112–15 verification and validation of, 115–17
design for reliability (DFR), 110 design for Six Sigma (DFSS), 110 design of experiments (DOE), 457–92
efficiency in, 457 factorial (See factorial experiments) objectives of, 457, 458 one-factor, 464–70 principles of, 463–64 and statistical control, 491–92
steps in process of, 461–63, 461t terminology in, 457–60
destructive sampling, 149 destructive testing, 191 detailed phase of design, 109 detection
in FMEA, 510, 513t, 514t in preventive action, 295 in risk analysis, 518
df. See degrees of freedom DFR. See design for reliability DFSS. See design for Six Sigma Diaconis, P., 224 diagnosis, in reaction plans, 138 dial indicators, 183 dimensional inspection, 116 direct computer- controlled coordinate
measuring machines, 189 discrete data
definition of, 299 hypothesis tests for, 392–98
discrete distributions, 321–47 common types of, 333–38 definition of, 321 formulas for expected value and variance
for, 324 summary of types of, 338, 338t theoretical probability functions for, 321–24
discrete sample spaces, 313 disposition
in material control, 146 in reaction plans, 139
distribution phase of design, 109 DMAIC process, 269–70
case study of, 296–97 phases of, 269
DMRCS process, 270 document, definition of, 80 documentation
in audits, 90–91 in configuration control, 177–78 control of, 81 definition of, 80 obsolete, 81 process for generation of, 81 project, 33 of quality system, 79, 80–81, 80f in risk management, 522–23
DoD. See Department of Defense Dodge, H. F., 5t Dodge-Romig tables, 168 Dodge’s continuous sampling plan, 173, 175 DOD-STD-480A standard, 178 DOD-STD-481 standard, 178
H1518_Burke.indd 605 6/29/17 11:49 AM
606 Index
DOD-STD-973 standard, 178 DOE. See design of experiments Does, R. J. M. M., 297 dot plots, 309, 309f double sampling plans, 158–63, 159t, 160f,
160t, 161f, 163f, 166 DPMO. See defects per million opportunities drawings, technical, 112–15 driving forces, in force field analysis, 232–33 Dudding, B. P., 5t Duffy, G. L., 286 Duncan, A. J., 6t Durivage, M. A., 294
E Ebeling, C. E., 120 Economic Control of Quality of Manufactured
Product (Shewhart), 8 economics of quality, 92–93 eddy current testing, 192 education, vs. training, 106 Edwards, George, 4 Edwards Medal, 4 EEO. See equal employment opportunity effectiveness, verification of
in corrective action, 294 in preventive action, 296
effects model, 465 eight disciplines (8D) model, 286–87 80/20 rule, 224–25 elimination, in preventive action, 295 Elsayed, E. A., 124 emergency changes, 177 employees, as stakeholders, 18 empowerment, lack of, as barrier to quality
improvement, 72 empty set, 314 Enterprise Resource Planning (ERP), 278 enterprise risk management, 494 environmental stress screening (ESS), 496–97 Environmental Stress Screening of Electronic
Hardware (ESSEH) committee, 497 equal employment opportunity (EEO)
laws, 40 equipment, for team success, 46 equipment limitation, 198, 198f ERP. See Enterprise Resource Planning error(s)
in data collection, 301–2 in experimental design, 458, 460 local control of, 464 margin of, 352 in measurement, 197–99, 202–3
standard, 343, 347 type I, 359, 381–82, 382f type II, 359, 381–82, 382f
error sum of squares, 385–86, 408, 467 error variance, 386, 405 error-proofing principles, 295 ESS. See environmental stress screening ESSEH. See Environmental Stress Screening
of Electronic Hardware estimated effects, 474, 476–81, 478t estimates
least squares, 402–3 point, 347–48
estimation, project, 26–32 ethics
code of, 40, 41f, 43 dilemmas in, 41–43
event(s) complement of, 314 definition of, 244, 313 independent, 319–20 intersection of, 314 mutually exclusive, 314–21 sample, 313–15 union of, 314
event trees, 238–39, 500 EWMA. See exponentially weighted moving
average excess motion, 272 excess movement of material, 273–74 excess processing, 275 executives
leadership of, in Six Sigma, 261 as stakeholders, 18
expected frequencies, 393–97, 394t, 397t expected values, formulas for, 324 experimental error, 458, 460 experiments
analysis of, 457–63 definition of, 313 design of (See design of experiments)
exponential distribution, 328–29, 329f, 333t, 549–50
exponentially weighted moving average (EWMA) control charts, 439–41, 441f, 442
external benchmarking, 17 external failure costs, 94–95, 97t–98t external quality audits, 89 extrapolation, 404
F F distribution, 343, 533–44 facilitation, in preventive action, 295
H1518_Burke.indd 606 6/29/17 11:49 AM
Index 607
facilitators, 49–55 as leadership role, 43 methods and tools of, 52–55 roles and responsibilities of, 49–52
fact, management by, 299 factor(s)
blocking, 465–70 control, 458 definition of, 458, 459 noise, 458, 459 nuisance, 464, 465 qualitative, 458 quantitative, 458
factorial experiments, 470–91 definition of, 389, 458 fractional, 487–91 full (See full- factorial experiments) vs. randomized block designs, 472 two-factor, 470–71, 471t 2k, 485–87 two-level, 472–85, 473f, 473t
fail-safe devices, 295, 295t failure(s). See also reliability
catastrophic, 118 critical, 135 main factors leading to, 118
failure analysis, in corrective action, 289–91 failure costs, 93, 94–95, 97t–98t
external, 94–95, 97t–98t internal, 94, 97t–98t
failure density, 120–24, 121t, 123f failure mode effects and criticality analysis
(FMECA), 514–17, 517f failure modes and effects analysis (FMEA),
501–14 in control plans, 137–38 definition of, 501 design, 502, 506–14, 507f, 511t–12t, 515f in evaluation of risk, 520–21 vs. fault tree, 238 inputs and outputs of, 505 planning for, 503–4 process, 502–3, 506–14, 508f, 511t–14t,
516f service delivery, 503 standards for, 503 steps in, 503 system, 502 team members in, 504–5 uses for, 502
failure rate, constant, 124, 129, 132, 133f. See also hazard (failure) rate function; reliability
failure reporting, closed- loop, 135
failure time distribution function. See probability density functions
Family Educational Rights and Privacy Act (FERPA), 42
fault tree (FT), 237–39, 238f, 497–500, 500f fault tree analysis (FTA), 497–500, 500f Fazel, F., 72 FDA. See Federal Drug Administration Federal Drug Administration (FDA), 34, 494 Feigenbaum, A. V., 6t, 10, 11, 256 Feigenbaum Medal, 4 FERPA. See Family Educational Rights and
Privacy Act financial measures, in balanced scorecard, 20 fishbone diagrams. See cause- and-effect
diagrams Fisher, R. A., 5t, 464, 491–92 fitness for use, quality as, 3, 10 fitted regression line, 402–4, 404f 5S methodology, 257, 277–78 fixed bridge configuration, 189, 190f fixed limit gages, 182–83, 182f fixed significance level, 380 fixes, quick, 73 flow diagrams, 290 flowcharts, 218–19, 220f
and process maps, 248 symbols used in, 218–19, 219f, 248
FMEA. See failure modes and effects analysis FMECA. See failure mode effects and
criticality analysis Food and Drug Administration, 144–45 force field analysis, 232–33, 233f Ford, Henry, 2, 5t fraction nonconforming, 108, 178 fraction nonconforming control charts. See p
charts fractional factorial experiments
definition of, 487 two-level, 487–91, 489t
Freedman, D., 224 frequencies
expected, 393–97, 394t, 397t observed, 393, 394t, 397
frequency distribution, 309–11, 310t, 311t FT. See fault tree FTA. See fault tree analysis full-factorial experiments
definition of, 389, 459, 470 examples of, 459–60, 459t, 460t vs. fractional factorial experiments,
487–88 functional benchmarking, 253 functional objectives, 16
H1518_Burke.indd 607 6/29/17 11:49 AM
608 Index
g gage(s), 182–83, 182f gage blocks, 183 gage repeatability and reproducibility (R&R)
studies, 204–16 ANOVA method in, 207–16, 208t, 211t, 212t assumptions in, 214–15 control charts in, 213–14, 213f and control plans, 138 design of, 205 example of, 209–13, 209t–12t interpretation of, 214 purposes of, 204 tabular method in, 205–7, 210–15, 210t, 212t
Gale, B. T., 61 gantry-type configuration, 189, 190f Gantt, Henry, 5t Gantt charts, 27–28, 29f Garvin, D. A., 2 gates, in fault tree analysis, 497–98, 498f gauge. See gage gauging
automatic, 300–301 definition of, 191
GD&T. See geometric dimensioning and tolerancing
general addition rule, 316, 321 general multiplication rule, 319–21 generators, in fractional factorial
experiments, 488 generic benchmarking, 253 geometric dimensioning and tolerancing
(GD&T), 112, 113f, 114f geometric distribution, 337 geometric symbols, 112, 113f Gilbreth, Frank, 5t given conditions, 318 global benchmarking, 253 Goal, The (Goldratt), 11 goal trees, 238–39, 240f goals
in deployment of quality management system, 15–16
organizational, 16 SMART, 14 in strategic planning, 14
Goldratt, Eliyahu, 10, 11 The Goal, 11
go/no-go gages, 182–83, 182f goodness-of-fit tests, 392–95, 393t, 394t Gosavi, A., 197 Gosset, W. S., 5t
Grant, Eugene, 6t graphs. See also specific types
data summarized in, 303–4, 309–11 in experimental design, 462 of probability distributions, 344–47
Gryna, F. M. contributions of, 6t on 80/20 rule, 225 on quality information systems, 37–38 on traceability, 145 on training, 102, 104
guide words, in hazard and operability analysis, 501, 501t
H HACCP. See hazard analysis and critical
control points Hallock, M. L., 219 hardware, in information systems strategy
matrix, 36f, 37 Hayes, B. E., 58 hazard (failure) rate function, 118–24, 121t,
123f Weibull, 132–33, 134f
hazard analysis and critical control points (HACCP), 139–42
hazard and operability analysis (HAZOP), 500–501, 501t
HAZOP. See hazard and operability analysis Health Insurance Portability and
Accountability Act (HIPAA), 42 healthcare industry
FMEA in, 503 material identification in, 143
heijunka box, 273, 274f heparin, 144–45 Hibino, S., 254 Hill, H. M., 105 HIPAA. See Health Insurance Portability and
Accountability Act histograms, 223–24, 225f, 309, 309f horizontal-arm configuration, 189, 190f hot standby redundancy, 127 house of quality diagrams, 59, 60f, 61, 62f Hunter, J. Stuart, 6t hypergeometric distribution, 336–37, 338, 338t hypothesis tests, 359–83
for discrete data, 392–98 for paired data, 377–79, 377t, 378t power of, 382 p-value approach to, 380–83 in simple linear regression, 405–7
H1518_Burke.indd 608 6/29/17 11:49 AM
Index 609
on single population mean, 360–62, 361t, 363t
on single population proportion, 365–66, 365t
on single population variance, 363–64, 364t steps in, 359–60 on two population means, 366–72,
368t, 370t on two population proportions, 374–77,
375t on two population variances, 372–74, 372t
i IATF. See International Automotive Task
Force IATF 16949:2016, 87 identification phase of risk management, 494 identity column, 477 IEC. See International Electrotechnical
Commission implementation
of action plans, 30, 32f of audits, 90–91 of corrective action, 292–93 of quality cost program, 95 of Six Sigma, 259, 261–64
implementation schedule, 30, 32f improvement
continuous (See continuous improvement) as element of quality system, 75, 76, 78–79 quality (See quality improvement) verbal forms of, 108
inclusion-exclusion formula, 499 incoming inspection, 67 incremental improvement. See kaizen independent events
definition of, 319, 321 probability of, 319–21
independent variables, in experimental design, 458–59
individuals control charts, 423–25, 424t, 425f Industry 4.0, 35 inferential statistics, 304, 312–13, 359 information systems. See quality information
systems information systems strategy matrix,
36–37, 36f infrastructure, Six Sigma, 262–64 innovation, as type of creativity, 291–92 input(s)
design, 110 FMEA, 505
input variables. See independent variables input–output requirements matrix, 60, 61f in-service review, 112 inspection(s)
vs. audits, 66, 88 definition of, 66, 191 design verification through, 115–16 mass, 4 100% (See 100% inspections) sampling, 66–67 (See also acceptance
sampling) in supplier management, 66–67 uses for, 148–49
installation qualification (IQ), 117 instantaneous failure rate function. See
hazard (failure) rate function Institute of Environmental Sciences, 497 instructions
in product and process control, 142 work, 81, 142
integrity in data collection, 301–3 sample, 176–78
interaction plots in two- level factorial experiments, 478,
479f, 480f, 485 in two- way ANOVA, 389, 389f, 391, 392f
interactions, in experimental design, 477–78 intercept, in linear regression, 402 internal benchmarking, 16–17, 253 internal failure costs, 94, 97t–98t internal quality audits, 89 internal rate of return (IRR), 23–24 International Automotive Task Force
(IATF), 87 International Bureau of Weights and
Measures, 194 International Electrotechnical Commission
(IEC), 109 International Organization for
Standardization. See ISO International System of Units. See Systems
International interpersonal conflict, 54–55 interquartile range (IQR), 303 interrelationship digraphs, 234–36, 237f intersection of events, 314 interval estimation, 348. See also confidence
intervals interval scales, 300 inventory, wasteful, 272–73, 273f inventory control systems, 278 IQ. See installation qualification
H1518_Burke.indd 609 6/29/17 11:49 AM
610 Index
IQR. See interquartile range IRR. See internal rate of return Ishikawa, K.
contributions of, 6t, 10 on quality characteristics, 230 on quality control tools, 217–18
Ishikawa diagrams. See cause- and-effect diagrams
Ishikawa Medal, 4 ISO
history of, 82 list of standards of, 561
ISO 4287:1997, 183–84 ISO 9000 series, 82–87, 145 ISO 9000:2000, 7t ISO 9000:2015, 82–84
on audits, 65 definition of quality in, 3 on design verification, 115 on documents, 80 history of, 7t, 83
ISO 9001:2015, 82, 84–86 changes to previous edition in, 84–85 clauses of, 85–86 on planning of quality system, 77, 78 on risk management, 493 on terms for improvement, 108 on traceability, 145
ISO 9004:2009, 82, 86–87 annexes to, 87 changes to previous edition in, 87 on quality management, 76–77
ISO 19011:2011, 89 ISO 31000:2009, 493, 523 ISO/R 468, 183–84, 184f
J Japan
concept of quality in, 230 Deming’s work in, 4, 8 history of quality in, 4, 6t–7t, 8–10 Juran’s work in, 4, 9 quality training in, 102
JIT. See just- in-time Joint Commission, 503 Judge, T., 43, 52, 56 Juran, Joseph M.
career and contributions of, 1, 4, 6t, 9–10 on cost of quality, 95 on DMAIC process, 270 on 80/20 rule, 225 on history of quality, 4
in Japan, 4, 9 Juran’s Quality Handbook, 10 Managerial Breakthrough, 10 in origins of total quality management,
11, 256 on quality as fitness for use, 3, 10 on strategic planning, 15
Juran Medal, 4 Juran trilogy, 10 Juran’s Quality Handbook, 10 just-in-time (JIT), 11, 257
K kaizen, 256–57
components of, 256–57 definition of, 256, 276 and lean principles, 276 and non- value-added activities, 272 origins of, 230
kaizen blitz, 256 kanban systems, 278, 279f Kaplan, Robert, 20 Karsh, B., 219 Kay, L., 296–97 Keeps, E. J., 103 Kerns, D. T., 253 Kilmann, R. H., 54 kilogram, 195 Kirkpatrick, D. L., 58, 105 Knowles, M. S., 104 Kolarik, W. J., 230 Kotz, S., 451 k-out-of-n systems, reliability of, 126–27
L labels, in calibration control system, 202 Laford, R. J., 68 Larsen, G., 205 Latin square designs, 472 layout devices, 182 leadership, 43–49
communication skills in, 57 definition of, 43 in implementation of Six Sigma, 261 lack of, as barrier to quality
improvement, 73 of quality initiatives, 48–49 theories of, 43 unofficial, 43
lean manufacturing, 11. See also lean thinking
H1518_Burke.indd 610 6/29/17 11:49 AM
Index 611
Lean Six Sigma (LSS) case studies of, 296, 297 theory of constraints combined with, 259
lean thinking, 11, 271–86 elimination of non- value-added activities
in, 272–76 tools for, 276–84
learning. See also training adults’ approach to, 104 in balanced scorecard, 20
least squares circle (LSC), 186, 187f least squares estimates, 402–3 Lee, H., 474 left-skewed distribution, 309 legal measurements, 193 length measurements, 180–81, 181t, 183,
196–97 levels of factors, in experimental design,
458, 459 life cycles
process, 108–9 product, 76, 77f, 108–9
limit gages, 182–83, 182f linear correlations, simple, 398–401, 399f linear measuring systems, 178–79 linear regression, 401–10
ANOVA for testing significance of, 408–9, 408t, 409t
assumptions in, 410 hypothesis tests in, 405–7 multiple, 410–12 predictions in, 402–4 simple, 401–10 terminology in, 401–2
linear responsibility matrix (LRM), 27 linearity, in accuracy of measurement, 203–4 liquid penetration, 192 listening, active, 57 local control of error, 464 locating devices, 182 lognormal distribution, 332–33 Long, C. S., 37–38 lost creativity, 275 lot size, in acceptance sampling, 154–58,
157f, 157t lot tolerance percent defective (LTPD), 150 Lovelace, C., 451 LRM. See linear responsibility matrix LSC. See least squares circle L-shaped bridge configuration, 189, 190f LSS. See Lean Six Sigma LTPD. See lot tolerance percent defective Lu, L., 270
M MA. See moving average magnetic particle testing, 192 magnification of senses, 295 main effects, 474, 476–82, 476f, 478t, 485 maintainability, as measure of reliability, 117,
129–31 maintenance
corrective, 130 predictive, 130–31 preventive, 130 total productive, 282
major defects, definition of, 147, 164 Makino, T., 133 Malcolm Baldrige National Quality Award
(MBNQA), 87–88 Malshe, A., 194 management by fact, 299 manual, quality, 80–81 manual coordinate measuring machines, 189 manufacturing
continuous flow, 271–72 lean, 11
MAPs. See measurement assurance protocols
margin of error, in confidence intervals, 352 marginal conditions, 101 marketing
functions of, 75 meaning of quality in, 75
Maslow’s hierarchy of needs, 44 mass inspection, 4 material, excess movement of, 273–74 material control, 142–48
classification in, 142, 146–47 definition of, 142 identification in, 143–45 material review board in, 147–48 segregation in, 146 traceability in, 143–46
material review board (MRB), 147–48 matrix charts, 103 matrix diagrams, 233–34, 235f maximum inscribed circle (MIC), 186, 187f maximum material condition (MMC), 112,
113f, 114–15 “may,” 108 Mayo, Elton, 4 MBNQA. See Malcolm Baldrige National
Quality Award MCC. See minimum circumscribed circle McClaskey, D. J., 105
H1518_Burke.indd 611 6/29/17 11:49 AM
612 Index
mean, 305–6 population (See population means) sample, 305, 343–44
mean squares (MS), 386–87 for treatments, 386–87
mean time between failures (MTBF), 128 mean time to failure (MTTF), 20, 128–29 mean time to repair (MTTR), 20, 129–30 means model, 384–85, 465 measurement(s), 178–216. See also specific types
calibration in, 200–202 classes of, 193 control of, 79 definition of, 107, 178, 193 design of systems of, 178–79 elements of systems of, 178 error in, 197–99, 202–3 in experimental design, 462 gage R&R studies on, 204–16 project, 27 of quality characteristics, 107 scales of, 299–300 standards of, 193–97, 194f in testing, 191–92 tools for, 179–90 uncertainty in, 197–200
measurement assurance, 202 measurement assurance protocols (MAPs), 202 measurement system analysis (MSA), 202–16
gage R&R studies in, 204–16 terminology in, 202–4
median, 306–7 median ranks, 553–54 meetings, team, facilitators’ role in, 50–51 Menesatti, P., 411 mentoring, 102 meter, 196–97 metrology, 193–202. See also measurement
definition of, 193 standards in, 179, 193–97 surface, 183 uncertainty in, 197–200
MIC. See maximum inscribed circle micrometers, 181 milestone charts. See Gantt charts MIL-HDBK-61A(SE), 143 MIL-STD-105E standard, 164 MIL-STD-1235B standard, 173, 175 MIL-STD-1629A standard, 503, 514 MIL-STD-1916 standard, 164 minimum circumscribed circle (MCC),
186, 187f minimum radial separation (MRS),
186–87, 187f
Minitab, rules used by, 443–44 minor defects, definition of, 147, 164 Minuteman Launch Control System, 497 mission statements, in project planning, 26 mitigation
in preventive action, 295 risk, 522–23
mitigation phase of risk management, 494 MMC. See maximum material condition mode, 307 model fitting, 462, 481–82 models
in experimental design, 462–63 reliability, 131–35
Mohanty, S., 296–97 monitoring
in project management, 32 risk, 523
Montgomery, D. C. on dimensions of quality, 2 on pre- control charts, 447 on process capability studies, 450 on process performance indices, 456 on quality management, 79, 493 on standards, 82 on total quality management, 256
motivation, lack of, as barrier to quality improvement, 73
moving average (MA) control charts, 441– 42 moving average smoothing, 412–14 moving bridge configuration, 190f moving ranges, 423–25 MRB. See material review board MRS. See minimum radial separation MS. See mean squares MSA. See measurement system analysis MTBF. See mean time between failures MTTF. See mean time to failure MTTR. See mean time to repair muda (waste), 272–75
definition of, 256, 272, 276 types of, 256, 272–75
multinomial distribution, 337 multiple linear regression, 410–12 multiple sampling plans, 161, 166 multiplication rule, general, 319–21 multivoting, 54, 54t mutually exclusive events, 314–21
n Nadler, D. T., 253 Nadler, G., 254 naming conventions, 37
H1518_Burke.indd 612 6/29/17 11:49 AM
Index 613
National Institute of Standards and Technology (NIST), 88, 193, 202, 257
NATO standards, 560 natural tolerance limits, 452 natural variation, 415 Naval Air Systems Command (NAVAIR),
109, 256 Nazli, E. K., 59 NDT. See nondestructive testing needs
assessment of, in Six Sigma, 259–60, 260t customer, 58, 60, 79 Maslow’s hierarchy of, 44
negative binomial distribution, 337 negative correlation, 227, 228f, 399 negligible effects, 485 negotiation, 55 Nelson, Lloyd S., 7t Nepal, B., 296–97 net present value (NPV), 22–25 networking, in information systems strategy
matrix, 36f, 37 NIST. See National Institute of Standards and
Technology no correlation, 227, 228f noise factors, in experimental design, 458, 459 nominal group technique, 53–54, 53t nominal scales, 299–300 nonconformities
in audits, 91–92 control charts for, 426–35 control of, 78 vs. defects, 147 definition of, 147 in material control, 147–48 as quality characteristic, 107–8
nondestructive testing (NDT), 191–92 non-value-added activities, 256, 272–76 normal distribution, 324–26, 333t
bivariate, 331–32, 398 definition of, 324–25 probability density function for, 325, 325f probability plots for, 345, 345f standard (See standard normal distribution)
normal inspection, 164–66, 165f normal probability plots, 345, 345f, 410, 451,
483–87, 483f, 486f normal scores, 555–56 normal use phase of design, 109 North Atlantic Treaty Organization (NATO)
standards, 560 Norton, David, 20 np charts, 426, 430–31, 431f NPV. See net present value
nuisance factors, 464, 465 null hypothesis, 359–60 number nonconforming control charts. See
np charts numeric studies, 312
o objectives
in dashboards, 21 in deployment of quality management
system, 15–16 functional, 16 measurable, 15 organizational, 16 project, 26 in quality system, planning for, 77 SMART, 14 in strategic planning, 14
observed frequencies, 393, 394t, 397 observed significance level. See p-values obsolescence and disposal phase of
design, 109 OC. See operating characteristic Occupational Safety and Health
Administration (OSHA), 42 occurrence
in FMEA, 510, 512t, 513t in FMECA, 514–17 in risk analysis, 518
offensive approach to risk, 495 Ohno, Taiichi, 6t on-condition maintenance. See predictive
maintenance 100% inspections
in continuous sampling plans, 173–76 definition of, 66, 148 nondestructive testing in, 192 uses for, 66, 148–49
one-factor experiments, 464–70, 466t, 467t one-way ANOVA, 383–89, 385t, 387t, 388t,
464–65 opening conference, 68–69 operating characteristic (OC) curves,
152–54, 152f acceptance number and, 154 for double sampling plans, 159–60,
160f, 160t lot size and, 154, 157f sample size and, 154, 155f type A, 152 type B, 152
operational qualification (OQ), 117 operator certification, 294
H1518_Burke.indd 613 6/29/17 11:49 AM
614 Index
operator fallibility, 198, 198f opportunities
defects per million, 453 definition of, 453 in quality system, planning for, 77 in SWOT analysis, 12
opposing forces, in force field analysis, 232–33
optical projectors, 183 OQ. See operational qualification OR gate, 497–98, 498f oral communication, 56–57 ordinal scales, 300 organizational goals, 16 organizational hierarchy, teams in, 45, 45f organizational objectives, 16 organizational strategy, 16 OSHA. See Occupational Safety and Health
Administration out of statistical control, 414, 443 output variables. See dependent variables outputs, FMEA, 505 overadjustment, 415 overcontrol, 415 overproduction, 272 oversight, risk, 494–96
P p charts, 426–30, 429f, 430f Page, E. S., 6t, 436 paired data, hypothesis tests and confidence
intervals for, 377–79, 377t, 378t Palady, P., 520 paper standards, 197 parallel activities, 244 parallel systems
redundancy in, 127 reliability of, 125–26 structure of components of, 125–26, 126t
parameters condition, 131 definition of, 312, 347
Pareto, Vilfredo, 10, 224–25 Pareto charts, 10, 224–25, 226f Pareto principle, 10, 100, 304 Parsowith, B. S., 81 parts per million (ppm) defective, 453 part-to-part variation, 304 Pascal distribution, 337 payback period, 22–23 PC. See peak count pdfs. See probability density functions PDPC. See process decision program charts
PDSA. See plan- do-study-act peak count (PC), 184 Pearlson, K. E., 36 percentiles, in probability plots, 344–45 performance capability. See process and
performance capability performance indices, 456 performance measures, 18–22
balanced scorecard in, 20 dashboard in, 21–22 guidelines for, 19 resistance to problematic behaviors in,
19–20 for suppliers, 71
Perry, B., 58 personally identifiable information (PII),
41–42 PERT. See program evaluation and review
technique Phillips, J. J., 105 philosophies, quality, 3 physical artifacts, 197 PII. See personally identifiable information Pinar, I. H., 59 plan-do-study-act (PDSA) cycle, 257, 258f planned customer- driven quality, 63–64 planning
of audits, 90–91 control, 137–39 in experimental design, 461–62 for FMEA, 503–4 project, 26–32 quality (See quality planning) of quality system, 75, 76, 77–78 in risk management, 494–96 strategic (See strategic planning)
PLCs. See programmable logic controllers plug gages, 182 pmfs. See probability mass functions point estimates, 347–48 Poisson distribution, 335–36, 338t
in acceptance sampling, 151–52, 154 definition of, 335 probabilities for, 551–52 rules for, 338
poka-yoke, 257, 279–80, 279f, 294 policy deployment, 33–34 policy statement, in documentation of
quality system, 80–81 politics, as barrier to quality improvement, 72 pooled standard deviation, 369–70 pooled variance, 369–70 population, definition of, 311–12 population correlation coefficient, 398–400
H1518_Burke.indd 614 6/29/17 11:49 AM
Index 615
population means confidence intervals for single, 350–55 confidence intervals for two, 366–72 definition of, 306 hypothesis tests on single, 360–62, 361t,
363t hypothesis tests on two, 366–72, 368t, 370t
population median, 307 population mode, 307 population proportions
confidence intervals for single, 356–58 confidence intervals for two, 374–77 hypothesis tests on single, 365–66, 365t hypothesis tests on two, 374–77, 375t
population standard deviation, 309 population variances
confidence intervals for single, 355–56 confidence intervals for two, 372–74 definition of, 308 hypothesis tests on single, 363–64, 364t hypothesis tests on two, 372–74, 372t
Porras, J. I., 58 positional tolerances, 114–15, 115f positive control (positrol), 293, 293t positive correlation, 227, 228f, 399 Potential Failure Modes and Effects Analysis
manual, 503 power, of hypothesis tests, 382 Pp process performance index, 456 Ppk process performance index, 456 ppm. See parts per million PQ. See process qualification practical interpretation, of confidence
intervals, 349 practical significance, vs. statistical
significance, 382–83 precision
in confidence intervals, 352 in measurements, 199, 203, 204
precision spindle instruments, 186 precision-to-tolerance ratios (PTRs), 214 pre-control charts, 445–47, 447f predicted values, in linear regression,
402–4 predictions, in linear regression, 402–4 predictive maintenance, 130–31 preliminary design review, 111–12 prevention costs, 93, 94–95, 96t–97t preventive action, in continuous
improvement, 294–96 preventive maintenance, 130 prioritization matrices, 239–41 priority changes, 177 proactive approach to risk, 495
probabilistic interpretation, of confidence intervals, 349
probability conditional, 318–19, 321 definition of, 312, 314 in fault tree analysis, 499–500 of independent events, 319–21 properties of, 315–16, 499 summary of key rules of, 321 terms and concepts in, 313–21
probability density functions (pdfs), 322–23 for continuous uniform distribution, 330 definition of, 322 for exponential distribution, 328, 329f graphs of, 322, 322f for normal distribution, 325, 325f properties of, 322 in reliability measures, 118–24, 121t in reliability models, 132, 132f for standard normal distribution, 326–27,
326f, 327f for Weibull distribution, 329–30, 330f
probability distributions, 321–47 central limit theorem on, 339–40 continuous (See continuous distributions) discrete (See discrete distributions) formulas for expected value and variance
for, 324 graphical displays of, 344–47 of sample statistics, 341–44 theoretical probability functions for,
321–24 probability mass functions (pmfs), 323–24,
323f probability plots, 344–47
definition of, 344–45 interpretation of, 346–47 normal, 345, 345f Weibull, 345–46, 346f
problem correction, in corrective action, 291–93
problem identification, in corrective action, 288–89
problem statement, in hypothesis testing, 360 problem-solving methods, 286 procedural conflict, 54–55 procedures, in documentation of quality
system, 81 process(es)
definition of, 414 life cycle of, 108–9 meaning of quality in, 76 quality characteristics of, 107–8
process and performance capability, 449–56
H1518_Burke.indd 615 6/29/17 11:49 AM
616 Index
process audits, 66, 89 process behavior charts, 8. See also Shewhart
control charts process capability indices, 453–55 process capability ratios, 455, 456t process capability studies, 450–51 process certification, in corrective action,
293–94, 294t process control. See product and process
control; statistical process control process control charts. See control charts process decision program charts (PDPC),
241–44, 242f, 243t, 249, 250f process design
definition of, 502 FMEA applied to, 502–3, 506–14, 508f,
511t–14t, 516f process environment, errors in, 198, 198f process flowcharts, 248 process improvement teams, 45, 45f, 46 process logs, 443, 445 process maps, 248–50, 249f, 250f process performance, vs. specifications,
451–53 process performance indices, 456 process qualification (PQ), 117 process stability, 450–51 process surveys, 67 process value chain (PVC) diagrams,
251–52, 252f process variability, 405, 417, 420 producer’s risk, in acceptance sampling, 150 product(s)
life cycle of, 76, 77f, 108–9 quality characteristics of, 107–8 as systems, 124
product and process control, 137–216 acceptance sampling in (See acceptance
sampling) as element of quality system, 75, 76,
78–79 material control in, 142–48 measurement in (See measurement) methods of, 137–42 testing in, 191–92
product audits, 66, 89 product designs
definition of, 502 FMEA applied to, 502, 506–14
product recalls, 118, 144, 146 Production Part Approval Process Manual
(AIAG), 65 production phase of design, 109
production readiness review, 112 profile limits, 112, 114f program, quality, definition of, 80 program evaluation and review technique
(PERT), 27, 244 programmable logic controllers (PLCs),
35–37 project justification, 26 project management, 22–34
documentation in, 33 monitoring and measuring in, 32 planning and estimation in, 26–32 of policy deployment, 33–34 selection of project in, 22–25
project risk management, 494 project scope, 26 project statement, 26 prototype phase of design, 109 protractors, 181 PTRs. See precision- to-tolerance ratios pull system, 271–72 purchasing, meaning of quality in, 76 push system, 271 p-values, in hypothesis tests, 380–83 PVC. See process value chain
Q QFD. See quality function deployment QIS. See quality information system QMS. See quality management system qualification
installation, 117 operational, 117 process, 117
qualitative factors, in experimental design, 458
qualitative measurements, 178 quality
creation of, 230 definitions of, 2–3, 73, 229, 493 dimensions of, 2 evolution of concept, 229–30 experience of, 230 as fitness for use, 3, 10 history of, 4–10, 5t–7t impact of, 2, 3t philosophies of, 3 as science vs. art, 229
quality assurance definition of, 79 risk management compared with, 493
quality audits, 89. See also audits
H1518_Burke.indd 616 6/29/17 11:49 AM
Index 617
quality characteristics classification of, 107–8 definition of, 107 substitute vs. true, 229–30
quality control definition of, 79 seven basic tools for, 217–29 total, 10
quality costs. See cost of quality quality function deployment (QFD), 59–63
application of, 60–61, 61f in assessment of need for training, 103 benefits of, 61 definition of, 59 in design, 110 tools of, 59–60
quality improvement barriers to, 72–74 corrective action in, 101 definition of, 79
quality information system (QIS), 34–40 definition of, 34 example of, 38–40, 39f functions of, 34–35 tools for, 36–38
quality initiatives, leadership of, 48–49 quality management
vs. risk management, 493 tools for, 229–55 (See also specific tools)
quality management system (QMS), 12–40 deployment techniques in, 15–34 quality information system in, 34–40 strategic planning in, 12–15
quality manual, 80–81 quality planning
definition of, 79, 80 tools for, 229–55 (See also specific tools)
quality program, definition of, 80 Quality Progress (magazine), 3 quality records, 79, 81 quality standards, 82–88. See also specific
standards industry-specific, 87 international, 82–87 list of, 559–61 national, 82 in supplier management, 71–72
quality system, 75–106 audits of, 88–92 cost of, 92–102 definition of, 75 design of, 79 documentation of, 79, 80–81, 80f
elements of, 75–79, 77f standards for, 82–88 training in, 102–6
quantitative concepts, 311–21 quantitative factors, in experimental
design, 458 quantitative measurements, 178 quantitative methods and tools, 299–492. See
also specific types quartiles, in box plots, 303–4 questionnaires, customer, 58 quick fixes, 73
r radio frequency identification (RFID), 143–44 radiographic techniques, 192 Ramey, S., 259 RAND, 518 random error, 197, 202–3 random error term, 401 random experiments, 313 random sampling, simple, 302 random variables, standard normal, 326 randomization
in experimental design, 463–64 in gage R&R studies, 215
randomized block designs, 465–70, 466t, 467t, 472
ranges interquartile, 303 moving, 423–25 sample, 308
rate of return, internal, 23–24 ratio scales, 300 rational subgrouping, 415–16 RCDQ. See reactive customer- driven quality reaction plans, 138–39 reactive customer- driven quality (RCDQ), 63 readability, of measurements, 199–200 recalls, 118, 144, 146 records. See documentation; quality records rectifying sampling, 149 recurrence control, in corrective action,
293–94, 293t, 294t reduced inspection, 164–66, 165f redundancy
in parallel systems, 127 in preventive action, 295 and reliability, 127–28
regression. See linear regression regression coefficients, 401–10 regression line, fitted, 402–4, 404f
H1518_Burke.indd 617 6/29/17 11:49 AM
618 Index
regression sum of squares, 408 rejection regions, 359 reliability, 117–35
availability in, 129–31 in customer perceptions, 118 definition of, 118 design for (DFR), 110 functions for quantifying, 118–24 maintainability in, 117, 129–31 models of, 131–35 of systems, 124–31 testing of, 117
reliability allocation, 19 reliability engineering, performance
measures in, 19 reliability function, 118–24, 121t, 122f, 132, 133f repair, standard, 148 repeatability, definition of, 204. See also gage
repeatability and reproducibility repeated measures, vs. replication, 464 replacement, in preventive action, 295 replication
in experimental design, 464 in gage R&R studies, 215
reporting systems closed-loop failure, 135 defect, 100
reports on audits, 91–92 on cost of quality, 100 on supplier surveys, 69
reproducibility, definition of, 204. See also gage repeatability and reproducibility
residual risk, 523 residuals, 403 resistant measure, 307 resolution III designs, 489, 491 resolution IV designs, 489–91 resolution V designs, 491 resource requirements matrix (RRM), 27,
28–30 response, in experimental design, 457, 459 response values, 459 ReVelle, J. B., 278 reviews
of design, 111–12 of projects, 32 of quality manual, 81
revisions, document, control of, 81 rework, as non- value-added activity, 272 RFID. See radio frequency identification right-skewed distribution, 309 ring gages, 182
risk definition of, 493 in quality system, planning for, 77 residual, 523 in supplier management, 71–72
risk acceptance, 522–23 risk assessment, 496–521
analysis of risk in, 496, 517–20 definition of, 22 evaluation of risk in, 496, 520–21 identification of risk in, 496–517
risk attitudes, 494 risk control, 522–23 risk identification, 496–517
environmental stress screening in, 496–97
fault tree analysis in, 497–500 FMEA in, 501–14 FMECA in, 514–17 hazard and operability analysis in,
500–501 purpose of, 496
risk management, 493–523 assessment of risk in, 496–521 control of risk in, 522–23 definition of, 493 identification of risk in, 496–517 oversight of risk in, 494–96 planning in, 494–96 principles of, 493 vs. quality management, 493 steps in process of, 494–95, 495f
risk matrices, 518–20, 519t risk mitigation, 522–23 risk monitoring, 523 risk oversight, 494–96 risk priority number (RPN)
in analysis of risk, 518 definition of, 518 in evaluation of risk, 520–21, 520t in FMEA, 510 in product and process control, 138
risk reduction, 522–23 risk statements, 496 risk syntax, 496 Robbins, S. P., 43, 52, 56 Roberts, S. W., 6t, 439 robust processes and products, in preventive
action, 296 robust tests, 313 Rodriguez-Perez, J., 139, 142 role-playing, 103 rolled throughput yield (RTY), 268
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Index 619
Romig, H. G., 5t root cause analysis, in corrective action,
289–91 Rother, M., 283 roundness measurement, 185–87, 186f, 187f routine changes, 177 RPN. See risk priority number RRM. See resource requirements matrix RTY. See rolled throughput yield Rule of Ten, 179 rulers, steel, 181 run charts
as quality control tool, 225–27, 227f in time series analysis, 412–13, 412f
Russell, J. P., 178
S SAE. See Society of Automotive Engineers SAE J1739 standard, 503 safety, in quality system, planning for, 78 Salegna, G., 72 sample, definition of, 311–12 sample events, 313–15 sample homogeneity, 302 sample mean
definition of, 305, 343 sampling distribution of, 343–44
sample median, 306–7 sample mode, 307 sample range, 308 sample size, in acceptance sampling, 154–58,
155f, 158f sample spaces, 313–15 sample standard deviation, 309 sample variance, 308 sampling distributions, 341–44
definition of, 341 of sample mean, 343–44 types of, 341–43
sampling inspections, 66–67. See also acceptance sampling
advantages of, 149 types of, 66–67 uses for, 148–49
sampling methods, 302–3 Sarbanes-Oxley (Sarbox) legislation, 40 Sauer Danfoss Company, 143–44 Saunders, C. S., 36 SCADA. See supervisory control and data
acquisition scatter diagrams, 227–29, 228f, 228t, 229f science vs. art, quality as, 229
scientific framework for quality, 230 scientific management, 4, 5t scientific measurements, 193 scientific method, 286 scope
of audits, 89 of projects, 26
scorecard, balanced, 20 scores, normal, 555–56 Scott, D., 224 scrap, 99 screening. See 100% inspections SDWTs. See self- directed work teams self-directed work teams (SDWTs), 46 senses, magnification of, 295 sensitivity, in measurements, 199–200 sensitivity analysis, 24 sequential activities, 244 sequential sampling plans, 172–73, 172f series systems
reliability of, 124–25 structure of components of, 124–25, 125f
serious defects, definition of, 147 service delivery
definition of, 503 FMEA applied to, 503
set empty, 314 universal, 313
setup instructions, in product and process control, 142
seven deadly diseases, 9 severity
classification of defects by, 147, 148, 164 in FMEA, 510, 511t–12t in FMECA, 514–17 in risk analysis, 518
“shall,” 108 shape, of data, 309–11 Shewhart, Walter A.
career and contributions of, 1, 4, 5t, 8 control chart invented by, 4, 8, 416 on definition of quality, 229, 230 Economic Control of Quality of Manufactured
Product, 8 on plan- do-study-act cycle, 257
Shewhart control charts assumptions in, 416, 442 as process behavior charts, 8 uses for, 416, 435, 442
Shewhart cycle. See plan- do-study-act cycle Shewhart Medal, 4, 8 Shingo, Shigeo, 6t, 281
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620 Index
Shook, J., 283 short-run statistical process control,
448–49, 449f “should,” 108 SI. See Systems International significance level
fixed, 380 in hypothesis testing, 359–60, 380–83 observed (See p-values) statistical vs. practical, 382–83
simple linear correlations, 398–401, 399f simple linear regression, 401–10
hypothesis tests in, 405–7 predictions in, 402–4 terminology in, 401–2
simple random sampling, 302 Simpson, O. J., 176 sine bars, 181 single minute exchange of dies (SMED),
281–82 single sampling plans, 151–54, 152f, 153t,
155f, 155t, 166 SIPOC diagrams, 249–50, 251f Six Sigma, 11, 259–71
as business strategy, 259, 261 case studies of, 296–97 design for, 110 DMAIC process in, 269–70 history of, 7t implementation of, 259, 261–64 Lean, 259, 296, 297 metrics of, 259, 264–69, 264f–67f, 269f needs assessment in, 259–60, 260t sustaining and communicating change in,
259, 270–71 skewed distribution, 309 skip-lot inspections, 66 slope, in linear regression, 402 SMART goals and objectives, 14 SMED. See single minute exchange of dies snap gages, 182 Society of Automotive Engineers (SAE),
65, 503 software
in information systems strategy matrix, 36f, 37
V model for development of, 36–37, 37f sorting. See 100% inspections source inspection, 67 spaces, sample, 313–15 sparsity-of-effects principle, 485 SPC. See statistical process control
special addition rule, 315–17, 320, 321 special cause variation, 414–15 special checking and control devices, 295 special multiplication rule, 321 specification limits, 446, 450–54 specification phase of design, 108–9 specifications
documentation of, 80 functions of, 75–76 meaning of quality in, 75–76 in process capability studies, 450 vs. process performance, 451–53 in supplier management, 71–72 technical, 112–15
sponsors, team, 44–45, 47 spread of data, 308–9 SQA. See supplier quality assurance SQM. See supplier quality management SQP. See strategic quality planning SS. See sum of squares stability
in accuracy of measurement, 203–4 process, definition of, 450–51
stakeholders analysis of, 17–18 identification of, 18 in project planning, 26
standard(s). See also quality standards for acceptance sampling, 158, 163–67,
169–72 definition of, 179 for FMEA, 503 list of, 559–61 of measurement, 193–97, 194f
standard deviation confidence intervals on, 355–56 definition of, 309 pooled, 369–70
standard error, 343, 347 standard normal distribution, 326–28
probability density function for, 326–27, 326f, 327f
Z-values in, 528–32 standard normal random variables, 326 standard repair, 148 standard work charts, 280, 280f standard work principle, 280 standardized control charts, 448–49 standby redundancy
cold, 128 hot, 127 warm, 128
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Index 621
standby systems reliability of, 127–29 structure of components of, 127–28, 128f
StarLink corn seed, 144 statistic, definition of, 312, 347 statistical control
definition of, 414 and experimental design, 491–92 out of, 414, 443
statistical decision making, 347–98. See also specific methods
statistical process control (SPC), 414–49 common and special causes of variation
in, 414–15 as continuous improvement methodology,
10–11 control charts in (See control charts) definition of, 414 objectives and benefits of, 414 rational subgrouping in, 415–16 selection of variables in, 415 short-run, 448–49
statistical quality control, 8 statistical significance, vs. practical
significance, 382–83 statistical tests, assumptions or conditions
in, 313 statistical tolerance intervals, 358, 527 statistics
descriptive, 304–11 drawing conclusions in, 312–13 inferential, 304, 312–13, 359 terminology of, 311–12
steady state availability, 130 steel rulers, 181 stem-and-leaf diagrams, 303, 303f Stevenson, W., 225, 226 stockholders, 18 Stolovitch, H. D., 103 strategic planning, 12–15
analysis phase of, 12 effectiveness tests for, 12–15, 13f lack of, as barrier to quality
improvement, 73 vs. strategic quality planning, 63–64
strategic quality planning (SQP), 63–64 stratified sampling, 302 strengths, in SWOT analysis, 12 strong claim, 360 Student’s t distribution, 342, 342f Sturges, H. A., 224 substantive conflict, 54–55
substitute quality characteristics, 229–30 sum of squares (SS)
due to different factor levels, 467 due to different levels of the blocking
factor, 467 due to treatments, 385–86 error, 385–86, 408, 467 regression, 408 total, 385, 408
Sumithra, B., 345 supervisory control and data acquisition
(SCADA) systems, 35–36 supplier(s)
communication with, 65 in design reviews, 111 as stakeholders, 18
supplier quality assurance (SQA), 71 supplier quality management (SQM),
65–72 performance measures in, 71 risk in, 71–72 techniques for, 65–71
surface metrology, 183 surface plates, 182 surface texture measurement, 183–85,
184f, 185f surveys
customer, 58 definition of, 67 process, 67 in supplier management, 67–69 system, 67
survival function. See reliability function SWOT (strengths, weaknesses, opportunities,
and threats) analysis, 12 symbols
in fault trees, 497–98, 498f in flowcharts, 218–19, 219f, 248 geometric, 112, 113f in matrix diagrams, 234 in process maps, 248 in value stream maps, 283
system(s). See also specific types definition of, 34, 502 FMEA applied to, 502 general process of, 457, 457f products as, 124 reliability of, 124–31
system audits, 66, 70 system availability, 129 system surveys, 67 systematic errors, 197–98, 202–3
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622 Index
systemic diagrams, 238–39 systems change, 47 Systems Engineering Technical Review
Process, 109 Systems International (SI), 193, 195–97,
195t, 196t systems requirements review, 111
T t distribution
Student’s, 342, 342f values of, 557–58
tabular method, 205–7, 210–15, 210t, 212t Taguchi, Genichi, 6t, 10 Tague, N., 502 takt time, 276, 284, 285–86 tally sheets. See check sheets Taylor, Frederick, 4, 5t teams, 44–55
facilitators in, 49–55 in FMEA, 504–5 need for, 44 in project planning, 26 selection of members for, 26, 46 sponsors in, 44–45, 47 stages of development of, 47–48 support mechanisms for, 46–47 types of, 45–46
technical drawings and specifications, 112–15
technical measurements, 193 technical risks, management of, 494 telecommunications industry, 56, 87 Ten, Rule of, 179 Tennessee Eastman Company, 105–6 term, vs. concept, definitions of, 84 test statistic
definition of, 360 in hypothesis testing, 360
testing. See also specific types definition of, 191 design validation through, 116–17 design verification through, 115–16 destructive, 191 functions of, 179 nondestructive, 191–92 in product and process control, 191–92 risk, 523
theory of constraints (TOC) as continuous improvement methodology,
10–11, 258–59 definition of, 258 steps to improvement in, 258
Thomas, K. W., 54 threats, in SWOT analysis, 12 tightened inspection, 164–66, 165f time series analysis, 412–14 time-dependent task diagrams, 27 TOC. See theory of constraints tolerance(s), 112
bilateral, 112 geometric, 112, 113f positional, 114–15, 115f unilateral, 112
tolerance intervals definition of, 358 statistical, 358, 527
tolerance limits, natural, 452 tools. See specific types total productive maintenance (TPM), 282 total quality control, coining of term, 10 total quality cost, 95 total quality management (TQM)
as continuous improvement methodology, 10–11, 256
definition of, 11 origins of, 11, 256
total sum of squares, 385, 408 Townsend, A., 183 Toyoda, Eiji, 6t Toyota, 6t, 273, 281 TPM. See total productive maintenance TQM. See total quality management traceability
in material control, 143–46 of measurements, 200
training, 102–6 development of, 103–6 vs. education, 106 evaluation of, 104–5, 105t in quality system, 102–6 reasons for failure of, 104 scale of, 102 in Six Sigma, 263, 269 for team success, 46–47
trait theory of leadership, 43 translations, of ISO 9000:2015, 84 treatment combinations, in experimental
design, 458 treatments
in experimental design, 458 mean squares for, 386–87 sum of squares due to, 385–86
tree diagrams, 237–39, 238f, 240f trend analysis
definition of, 413 in time series analysis, 412–14
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Index 623
true quality characteristics, 229–30 trust, lack of, as barrier to quality
improvement, 72 t-tests
in factorial experiments, 481, 482t, 487, 487t for paired data, 377–78 for single population mean, 362 statistical software output of, 381t
Tukey, John, 303 two-factor experiments, 470–71, 471t 2k factorial design, 485–87 two-level factorial experiments, 472–85,
473f, 473t two-level fractional factorial experiments,
487–91, 489t two-way ANOVA, 389–92, 391t two-way contingency tables, 395 type I errors
definition of, 359 in hypothesis tests, 359, 381–82, 382f
type II errors definition of, 359 in hypothesis tests, 359, 381–82, 382f
u u control charts, 426, 433–35, 435f ultrasonic testing, 192 unbiased point estimators, 347 underadjustment, 415 undercontrol, 415 ungrouped frequency distribution, 309, 310t unilateral tolerance, 112 union of events, 314 universal set, 313 “useful many,” 10
v V model for software development, 36–37, 37f validation, design, 115–17 value chain diagrams. See value stream maps value stream maps (VSMs), 282–84, 282f, 283f Van Patten, J., 278 variability, measures of, 308–9 variables
dependent, in experimental design, 457–59
independent, in experimental design, 458–59
quality characteristics as, 107 relationships between, 398–414 selection of, in statistical process
control, 415
variables control charts, 417–25, 525 variables sampling plans, 150, 168–72, 168t variance. See also population variances
constant, 384, 389 error, 386, 405 formulas for, 324 pooled, 369–70 sample, 308
variation allowable, in parts and components, 112 common and special causes of, in
statistical process control, 414–15 verbal (oral) communication, 56–57 verification
design, 115–16 of effectiveness, in corrective action, 294 of effectiveness, in preventive action, 296 in reaction plans, 139
vernier calipers, 181 Vining, G., 296 visual communication, 56 visual control, 257, 276–78, 277f visual factories, 276 visual inspection, design verification
through, 116 “vital few,” 9–10, 224 voice of the customer (VOC), 59, 110 VSMs. See value stream maps
W waiting, 272 Wald, A., 172 Wal-Mart, 17 warm standby redundancy, 128 waste. See muda Watson, G. H., 253 Watson, H. A., 497 WBS. See work breakdown structure weak claim, 360 weaknesses, in SWOT analysis, 12 Weber, E. U., 494 webinars, 104 Weibull distribution, 329–30, 333t
cumulative distribution function for, 330 definition of, 329 probability density function for, 329–30,
330f probability plots for, 345–46, 346f in reliability models, 132–33, 134f
Welch, Jack, 262 West, J. E., 83 Westcott, R. T., 87 Wheeler, D. J., 456
H1518_Burke.indd 623 6/29/17 11:49 AM
624 Index
Whitehouse, D., 180 Whitney, Eli, 5t Wijma, J., 296 Wilson, K. B., 6t WIP. See work in progress Wood, R. C., 61 work breakdown structure (WBS), 26, 27, 28f work groups, 45 work in progress (WIP), 272 work instructions
in documentation of quality system, 81 in product and process control, 142
World War II, history of quality in, 4 written communication, 56–57
x x– and R control charts, 417–19, 417t, 418t,
419f x– and s control charts, 420–22, 421t, 422f Xerox, 16 x-ray techniques, 192
z zero defects, 10 Z-values, 325–26, 528–32 Zwetsloot, I. M., 297
H1518_Burke.indd 624 6/29/17 11:49 AM
- Table of Contents
- List of Figures
- List of Tables
- Preface to the Fourth Edition
- Preface to the Third Edition
- Preface to the Second Edition
- Preface to the First Edition
- How to Use this Book
- Acknowledgments
- List of Acronyms
- Certified Quality Engineer (CQE) Body of Knowledge
- Chapter 1
- Chapter 2
- Chapter 3
- Chapter 4
- Chapter 5
- Chapter 6
- Chapter 7
- Appendix A
- Appendix B
- Appendix C
- Appendix D
- Appendix E
- Appendix F
- Appendix G
- Appendix H
- Appendix I
- Appendix J
- Appendix K
- Appendix L
- Appendix M
- Appendix N
- Appendix O
- Appendix P
- Glossary
- References
- Index