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Statistics in

Criminal Justice

Fourth Edition

David Weisburd

Chester Britt

Statistics in Criminal Justice

Statistics in

Criminal Justice

Edition

David Weisburd

Hebrew University of Jerusalem, Jerusalem, Israel

Fourth

Chester Britt

Northeastern University, Boston, MA, USA

and

and George Mason University, Fairfax, VA, USA

David Weisburd Chester Britt

Institute of Criminology School of Criminology and Criminal Justice Faculty of Law Northeastern University

Hebrew University of Jerusalem Jerusalem, Israel

ISBN 978-1-4614-9169-9 ISBN 978-1-4614-9170-5 (eBook) DOI 10.1007/978-1-4614-9170-5 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: © Springer Science+Business Media New York 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Boston, MA, USA

2013952914

and

George Mason University Fairfax, VA, USA

Department of Criminology, Law and Society

Additional material to this book can be downloaded from http://extras.springer.com

For Bryan, who made the desert bloom, used

sun to brighten the night, and brought such joy

to family and friends

D. W.

v

For my parents, Chester and Lila, who have been

a constant source of support

C. B.

iv C H A P T E R N U M B E R : C H A P T E R T I T L E

Contents

1

Statistics Are Used to Solve Problems 4

The Uses of Statistics 7

Measurement: The Basic Building Block of Research 13 Science and Measurement: Classification as a First Step in Research 14

Levels of Measurement 15

Relating Interval, Ordinal, and Nominal Scales: The Importance of Collecting Data

at the Highest Level Possible 22

What Is a Good Measure? 23

Representing and Displaying Data 36 What Are Frequency Distributions and Histograms? 37

Extending Histograms to Multiple Groups: Using Bar Charts 43

Using Bar Charts with Nominal or Ordinal Data 50

Pie Charts 51

Time Series Data 52

Describing the Typical Case: Measures of Central Tendency 65 The Mode: Central Tendency in Nominal Scales 66

The Median: Taking into Account Position 68

The Mean: Adding Value to Position 74

Statistics in Practice: Comparing the Median and the Mean 82

How Typical Is the Typical Case?: Measuring Dispersion 94

The Purpose of Statistics Is to Clarify 3

Measuring Dispersion in Interval Scales: The Range, Variance, and Standard Deviation 102

Introduction: Statistics as a Research Tool

P r e f a c e

C h a p t e r o n e

C h a p t e r t w o

C h a p t e r t h r e e

C h a p t e r f o u r

C h a p t e r f i v e

x i i i

vii

Basic Principles Apply Across Statistical Techniques 5

Measures of Dispersion for Nominal- and Ordinal-Level Data 95

The Logic of Statistical Inference: Making Statements About Populations from Sample Statistics 125

The Dilemma: Making Statements About Populations from Sample Statistics 126

The Research Hypothesis 129

The Null Hypothesis 131

Risks of Error in Hypothesis Testing 133

Risks of Error and Statistical Levels of Significance 135

Departing from Conventional Significance Criteria 1 7

Defining the Observed Significance Level of a Test: A Simple Example Using the Binomial Distribution 145

The Fair Coin Toss 147

Different Ways of Getting Similar Results 151

Solving More Complex Problems 154

The Binomial Distribution 155

Using the Binomial Distribution to Estimate the Observed Significance Level of a Test 159

Steps in a Statistical Test: Using the Binomial Distribution to Make Decisions About Hypotheses

Chi-Square: A Test Commonly Used for Nominal-Level Measures

Extending the Chi-Square Test to a Relationship Between Two Ordinal Variables: Identification with Fathers

The Normal Distribution and Its Application to Tests

171 The Problem: The Impact of Problem-Oriented Policing on Disorderly Activity at Violent-Crime Hot Spots 172

Assumptions: Laying the Foundations for Statistical Inference 174

Selecting a Sampling Distribution 180

Significance Level and Rejection Region 182

The Test Statistic 187

Making a Decision 187

197 Testing Hypotheses Concerning the Roll of a Die 198

Relating Two Nominal-Scale Measures in a Chi-Square Test 206

Extending the Chi-Square Test to Multicategory Variables: The Example of Cell Allocations in Prison 212

and Delinquent Acts 217

The Use of Chi-Square When Samples Are Small: A Final Note 222

of Statistical Significance 234 The Normal Frequency Distribution, or Normal Curve 235

Applying Normal Sampling Distributions to Nonnormal Populations 247

Comparing a Sample to an Unknown Population: The Single-Sample z-Test for Proportions 252

Comparing a Sample to an Unknown Population: The Single-Sample t-Test for Means 257

C h a p t e r s i x

C h a p t e r s e v e n

C h a p t e r e i g h t

C h a p t e r n i n e

C h a p t e r t e n

C O N T E N T Sviii

3

Distinguishing Statistical Significance and Strength of Relationship:

Measuring Association for Interval-Level Data: Pearson’s Correlation Coefficient

Testing the Statistical Significance of Pearson’s r

Testing the Statistical Significance of Spearman’s r

An Introduction to Bivariate Regression

Multivariate Regression

Comparing Means and Proportions in Two Samples 269 Comparing Sample Means 270

Comparing Sample Proportions: The Two-Sample t-Test for Differences of Proportions 282

The t-Test for Dependent Samples 288

A Note on Using the t-Test for Ordinal Scales 293

Comparing Means Among More Than Two Samples: Analysis of Variance Analysis of Variance 307

Defining the Strength of the Relationship Observed 328

Making Pairwise Comparisons Between the Groups Studied 331

A Nonparametric Alternative: The Kruskal-Wallis Test 334

Measures of Association for Nominal and Ordinal Variables 351

The Example of the Chi-Square Statistic 352

Measures of Association for Nominal Variables 355

Measures of Association for Ordinal-Level Variables 367

Choosing the Best Measure of Association for Nominal- and Ordinal-Level Variables 385

398 Measuring Association Between Two Interval-Level Variables 399

Pearson’s Correlation Coefficient 401

Spearman’s Correlation Coefficient 419

421

428

439 Estimating the Influence of One Variable on Another: The Regression Coefficient 440

Prediction in Regression: Building the Regression Line 445

Evaluating the Regression Model 453

The F-Test for the Overall Regression 467

481 The Importance of Correct Model Specifications 482

Correctly Specifying the Regression Model 494

C h a p t e r e l e v e n

C h a p t e r t w e l v e

C h a p t e r t h i r t e e n

C h a p t e r f o u r t e e n

C h a p t e r f i f t e e n

C h a p t e r s i x t e e n

C O N T E N T S ix

306

Logistic Regression

Multivariate Regression: Additional Topics Non-linear Relationships 516

Interaction Effects 522

An Example: Punishment Severity 533

An Example: Race and Punishment Severity 525

The Problem of Multicollinearity 534

548 Why Is It Inappropriate to Use OLS Regression for a Dichotomous Dependent Variable? 550

Logistic Regression 555

Interpreting Logistic Regression Coefficients 567

Comparing Logistic Regression Coefficients 577

Evaluating the Logistic Regression Model 583

Statistical Significance in Logistic Regression 587

C h a p t e r s e v e n t e e n

C h a p t e r e i g h t e e n

C O N T E N T Sx

514

C h a p t e r n i n e t e e n

Special Topics: Randomized Experiments 674

Sample Size, Equivalence, and Statistical Power 683

C h a p t e r t w e n t y o n e

Statistical Power

Examining Interaction Terms in Experimental Research 695

The Structure of a Randomized Experiment

The Main Advantage of Experiments: Isolating Causal Effects 677

Internal Validity 682

and Block Randomization 691

Using Covariates to Increase Statistical Power in Experimental Studies 693

C h a p t e r t w e n t y

Multilevel Regression Models 637 Variance Components Model 640

Random Intercept Model 646

Random Coefficient Model 655

Adding Cluster (Level 2) Characteristics 660

Special Topics: Confidence Intervals

Constructing Confidence Intervals

Confidence Intervals 704

C h a p t e r t w e n t y t w o

676

Multivariate Regression with Multiple Category Nominal or Ordinal Measures: Extending the Basic Logistic Regression Model 601

Multinomial Logistic Regression 603

Ordinal Logistic Regression 615

Substantive Example: Severity of Punishment Decisions 619

702

708

C O N T E N T S xi

C h a p t e r t w e n t y t h r e e

Critical Values of �2

Critical Value for P (Pcrit

Factorials 759

Distribution 760

Areas of the Standard Normal Distribution 761

Critical Values of Student’s t Distribution 762

Critical Values of the F-Statistic 763

), Tukey’s HSD Test 766

Critical Values for Spearman’s Rank-Order Correlation Coefficient 767

Fisher r-to-Z* Transformation 768

Glossary 770

Index 778

Appendix 1

Appendix 2

Appendix 3

Appendix 4

Appendix 5

Appendix 6

Appendix 7

Appendix 8

Special Topics: Statistical Power Statistical Power

Estimating Statistical Power and Sample Size for a Statistically Powerful Study 738

Summing Up: Avoiding Studies Designed for Failure 747

726 728

Components of Statistical Power 731

Preface

Oliver Wendell Holmes, the distinguished associate justice of the

Supreme Court, was noted for his forgetfulness. On a train leaving

Washington, D.C., he is said to have been approached by a

awkward moments, the conductor recognized the distinctive-

however, is said to have looked sternly at the conductor and

responded, “Young man, the problem is not where is my ticket;

the problem is where am I going.”

basic understanding of statistics in this field. In the first chapter, the main

themes of the text are outlined and discussed. This preface describes

how the text is organized.

The text takes a building-block approach. This means that each chap-

ter helps prepare you for the chapters that follow. It also means that the

level of sophistication of the text increases as the text progresses. Basic

concepts discussed in early chapters provide a foundation for the intro-

duction of more complex statistical issues later. One advantage to this

approach is that it is easy to see, as time goes on, how much you have

learned about statistics. Concepts that would have seemed impossible to

of the book now, you will see equations that are quite forbidding. How-

ever, when you come to these equations after covering the material in

earlier chapters, you will be surprised at how easy they are to under-

stand.

Throughout the text, there is an emphasis on comprehension and not

his case and his pockets, could not locate his pass. After a few

conductor who requested his ticket. Holmes, searching through

the rail company the ticket when he found it. Justice Holmes,

looking and well-known jurist and suggested that he just send

For the student of statistics, a textbook is like a train ticket. Not only does

it provide a pass the student can use for entering a new and useful area

of study; it also defines the route that will be taken and the goals that

are important to achieve. Different textbooks take different approaches

and emphasize different types of material. Statistics in Criminal Jus-

tice emphasizes the uses of statistics in research in crime and justice.

This text is meant for students and professionals who want to gain a

examine real-life criminal justice problems. In the opening chapters of the

sible but sophisticated understanding of statistics that can be used to

simple when you encounter them later on. If you turn to the final chapters

understand, had they been introduced at the outset, are surprisingly

xiii

computation. This approach is meant to provide readers with an acces-

book, basic themes and materials are presented. Chapter 1 provides an

introduction to how we use statistics in criminal justice and the problems

we face in applying statistics to real-life research problems. Chapters 2

through 5 introduce basic concepts of measurement and basic methods

build on the themes covered in these early chapters.

One of the fundamental problems researchers face is that they seek

to make statements about large populations (such as all U.S. citizens)

but are generally able to collect information or data on only a sample,

or smaller group, drawn from such populations. In Chapters 6 through

12, the focus is on how researchers use statistics to overcome this

special problems are encountered in criminal justice research, and how

should the researcher approach them? Some texts skip over the basics,

Having examined how we can make statements about populations

from information gained from samples, we turn to how we describe the

strength of association between variables. In the social sciences, it is

often essential not only to determine whether factors are related but also

to define the strength and character of those relationships. Accordingly,

in Chapters 13 and 14, we look at measures of association, and in Chap-

remember that the more advanced statistics presented in later chapters

Many of the statistics provided here will be familiar to you; however,

ing statements about populations based on samples? What are the

problem. What is the logic that underlies the statistics we use for mak-

for graphically representing data and using statistics to describe data.

different types of statistical procedures or tests that can be used? What

moving students from test to test before they understand the logic

behind the tests. The approach here is to focus in greater detail on

relatively simple statistical decisions before moving on to more com-

plex ones.

P R E F A C Exiv

In the concluding chapters, we look at three special topics. Chapter

estimates that you obtain from a sample. Because our emphasis is on

research in criminal justice, we conclude the text with a chapter that

examines methods for evaluating and improving the design of a research

project. The statistical concept that is central to Chapter 23—statistical

ters 15 through 20, we examine bivariate and different types of multi-

a method for assessing how much trust you can place in the specific

power—follows directly from the concepts developed in prior chapters.

it has become a central concern in criminal justice research and accord-

Statistical power is often ignored in introductory statistics texts. However,

While it is always difficult in statistics to decide where an introduc-

tory text should stop, with an understanding of these techniques you

variate regression. These are likely to be new topics for you, though they

are statistics commonly used in criminal justice.

criminal justice interventions. Chapter 22 describes confidence intervals,

21 focuses on the design of randomized experiments. Randomized experi-

ments allow criminal justice researchers to be confident about the causal

relationships between variables, and are often used in the evaluation of

in this text.ingly is given emphasis

point in learning statistics, but also to leave you with the confidence and

A working knowledge of computers is not required to understand

the statistical concepts or procedures presented in the text. However,

computers have become a very important part of research in statistics,

and thus we provide computer exercises for relevant chapters and a

tools to tackle more complex problems on your own. Each chapter starts

with a statement of the basic concepts and problems addressed and ends

with a full chapter summary. There is also a list of equations, when

relevant, at the end of the chapter. These materials should help you to

review what you have learned and to identify the basic knowledge you

need to move on to subsequent chapters.

All of the chapters contain a list of key terms with short definitions.

The key terms appear in boldface the first time they are mentioned in

the chapter. Sometimes a term may have been briefly explained in an

earlier chapter, but is designated as a key term in the chapter where

the concept is more central. A general glossary of key terms appears at

at the end. The questions are designed to make you think about the

subjects covered in the chapter. Sometimes they are straightforward,

following directly from the text. Sometimes they ask you to develop

ideas in slightly different ways than in the text. In constructing the ques-

tions, we sought to make working on statistical issues as much fun as

possible. In statistics, it is crucial to go over material more than once.

The questions are meant to reinforce the knowledge you have gained.

will have the basic tools to comprehend and conduct criminal justice

research. Of course, these tools constitute a building block for more

advanced methods. The goal of the text is not only to bring you to this

web site where you can access the data needed for those exercises

P R E F A C E xv

the end of the book. Chapters 2 through 23 each have a set of questions

Statistics in Criminal Justice will allow you to approach statistics in a

familiar context. It emphasizes the statistics and the problems that are

commonly encountered in criminal justice research. It focuses on under-

standing rather than computation. However, it takes a serious approach

to statistics, which is relevant to the real world of research in crime and

justice. The text is meant not only as an introduction for students but

also as a reference for researchers. The approach taken will help both

students who want an introduction to statistics and professionals who

seek a straightforward explanation for statistics that have become a

routine tool in contemporary criminal justice systems.

You are encouraged to use the web site. It will help you to see the

connection between the topics discussed in the chapters and statistical

computing.

(see the Computer Exercises at the end of Chapter 2 for details).

A c k n o w l e d g m e n t s

xvi

Professor Joseph Naus of the Department of Statistics of Rutgers Univer-

In the development of any academic enterprise, many students and

colleagues provide support and advice. We are particularly indebted to

of the new material for the third edition. A number of current and former

original edition of this book, and to Daniel Salem, a graduate of the Insti-

sity, who played a crucial advisory role in the preparation of the

Stephen Schnebly of the University of Illinios at Springfield, and Marc

work and provided comments over the previous editions, particularly the

tute of Criminology of the Hebrew University, who played a major role

of South Carolina was particularly helpful in helping us to refine some

in the production of the first edition. Robert Brame of the University

graduate students helped us in revising and editing the manuscript

across the four editions, including SueMing Yang and Kristen Miggans at

the University of Maryland, and Shomron Moyal, Rochelle Schnurr, Tal

Yonaton and Gali Weissman of the Hebrew University, and Arian Ferrer

We also want to thank the many scholars who have read and used our Emmanuelle Klossou, and Michael Rocque of Northeastern University.

comments and suggestions made by Todd Armstrong of Sam Houston

Swatt of Justice & Security Strategies, Inc. The final product reflects their

keen insights and thoughtful suggestions.

State University, Pamela Lattimore of the Research Triangle Institute,

xvii

A b o u t t h e A u t h o r s

David Weisburd is a Distinguished Professor of Criminology, Law and

Society at George Mason University (and Director of the Center for

Evidence Based Crime Policy) and Walter E. Meyer Professor of Law and

Criminal Justice at the Hebrew University. He also serves as a Senior

Fellow at the Police Foundation in Washington DC and is Chair of its

Research Advisory Committee. Professor Weisburd is an elected Fellow of

the American Society of Criminology and of the Academy of Experimental

Criminology. He is a member of the Science Advisory Board of the Office

of Justice Programs, the Steering Committee of the Campbell Crime and

Justice Group, the National Institute of Justice/Harvard Executive Session

in Policing, the Scientific Commission of the International Society of

Criminology, and the Committee on Law and Justice of the National

Research Council of the National Academy of Sciences (USA). Professor

Weisburd is one of the leading international researchers in crime and

justice. He is author or editor of more than twenty books and more than

100 scientific articles that cover a wide range of criminal justice research

topics, including crime at place, white collar crime, policing, illicit markets,

terrorism, criminal justice statistics and social deviance. Professor Weisburd

is the recipient of the 2010 Stockholm Prize in Criminology and the 2011

Klachky Family Prize for the Advancement of the Frontiers of Science.

Professor Weisburd is also the founding editor of the Journal of Experi-

mental Criminology.

Chester L. Britt is a researcher and scholar in criminology and criminal

from the University of Arizona, Professor Britt taught at the University

journals as Law and Society Review, Justice Quarterly, journal of Quanti-

and Journal of Research in Crime and Deliquency.

justice. He is Dean of the School of Criminology and Criminal Justice at

of Illinois, Penn State University, and Arizona State University. He served

Justice Quarterly from 2005 through 2007. He has also

co-edited a volume on criminological theory entitled Control Theories

of Crime and Delinquency. His research articles have appeared in such

Northeastern University in Boston. After receiving his Ph.D. in sociology

as the Editor of

tative Criminonology,

Introduction: Statistics as a Research Tool

Do Statisticians Have to Be Experts in Mathematics?

Are Computers Making Statisticians Redundant?

What Basic Principles Apply to Different Types of Statistics?

What Are the Different Uses of Statistics in Research?

C h a p t e r o n e

I n i t i a l h u r d l e s

K e y p r i n c i p l e s

What is Our Aim in Choosing a Statistic?

D. Weisburd and C. Britt, Statistics in Criminal Justice, DOI 10.1007/978-1-4614-9170-5_1,

© Springer Science+Business Media New York 2014

tool for answering questions. It allows us to take large bodies of infor-

mation and summarize them with a few simple statements. It lets us

come to solid conclusions even when the realities of the research world

make it difficult to isolate the problems we seek to study. Without statis-

tics, conducting research about crime and justice would be virtually im-

possible. Yet, there is perhaps no other subject in their university studies

that criminal justice students find so difficult to approach.

A good part of the difficulty lies in the links students make between

statistics and math. A course in statistics is often thought to mean long

hours spent solving equations. In developing your understanding of sta-

tistics in criminal justice research, you will come to better understand the

formulas that underlie statistical methods, but the focus will be on con-

cepts and not on computations. There is just no way to develop a good

understanding of statistics without doing some work by hand. But in the

age of computers, the main purpose of doing computations is to gain a

deeper understanding of how statistics work.

Researchers no longer spend long hours calculating statistics. In the

1950s, social scientists would work for months developing results that

can now be generated on a computer in a few minutes. Today, you do

not need to be a whiz kid in math to carry out a complex statistical

analysis. Such analyses can be done with user-friendly computer pro-

grams. Why then do you need a course in statistics? Why not just leave it

to the computer to provide answers? Why do you still need to learn the

basics?

The computer is a powerful tool and has made statistics accessible to

a much larger group of criminal justice students and researchers. How-

ever, the best researchers still spend many hours on statistical analysis.

Now that the computer has freed us from long and tedious calculations,

what is left is the most challenging and important part of statistical analy-

sis: identifying the statistical tools that will best serve researchers in inter-

preting their research for others.

2

THE PURPOSE OF STATISTICAL ANALYSIS is to clarify and not confuse. It is a

T H E P U R P O S E O F S T A T I S T I C S I S T O 3

need to choose statistics for research and interpret them. It is meant for

students of criminology and criminal justice. As in other fields, there are

specific techniques that are commonly used and specific approaches that

have been developed over time by researchers who specialize in this

area of study. These are the focus of this text. Not only do we draw our

examples from crime and justice issues; we also pay particular attention

to the choices that criminal justice researchers make when approaching

statistical problems.

Before we begin our study of statistics in criminal justice, it is useful

to state some basic principles that underlie the approach taken in this

text. They revolve around four basic questions. First, what should our

purpose be in choosing a statistic? Second, why do we use statistics to

answer research questions? Third, what basic principles apply across

very different types of statistics? And finally, what are the different uses

for statistics in research?

guage. In this sense, statistics provide a way for the initiated to share

ideas and concepts without including the rest of us. Of course, it is nec- essary to use a common language to report research results. This is one

reason why it is important for you to take a course in statistics. But the

reason we use statistics is to make research results easier—not more dif-

would be just to tell us about your subjects. You could describe each

offender and his or her criminal history without creating any real

confusion. But what if you wanted to report on 20 offenders? It would

take quite a long time to tell us about each one in some detail, and it

is likely that we would have difficulty remembering who was who.

It would be even more difficult to describe 100 offenders. With

thousands of offenders, it would be just about impossible to take this

approach.

This is one example of how statistics can help to simplify and clarify

the research process. Statistics allow you to use a few summary state- ments to provide a comprehensive portrait of a large group of offenders.

For example, instead of providing the name of each offender and telling

us how many crimes he or she committed, you could present a single

ficult—to understand. For example, if you wanted to provide a description

of three offenders you had studied, you would not need to search for statis-

tics to summarize your results. The simplest way to describe your sample

The goal of this text is to provide you with the basic skills you will

It sometimes seems as if researchers use statistics as a kind of secret lan-

C L A R I F Y

T h e P u r p o s e o f S t a t i s t i c s I s t o C l a r i f y

4 C H A P T E R O N E : I N T R O D U C T I O N

statistic that described the average number of crimes committed by the

people you studied. You might say that, on average, the people you

studied committed two to three crimes in the last year. Thus, although it

might be impossible to describe each person you studied, you could, by

using a statistic, give your audience an overall picture. Statistics make it

possible to summarize information about a large number of subjects with

a few simple statements.

Given that statistics should simplify the description of research results,

it follows that the researcher should utilize the simplest statistics appro-

priate for answering the research questions that he or she raises.

Nonetheless, it sometimes seems as if researchers go out of their way to

identify statistics that few people recognize and even fewer understand.

This approach does not help the researcher or his or her audience.

There is no benefit in using statistics that are not understood by those in-

terested in your research findings. Using a more complex statistic when a

simpler one is appropriate serves no purpose beyond reducing the num-

ber of people who will be influenced by your work.

The best presentation of research findings will communicate results in

a clear and understandable way. When using complex statistics, the re-

searcher should present them in as straightforward a manner as possible.

The mark of good statisticians is not that they can mystify their audi-

ences, but rather that they can communicate even complex results in a

way that most people can understand.

S t a t i s t i c s A r e U s e d t o S o l v e P r o b l e m s

Statistics develop because of a need to deal with a specific type of ques-

tion or problem. In the example above, you were faced with the

dilemma that you could not describe each person in a very large study

without creating a good deal of confusion. We suggested that an average

might provide a way of using one simple statistic to summarize a charac-

teristic of all the people studied. The average is a statistical solution. It is

a tool for solving the problem of how to describe many subjects with a

short and simple statement.

As you will see in later chapters, statistics have been developed to

deal with many different types of problems that researchers face. Some

of these may seem difficult to understand at the outset, and indeed it is

natural to be put off by the complexities of some statistics. However, the

solutions that statisticians develop are usually based on simple common

sense. Contrary to what many people believe, statistics follow a logic

that you will find quite easy to follow. Once you learn to trust your com-

mon sense, learning statistics will turn out to be surprisingly simple. Indeed,

5

encounter in this text. Stating them at the outset will help you to see

how statistical procedures in later chapters are linked one to another. To

understand these principles, you do not need to develop any computa-

tions or formulas; rather, you need to think generally about what we are

trying to achieve when we develop statistics.

The first is simply that in developing statistics we seek to reduce the

level of error as much as possible. The purpose of research is to provide

answers to research questions. In developing these answers, we want to

be as accurate as we can. Clearly, we want to make as few mistakes as

possible. The best statistic is one that provides the most accurate state-

ment about your study. Accordingly, a major criterion in choosing which

statistic to use—or indeed in defining how a statistic is developed—is the

amount of error that a statistic incorporates. In statistics, we try to mini-

mize error whenever possible.

Unfortunately, it is virtually impossible to develop any description

without some degree of error. This fact is part of everyday reality. For

example, we do not expect that our watches will tell perfect time or that

our thermostats will be exactly correct. At the same time, we all know

that there are better watches and thermostats and that one of the factors

that leads us to define them as “better” is that they provide information

with less error. Similarly, although we do not expect our stockbroker to

possible. Fear of statistics is a greater barrier to learning than any of the com-

putations or formulas that we will use. It is difficult to learn anything when

you approach it with great foreboding. Statistics is a lot easier than you think.

The job of this text is to take you step by step through the principles and

ideas that underlie basic statistics for criminal justice researchers. At the

beginning, we will spend a good deal of time examining the logic behind

statistics and illustrating how and why statisticians choose a particular

that the solutions statisticians use make very good sense.

solution to a particular statistical problem. What you must do at the outset

is take a deep breath and give statistics a chance. Once you do, you will find

our experience is that students who have good common sense, even

if they have very little formal background in this area, tend to become

A few basic principles underlie much of the statistical reasoning you will

be correct all of the time, we are likely to choose the broker who we

believe will make the fewest mistakes.

the best criminal justice statisticians. But in order to be able to use com-

B A S I C P R I N C I P L E S A P P L Y A C R O S S S T A T I S T I C A L T E C H N I Q U E S

mon sense, it is important to approach statistics with as little fear as

B a s i c P r i n c i p l e s A p p l y A c r o s s S t a t i s t i c a l T e c h n i q u e s

6 C H A P T E R O N E : I N T R O D U C T I O N

In choosing a statistic, we also use a second principle to which we

will return again and again in this text: Statistics based on more informa-

tion are generally preferred over those based on less information. This

principle is common to all forms of intelligence gathering and not just

those that we use in research. Good decision making is based on infor-

mation. The more information available to the decision maker, the better

he or she can weigh the different options that are presented. The same

goes for statistics. A statistic that is based on more information, all else

being equal, will be preferred over one that utilizes less information.

There are exceptions to this rule, often resulting from the quality or form

of the information or data collected. We will discuss these in detail in the

text. But as a rule, the best statistic utilizes the maximum amount of

information.

Our third principle relates to a danger that confronts us in using statis-

tics as a tool for describing information. In many studies, there are cases

that are very different from all of the others. Indeed, they are so different

that they might be termed deviant cases or, as statisticians sometimes call

them, “outliers.” For example, in a study of criminal careers, there may

be one or two offenders who have committed thousands of crimes,

whereas the next most active criminal in the sample has committed only

choice of statistics and your presentation of results.

In almost every statistic we will study, outliers present a distinct and

troublesome problem. A deviant case can make it look as if your offend-

ers are younger or older than they really are—or less or more criminally

active than they really are. Importantly, deviant cases often have the

most dramatic effects on more complex statistical procedures. And it is

precisely here, where the researcher is often preoccupied with other sta-

tistical issues, that deviant cases go unnoticed. But whatever statistic is

used, the principle remains the same: Outliers present a significant prob-

lem in choosing and interpreting statistics.

The final principle is one that is often unstated in statistics, because it

is assumed at the outset: Whatever the method of research, the researcher

must strive to systematize the procedures used in data collection and

analysis. As Albert J. Reiss, Jr., a pioneer in criminal justice methodolo-

gies, has noted, “systematic” means in part “that observation and record-

ing are done according to explicit procedures which permit replication

and that rules are followed which permit the use of scientific inference.”1

1A. J. Reiss, Jr., “Systematic Social Observation of Social Phenomenon,” in Herbert

Costner (ed.), Sociological Methodology (San Francisco: Jossey Bass, 1971), pp. 3–33.

a few hundred crimes. Although such cases form a natural part of the

research process, they often have very significant implications for your

T H E U S E S O F S T A T I S T I C S 7

While Reiss’s comment will become clearer as statistical concepts are

defined in coming chapters, his point is simply that you must follow

clearly stated procedures and rules in developing and presenting statisti-

follow a consistent logic from start to finish. You should not jump from

statistic to statistic merely because the outcomes are favorable to the the-

sis you raise. In learning about statistics, it is also important to go step by

step—and to be well organized and prepared. You cannot learn statistics

by cramming in the last week of classes. The key to learning statistics is

to adopt a systematic process and follow it each week.

Statistical procedures are built on all of the research steps that pre-

cede them. If these steps are faulty, then the statistics themselves will be

faulty. In later chapters, we often talk about this process in terms of the

assumptions of the statistics that we use. We assume that all of the rules

of good research have been followed up to the point where we decide

on a statistic and calculate it. Statistics cannot be disentangled from the

larger research process that comes before them. The numbers that we

systematic approach is crucial not only to the statistical procedures that

you will learn about in this text but to the whole research process.

three ways in which statistics are used in criminal justice. The first is

called descriptive statistics, because it helps in the summary and de-

scription of research findings. The second, inferential or inductive sta-

tistics, allows us to make inferences or statements about large groups

from studies of smaller groups, or samples, drawn from them. Finally,

we introduce multivariate statistics toward the end of the text. Multi-

variate statistics, as the name implies, allow us to examine a series of

variables at one time.

Descriptive Statistics

We are all familiar in some way with descriptive statistics. We use them

often in our daily lives, and they appear routinely in newspapers and on

television. Indeed, we use them so often that we sometimes don’t think

cal findings. It is important to approach statistics in a systematic way. You

cannot be sloppy or haphazard, at least if the statistic is to provide a good

answer to the research question you raise. The choice of a statistic should

In the chapters that follow, we will examine three types of statistics or

employed. Very complex statistics cannot hide bad research methods. A

use are only as good as the data collection techniques that we have

T h e U s e s o f S t a t i s t i c s

8 C H A P T E R O N E : I N T R O D U C T I O N

of them as statistics at all. During an election year, everyone is con-

cerned about the percentage support that each candidate gains in the

primaries. Students at the beginning of the semester want to know what

proportion of their grades will be based on weekly exercises. In deciding

whether our salaries are fair, we want to know what the average salary is

for other people in similar positions. These are all descriptive statistics.

They summarize in one simple statement the characteristics of many

people. As discussed above in the example concerning criminal histories,

descriptive statistics make it possible for us to summarize or describe

large amounts of information.

In the chapters that follow, we will be concerned with two types of

descriptive statistics: measures of central tendency and measures of

dispersion. Measures of central tendency are measures of typicality.

They tell us in one statement what the average case is like. If we could

take only one person as the best example for all of the subjects we stud-

ied, who would it be? If we could choose only one level of criminal ac-

tivity to typify the frequency of offending of all subjects, what level

would provide the best snapshot? If we wanted to give our audience a

general sense of how much, on average, a group of offenders stole in a

year, what amount would provide the best portrait? Percentages, propor-

tions, and means are all examples of measures of central tendency that

we commonly use. In the coming chapters, you will learn more about

these statistics, as well as more complex measures with which you may

not be familiar, such as correlation and regression coefficients.

Having a statistic that describes the average case is very helpful in de-

scribing research results. However, we might also want to know how

typical this average case is of the subjects in our study. The answer to

this question is provided by measures of dispersion. They tell us to what

extent the other subjects we studied are similar to the case or statistic we

have chosen to represent them. Although we don’t commonly use mea-

sures of dispersion in our daily lives, we do often ask similar questions

without the use of such statistics.

For example, in deciding whether our income is fair, we might want

to know not only the average income of others in similar positions, but

also the range of incomes that such people have. If the range was very

small, we would probably decide that the average provides a fairly good

portrait of what we should be making. If the range was very large, we

might want to investigate more carefully why some people make so

much more or less than the average. The range is a measure of disper-

sion. It tells us about the spread of scores around our statistic. In the

chapters that follow, we will look at other measures of dispersion—for

example, the standard deviation and variance, which may be less famil-

iar to you. Without these measures, our presentation of research findings

T H E U S E S O F S T A T I S T I C S 9

would be incomplete. It is not enough simply to describe the typical

case; we must also describe to what degree other cases in our study are

different from or similar to it.

Inferential Statistics

Inferential statistics allow us to make statements about a population, or

the larger group of people we seek to study, on the basis of a sample

drawn from that population. Without this very important and powerful

tool, it would be very difficult to conduct research in criminal justice.

The reason is simple. When we conduct research, we do so to answer

questions about populations. But in reality we seldom are able to collect

information on the whole population, so we draw a sample from it. Sta-

tistical inference makes it possible for us to infer characteristics from that

sample to the population.

Why is it that we draw samples if we are really interested in making

statements about populations? In good part it is because gaining informa-

tion on most populations is impractical and/or too expensive. For exam-

ple, if we seek to examine the attitudes of U.S. citizens toward criminal

justice processing, we are interested in how all citizens feel. However,

studying all citizens would be a task of gigantic proportion and would

cost billions of dollars. Such surveys are done every few years and are

called censuses. The last census in the United States took many years to

prepare and implement and cost over $5 billion to complete. If every re-

search study of the American population demanded a census, then we

would likely cost millions of dollars to complete a simple study of their

attitudes. This is because the most inexpensive data collection can still

cost tens of dollars for each subject studied. When you consider that the

National Institute of Justice, the primary funder of criminal justice re-

search in the United States, provides a total of about $100 million a year

for all research, it is clear that criminal justice research cannot rely on

studies of whole populations.

It is easy to understand, then, why we want to draw a sample or sub-

set of the larger population to study, but it is not obvious why we should

believe that what we learn from that sample applies to the population

from which it is drawn. How do we know, for example, that the atti-

tudes toward criminal justice expressed by a sample of U.S. citizens are

similar to the attitudes of all citizens? The sample is a group of people

would have very few research projects indeed. Even when we are interested

in much smaller populations in the criminal justice system, examination of

the entire population is often beyond the resources of the criminal justice

researcher. For example, to study all U.S. prisoners, we would have to

study over 1 million people.

Even if we wanted to look at the 100,000 or so women prisoners, it

10 C H A P T E R O N E : I N T R O D U C T I O N

drawn from the population; it is not the population itself. How much

surveys that now form so much a part of public life or the studies that

you read about in your other college classes. When a news organization

sions about populations—whether of offenders, criminal justice agents,

crime-prone places, or criminal justice events—on samples. Statistical in-

ference provides a method for deciding to what extent you can have

faith in such results. It allows you to decide when the outcome observed

in a sample can be generalized to the population from which it was

drawn. Statistical inference is a very important part of statistics and one

we will spend a good deal of time discussing in this text.

Taking into Account Competing Explanations: Multivariate Statistics

Multivariate statistics allow us to solve a different type of problem in re-

search. It is often the case that the issue on which we want to focus is

confounded by other factors in our study. Multivariate statistics allow us

to isolate one factor while taking into account a host of others. For ex-

ample, a number of criminal justice studies examine the impact of im-

prisonment on the future criminal behavior of offenders. In general, they

compare offenders who are found guilty in court and sentenced to

once they are released into the community, is different from that of non-

prisoners. Researchers conducting these studies face a very difficult re-

search problem. Prisoners and nonprisoners are often very different

types of people, and some of these differences are likely to affect their

impact of imprisonment on future offending. If we discover that prison-

ers, once released into the community, are more likely than nonprison-

ers to commit a crime, how can we tell whether this was a result of the

can we rely on such estimates? And to what extent can we trust such statis-

tics? You have probably raised such issues already, in regard to either the

conducts a survey of 1,000 people to tell us how all voters will vote in the

next election, it is using a sample to make statements about a population.

The criminal justice studies you read about also base their conclu-

prison with those who are found guilty but do not receive a prison sanction.

Such studies focus on whether the criminal behavior of prisoners,

criminal behavior in the community. For example, prisoners are more likely

than nonprisoners to have been arrested before, since a prior arrest is often an important factor in the judge’s decision to incarcerate a convicted off-

ender in the first place. And we know from research about criminal careers

that people with a prior history of arrest are much more likely than people

without such a history to commit a crime in the future. Accordingly, prisoners

are more likely to commit a crime in the future, irrespective of the fact that

they have served a prison sentence. This makes it very difficult to assess the

C H A P T E R S U M M A R Y 11

experience of imprisonment? It might be due to the simple fact that pris-

oners are more likely than nonprisoners to commit crimes in the first

place. Their more serious arrest histories would predict this result.

The complex task facing the criminal justice researcher is to isolate

the specific impact of imprisonment itself from all of the other possible

explanations for differences in reoffending between prisoners and non-

prisoners. Multivariate analysis provides a statistical solution to this prob-

lem. It allows the criminal justice researcher to isolate the impact of one

factor—in this case, imprisonment—from those of other factors, such as

prior criminal history, that might confound the researcher’s conclusions.

C h a p t e r S u m m a r y

Statistics seem intimidating because they are associated with complex

mathematical formulas and computations. Although some knowledge of

math is required, an understanding of the concepts is much more im-

portant than an in-depth understanding of the computations. Today’s

computers, which can perform complex calculations in a matter of sec-

onds or fractions of seconds, have drastically cut the workload of the

researcher. They cannot, however, replace the key role a researcher

plays in choosing the most appropriate statistical tool for each research

problem.

The researcher’s aim in using statistics is to communicate findings in a

clear and simple form. As a result, the researcher should always choose

the simplest statistic appropriate for answering the research question.

ing principles apply to all types of statistics: (1) In developing statistics,

we seek to reduce the level of error as much as possible. (2) Statistics

based on more information are generally preferred over those based on

less information. (3) Outliers present a significant problem in choosing

and interpreting statistics. (4) The researcher must strive to systematize

the procedures used in data collection and analysis.

There are three principal uses of statistics discussed in this book. In

descriptive statistics, the researcher summarizes large amounts of in-

formation in an efficient manner. Two types of descriptive statistics that

go hand in hand are measures of central tendency, which describe

the characteristics of the average case, and measures of dispersion,

which tell us just how typical this average case is. We use inferential

statistics to make statements about a population on the basis of a sam-

ple drawn from that population. Finally, in multivariate statistics, we

isolate the impact of one factor from others that may distort our results.

Statistics offer commonsense solutions to research problems. The follow-

12 C H A P T E R O N E : I N T R O D U C T I O N

K e y T e r m s

descriptive statistics A broad area of sta-

tistics that is concerned with summarizing

large amounts of information in an efficient

manner. Descriptive statistics are used to

describe or represent in summary form the

characteristics of a sample or population.

inferential, or inductive, statistics A

broad area of statistics that provides the re-

searcher with tools for making statements

about populations on the basis of knowl-

edge about samples. Inferential statistics

allow the researcher to make inferences re-

garding populations from information

gained in samples.

measures of central tendency Descrip-

tive statistics that allow us to identify the

typical case in a sample or population.

Measures of central tendency are measures

of typicality.

measures of dispersion Descriptive sta-

tistics that tell us how tightly clustered or

dispersed the cases in a sample or popula-

tion are. They answer the question “How

typical is the typical case?”

multivariate statistics Statistics that

examine the relationships among vari-

ables while taking into account the possi-

variable from others that may distort his

or her results.

ble influences of other confounding factors.

Multivariate statistics allow the resear-

cher to isolate the impact of one

Measurement: The Basic Building Block

of Research

How Do Criminal Justice Researchers Develop Knowledge?

How Do the Different Scales Interconnect?

C h a p t e r t w o

C r i m i n a l j u s t i c e r e s e a r c h a s a s c i e n c e

T h e f o u r s c a l e s o f m e a s u r e m e n t

D e f i n i n g a g o o d m e a s u r e

What are They?

What are Their Characteristics?

Which is the Most Appropriate Scale of Measurement?

What is Meant by the “Validity” of a Measure?

What is Meant by the “Reliability” of a Measure?

D. Weisburd and C. Britt, Statistics in Criminal Justice, DOI 10.1007/978-1-4614-9170-5_2,

© Springer Science+Business Media New York 2014

MEASUREMENT LIES AT THE HEART of statistics. Indeed, no statistic would

be possible without the concept of measurement. Measurement is also an

integral part of our everyday lives. We routinely classify and assign values

to people and objects without giving much thought to the processes that

underlie our decisions and evaluations. In statistics, such classification and

ordering of values must be done in a systematic way. There are clear rules

for developing different types of measures and defined criteria for decid-

ing which are most appropriate for answering a specific research question.

Although it is natural to focus on the end products of research, it is im-

portant for the researcher to remember that measurement forms the first

building block of every statistic. Even the most complex statistics, with

numbers that are defined to many decimal places, are only as accurate as

the measures upon which they are built. Accordingly, the relatively simple

rules we discuss in this chapter are crucial for developing solid research

findings. A researcher can build a very complex structure of analysis. But if

the measures that form the foundation of the research are not appropriate

for the analyses that are conducted, the findings cannot be relied upon.

We begin Chapter 2 by examining the basic idea of measurement in sci-

ence. We then turn to a description of the main types of measures in statis-

tics and the criteria used to distinguish among them. We are particularly

concerned with how statisticians rank measurement based on the amount

of information that a measure includes. This concept, known as levels of

measurement, is very important in choosing which statistical procedures are

appropriate in research. Finally, we discuss some basic criteria for defining

a good measure.

S c i e n c e a n d M e a s u r e m e n t : C l a s s i f i c a t i o n a s a F i r s t S t e p i n R e s e a r c h

Criminal justice research is a scientific enterprise that seeks to develop

knowledge about the nature of crimes, criminals, and the criminal justice

14

L E V E L S O F M E A S U R E M E N T 15

system. The development of knowledge can, of course, be carried out in

a number of different ways. Criminal justice researchers may, for exam-

ple, observe the actions of criminal justice agents or speak to offenders.

They may examine routine information collected by government or crim-

inal justice agencies or develop new information through analyses of the

content of records in the criminal justice system. Knowledge may be de-

veloped through historical review or even through examination of ar-

chaeological records of legal systems or sanctions of ancient civilizations.

The methods that criminal justice researchers use vary. What they

have in common is an underlying philosophy about how knowledge

may be gained and what scientific research can tell us. This philosophy,

which is predominant in scientific study in the modern world, is usually

called positivism.1 At its core is the idea that science is based on facts

and not values. Science cannot make decisions about the way the world

should be (although scientific observation may inform such decisions).

Rather, it allows us to examine and investigate the realities of the world

as we know it. The major tool for defining this reality in science is mea-

surement.

Measurement in science begins with the activity of distinguishing

groups or phenomena from one another. This process, which is gener-

ally termed classification, implies that we can place units of scientific

study—such as victims, offenders, crimes, or crime places—in clearly de-

fined categories. The classification process leads to the creation of vari-

ables. A variable is a trait, characteristic, or attribute that can be mea-

sured. What differentiates measurement in science from measurement in

our everyday lives is that there must be systematic criteria for determin-

ing both what each category of a variable represents and the boundaries

between categories. We now turn to a discussion of these criteria as they

relate to different levels of measurement.

L e v e l s o f M e a s u r e m e n t

Classification forms the first step in measurement. There are a number of

different ways we can classify the people, places, or phenomena we

wish to study. We may be content to simply distinguish one category

from another. But we may also be interested in how those categories re-

late to one another. Do some represent more serious crime or less seri-

ous crime? Can we rank how serious various crimes are in a clear and

1See D. Black, “The Boundaries of Legal Sociology,” in D. Black and M. Mileski (eds.),

The Social Organization of Law (New York: Seminar Press, 1973), pp. 41–47.

16 C H A P T E R T W O : M E A S U R E M E N T

defined order? Is it possible to define exactly how serious one crime is

relative to another?

As these types of questions suggest, measurement can be a lot more

complex than simply distinguishing one group from another. Recogniz-

ing this complexity, statisticians have defined four basic groups of mea-

sures, or scales of measurement, based on the amount of information

that each takes advantage of. The four are generally seen as occupying

different positions, or levels, on a ladder of measurement (see Figure

2.1). Following a principle stated in Chapter 1—that statistics based on

more information are generally preferred—measures that include more

information rank higher on the ladder of measurement.

Nominal Scales

At the bottom of the ladder of measurement are nominal scales. Nominal-

scale variables simply distinguish one phenomenon from another. Sup-

pose, for example, that you want to measure crime types. In your study,

you are most interested in distinguishing between violent crime and

other types of crime. To fulfill the requirements of a nominal scale, and

thus the minimum requirements of measurement, you need to be able to

take all of the crime events in your study and place them in one of two

categories: either violent crime or other crime. There can be no overlap.

In practice, you might come across many individual events that seem dif-

ficult to classify. For example, what would you do with a crime event in

which the offender first stole from his victim and then assaulted him?

This event includes elements of both violent and property crime. What

about the case where the offender did not assault the victim, but merely

Ratio

Interval

Ordinal

Nominal Categorization

Order + Categorization

True Zero + Set Intervals + Order + Categorization

Set Intervals + Order + Categorization

Ladder of MeasurementFigure 2.1

threatened her? Would you decide to include this in the category of

violent crime or other crime?

L E V E L S O F M E A S U R E M E N T 17

In criminology and criminal justice, we often make use of nominal-

scale variables. Many of these reflect simple dichotomies, like the distinc-

tion between violent and other crime. For example, criminologists often

seek to examine differences between men and women in their involve-

ment in criminality or treatment in the criminal justice system. It is com-

mon as well to distinguish between those who are sentenced to prison

and those who are not or those who commit more than one crime (“re-

cidivists”) and those who are only one-shot offenders.

It is often necessary to distinguish among multiple categories of a

nominal-level variable. For example, if you wanted to describe legal rep-

resentation in court cases, you would provide a very simplistic picture if

you simply distinguished between those who had some type of legal

representation and those who did not. Some of the offenders would be

likely to have private attorneys and others court-appointed legal repre-

sentation. Still others might gain help from a legal aid organization or a

public defender. In order to provide a full portrait of legal representa-

tion, you would likely want to create a nominal-scale variable with five

distinct categories: No attorney, Legal aid, Court appointed, Public de-

fender, and Private attorney. Table 2.1 presents a number of examples of

nominal-level scales commonly used in criminal justice.

Nominal-scale measures can include any number of different cate-

gories. The Uniform Crime Reporting system, which keeps track of ar-

rests in the United States, includes some 29 categories of crime. These

Nominal-Scale Variables Commonly Found in Criminal Justice Research

VARIABLE COMMON CATEGORIES

Gender Male, Female Race-Ethnicity Non-Hispanic Black, Non-Hispanic White, Hispanic (any race) Marital Status Single, Married, Separated, Divorced, Widowed Pretrial Release Status Detained, Released Type of Case Disposition Dismissed, Acquitted, Diverted, Convicted Method of Conviction Negotiated guilty plea, Nonnegotiated guilty plea, Bench trial, Jury trial Type of Punishment Incarceration, Nonincarceration

Table 2.1

In measurement, you must make systematic choices that can be applied

across events. You cannot decide one way for one event and another

way for another. In the situation described above, you might conclude

that the major issue in your study was the presence of violence. Thus,

all cases with any violent events would be placed in the violent cate-

gory. Similarly, you might conclude that violence had to include physical

victimization. Whatever your choice, to meet the requirements of mea-

surement you must define clearly where all events in your study are to

be placed.

18 C H A P T E R T W O : M E A S U R E M E N T

range from violent crimes, such as murder or robbery, to vagrancy and

vandalism. Although there is no statistical difficulty with defining many

categories, the more categories you include, the more confusing the de-

scription of the results is likely to be. If you are trying to provide a sense

of the distribution of crime in your study, it is very difficult to practically

describe 20 or 30 different crime categories. Keeping in mind that the

purpose of statistics is to clarify and simplify, you should try to use the

smallest number of categories that will accurately describe the research

problem you are examining.

At the same time, do not confuse collection of data with presentation

of your findings. You do not lose anything by collecting information in

the most detailed way that you can. If you collect information with a

large number of categories, you can always collapse a group of cate-

gories into one. For example, if you collect information on arrest events

utilizing the very detailed categories of the criminal law, you can always

combine them later into more general categories. But if you collect infor-

mation in more general categories (for example, just violent crime and

property crime), you cannot identify specific crimes such as robbery or

car theft without returning to the original source of your information.

Though nominal-scale variables are commonly used in criminology

and criminal justice, they provide us with very limited knowledge about

the phenomenon we are studying. As you will see in later chapters, they

also limit the types of statistical analyses that the researcher can employ.

In the hierarchy of measurement, nominal-scale variables form the low-

est step in the ladder. One step above are ordinal scales.

Ordinal Scales

What distinguishes an ordinal from a nominal scale is the fact that we as-

sign a clear order to the categories included. Now not only can we dis-

tinguish between one category and another; we also can place these

categories on a continuum. This is a very important new piece of infor-

mation; it allows us to rank events and not just categorize them. In the

case of crime, we might decide to rank in order of seriousness. In mea-

suring crime in this way, we would not only distinguish among cate-

gories, such as violent, property, and victimless crimes; we might also

argue that violent crimes are more serious than property crimes and that

victimless crimes are less serious than both violent and property crimes.

We need not make such decisions arbitrarily. We could rank crimes by

the amount of damage done or the ways in which the general popula-

tion rates or evaluates different types of crime.

Ordinal-scale variables are also commonly used in criminal justice

and criminology. Indeed, many important criminal justice concepts are

measured in this way. For example, in a well-known London survey of

L E V E L S O F M E A S U R E M E N T 19

2

Ranking crime by seriousness and measuring people’s fear of crime

tences, damage to victims, complexity of crime, or seriousness of prior

records of offenders, as illustrated in Table 2.2. What all of these vari-

ables have in common is that they classify events and order them along

a continuum. What is missing is a precise statement about how various

categories differ one from another.

Interval and Ratio Scales

Interval scales not only classify and order people or events; they also

define the exact differences between them. An interval scale requires

that the intervals measured be equal for all of the categories of the scale

examined. Thus, an interval-scale measure of prior record would not

simply rank prior record by seriousness; it would allow us to say how

much more serious one offender’s record was than another’s in a stan-

dard unit of measurement—for example, number of arrests, convictions,

or prison stays.

2See R. Sparks, H. Genn, and D. Dodd, Surveying Victims: A Study of the Measurement

of Criminal Victimization (New York: Wiley, 1977).

Ordinal Scale Variables Commonly Found in Criminal Justice Research

VARIABLE COMMON CATEGORIES

Level of Education Less than high school, Some high school, High school graduation, Some college or trade school, College graduate, Graduate/ professional school

Severity of Injury in an Assault None, Minor—no medical attention, Minor—medical attention required, Major—medical attention required with no hospitalization, Major—medical attention required with hospitalization

Attitude and Opinion Survey Strongly disagree, Disagree, No opinion, Agree, Strongly agree; Questions Very high, High, Moderate, Low, Very low

Bail-Release Decision Released on own recognizance, Released on bail, Detained— unable to post bail, Denied release

Type of Punishment Probation/community service, Jail incarceration, Prison incarceration, Death sentence

Table 2.2

victimization, fear of crime was measured using a simple four-level

ordinal scale. Researchers asked respondents: “Are you personally con-

cerned about crime in London as a whole? Would you say you are (1)

very concerned, (2) quite concerned, (3) a little concerned, or (4) not

concerned at all?”

are only two examples of the use of ordinal scales in criminal justice

research. We could also draw examples regarding severity of court sen-

20 C H A P T E R T W O : M E A S U R E M E N T

Most criminal justice variables that meet the criteria of an interval scale

also meet the criteria of a ratio scale. A ratio scale has all of the charac-

teristics of an interval scale but also requires that there be a non-arbitrary,

or true, zero value. This means simply that zero represents the absence

that the former has 15 more arrests than the latter. We have an important

piece of information that we would not have gained with an ordinal

scale. Now, not only can we say that the prior record of one offender is

more serious than that of another, but we can specify exactly how many

more arrests the offender has. This variable thus meets the requirements

of an interval scale. But it also meets the additional requirement of a

ratio scale that there be a true zero value, since we can state that some-

one with 20 arrests has 4 times as many arrests as someone with 5 ar-

rests. If the zero value were arbitrary, we could not make this statement.

This fact is best illustrated with an example. Suppose we alter our

measure of prior record to focus on the degree to which offenders ex-

ceed a specific threshold of prior offending. Let’s say that our threshold

is 4 prior arrests and we are interested only in offenders who have 4 or

more prior arrests. An offender with 5 arrests would gain a score of 1 on

this new measure, and an offender with 20 arrests would have a score of

16. An offender with 4 arrests would have a score of 0. This variable

meets the criteria of an interval scale because we can distinguish scores,

rank them, and define the exact difference between them. A score of 16

represents a more serious prior criminal record than a score of 1. In turn,

an offender with a score of 16 has 15 more arrests than an offender with

a score of 1. However, we cannot say that the offender with a score of

16 on this measure had 16 times as many prior arrests as the offender

with a score of 1. This is because the scale has an arbitrary zero point.

Zero represents not the absence of a prior record, but the fact that the

offender has 4 prior arrests. Thus, the scale is an interval scale but not a

ratio scale.

Nearly all the statistics that we use in criminal justice (and all those

that we describe in this text) are also appropriate for interval scales if

they are appropriate for ratio scales. For this reason, most statistics texts

do not differentiate between the scales in practice, even if they identify

how they differ in theory. We follow the same approach. For the rest of

the chapter and indeed the rest of this text, we will concentrate on the

differences among nominal, ordinal, and at least interval scales.

Criminal justice researchers use interval scales to present findings

about criminal justice agency resources, criminal sentences, and a whole

of the trait under study. To understand how interval scales differ from

ordinal scales and from ratio scales, it is useful to examine a concrete

example. We commonly measure prior offending in terms of the number

of arrests on an offender’s criminal history record. If we compare an

offender who has 20 arrests with one who has only 5 arrests, we know

L E V E L S O F M E A S U R E M E N T 21

host of other issues related to crimes and criminals. For example, we can

measure the amount spent by criminal justice agencies to pay the salaries

of police officers or to pay for the health care costs of prison inmates.

We can measure the financial costs of different types of crime by mea-

suring the amount stolen by offenders or the amount of time lost from

meet the requirements of at least an interval level of measurement.

Now that we have defined each step in the ladder of measurement,

we can summarize. As is illustrated in Table 2.4, as you move up the

ladder of measurement, the amount of information that is gained in-

creases. At the lowest level, you have only categorization. At the next

level, you add knowledge about the order of the categories included.

With interval scales, you not only classify and order your measure but

also define how much categories differ one from another. A ratio scale

requires all of these characteristics as well as a non-arbitrary, or true,

zero value.

Variables Commonly Found in Criminal Justice Research

That Are Measured on at Least Interval Scales

VARIABLE COMMON CATEGORIES

Age Years Education Years Income or Salary Dollars, etc. Number of Crimes in a

City/County State Nation Count Crime Rates for a

City/County/State/Nation Count of crimes, adjusted for the size of the population Self-Reported Delinquent Acts Count

Table 2.3

Summary of the Information Required for Each Level of Measurement

TRUE ZERO �

ORDER � SET INTERVALS � SET INTERVALS �

LEVEL OF CATEGOR- CATEGOR- ORDER � ORDER �

MEASUREMENT IZATION IZATION CATEGORIZATION CATEGORIZATION

Ratio X X X X

Interval X X X

Ordinal X X

Nominal X

Table 2.4

arrested. Table 2.3 provides examples of criminal justice variables that

of prison served or sentenced or the age at which offenders were first

work by violent crime victims. We can measure the number of years

22 C H A P T E R T W O : M E A S U R E M E N T

R e l a t i n g I n t e r v a l , O r d i n a l , a n d N o m i n a l S c a l e s : T h e I m p o r t a n c e o f C o l l e c t i n g D a t a a t t h e H i g h e s t L e v e l P o s s i b l e

Take, for example, the measurement of victimization. If you decided

to simply compare the types of victimization involved in a crime event,

you would measure victimization using a nominal scale. You might

choose the following categories: events involving loss of money or prop-

erty, events including physical harm, a combination of such events, and

all other events. But let us assume, for a moment, that at some time after

you collected your data, a colleague suggests that it is important to dis-

tinguish not only the type of event but also the seriousness of crimes

within each type. In this case, you would want to distinguish not only

whether a crime included monetary loss or violence but also the serious-

ness of each loss. However, because your variable is measured on a

nominal scale, it does not include information on the seriousness of loss.

Accordingly, from the information available to you, you cannot create an

ordinal-level measure of how much money was stolen or how serious

the physical harm was.

Similarly, if you had begun with information only on the order of

crime seriousness, you could not transform that variable into one that

defined the exact differences between categories you examined. Let’s

say, for example, that you received data from the police that ranked

($10,001 and above). If you decide that it is important to know not just

the general order of monetary harm but also the exact differences in

harm between crimes, these data are insufficient. Such information

would be available only if you had received data about harm at an in-

terval level of measurement. In this case, the police would provide in-

formation not on which of the four categories of harm a crime belonged

to, but rather on the exact amount of harm in dollars caused by each

crime.

is that you should measure variables in a study at the highest level of

One important lesson we can draw from the ladder of measurement

measurement your data allow. This is because each higher level of mea-

information at the outset, you may not be able to add it at the end of

surement cannot be transformed easily into measures higher on the

surement requires additional information. And if you fail to collect that

your study. In general, variables measured lower on the ladder of mea-

ladder. Conversely, variables measured higher on the ladder of mea-

surement can be transformed easily into measures lower on the ladder.

moderate monetary harm ($501–10,000), and serious monetary harm

monetary victimization for each crime into four ordinally scaled

categories: no monetary harm, minor monetary harm (up to $500),

W H A T I S A G O O D M E A S U R E ? 23

While you cannot move up the ladder of measurement, you can move

down it. Thus, for example, if you have information collected at an inter-

val level, you can easily transform that information into an ordinal-scale

measure. In the case of victimization, if you have information on the

exact amount of harm caused by a crime in dollars, you could at any

point decide to group crimes into levels of seriousness. You would sim-

ply define the levels and then place each crime in the appropriate level.

For example, if you defined crimes involving harm between $501 and

$10,000 as being of moderate victimization, you would take all of the

crimes that included this degree of victimization and redefine them as

falling in this moderate category. Similarly, you could transform this mea-

sure into a nominal scale just by distinguishing between those crimes

that included monetary harm and those that did not.

Beyond illustrating the connections among different levels of mea-

surement, our discussion here emphasizes a very important rule of

thumb for research. You should always collect information at the highest

level of measurement possible. You can always decide later to collapse

such measures into lower-level scales. However, if you begin by collect-

ing information lower on the ladder of measurement, you will not be

able to decide later to use scales at a higher level.

W h a t I s a G o o d M e a s u r e ?

In analysis and reporting of research results, measures that are of a

higher scale are usually preferred over measures that are of a lower

scale. Higher-level measures are considered better measures, based on

the principle that they take into account more information. Nonetheless,

this is not the only criterion we use in deciding what is a good variable

in research. The researcher must raise two additional concerns. First,

does the variable reflect the phenomenon to be described? Second, will

the variable yield results that can be trusted?

The first question involves what those who study research methods

call validity. Validity addresses the question of whether the variable

used actually reflects the concept or theory you seek to examine. Thus,

for example, collecting information on age in a sample is not a valid way

of measuring criminal history. Age, although related to criminal history,

is not a measure of criminal history. Similarly, work history may be re-

lated to criminality, but it does not make a valid measure of criminality.

But even if we restrict ourselves to variables that directly reflect criminal

history, there are often problems of validity to address.

Let’s say that you wanted to describe the number of crimes that of-

fenders committed over a one-year period. One option you might have

24 C H A P T E R T W O : M E A S U R E M E N T

The most valid measure of frequency of offending is the one that

most directly assesses how many crimes an individual has committed.

Associated with each of the three variables included on the rap sheet is

some degree of threat to validity. This means that each can be criticized

because it does not quite reflect the concept we wish to study. Incarcera-

tion, for example, is more a measure of seriousness of crime than fre-

quency of offending. This is because judges may impose a number of

different types of sanctions, and they are more likely to impose a prison

sentence for more serious crimes. Many crimes that result in a conviction

lead not to incarceration but rather to probation, fines, or community

service. Thus, if we use incarceration to measure frequency of offending,

we are likely to miss many crime events in an offender’s criminal record.

Accordingly, incarceration provides a biased picture of the number of of-

fenses committed by an offender. It is not a highly valid measure of this

concept.

Using this logic, criminologists have generally assumed that arrest is

the most valid measure of frequency of offending that can be gained

from official data sources, such as the FBI rap sheet. Arrests are much

closer in occurrence to the actual behavior we seek to study and are not

filtered by the negotiations found at later stages of the legal process.

While criminologists have assumed that arrests reflect criminal behavior

more accurately than convictions or incarceration, some legal scholars

contend that arrests are a less valid measure of criminality precisely be-

cause they come before the court reaches a conclusion regarding the in-

nocence or guilt of a defendant. They contend that someone has not

committed a crime until the legal system defines an act as such.

Self-report surveys are generally considered to provide the most valid

measure of frequency of offending. This is because an individual can be

asked directly how many crimes he or she has committed. But self-report

studies are often criticized in terms of another concern in measurement,

which is termed reliability.

Reliability addresses the question of whether a measure gains infor-

mation in a consistent manner. Will you get the same result if you repeat

measurement of the same case or person? If different people have similar

characteristics, will your measure reflect that similarity? Returning to the

above example of criminal history, we would ask not whether the mea-

sure reflects the concept of frequency of offending, but whether mea-

surement of the concept is reliable across different subjects.

is to examine their criminal history as it is recorded on the Federal

Bureau of Investigation’s (FBI) criminal history record, or rap sheet.

The rap sheet includes information on arrests, convictions, and incar-

cerations. Although each of these variables tells us something about a

person s criminal history, they are not all equally valid in terms of ans-

wering the research question we have proposed.

’

C H A P T E R S U M M A R Y 25

Self-reports, which allow us to ask valid questions about the number

of crimes that a person has committed, have been challenged on the basis

of their reliability. One problem is that people may lie about their crimi-

nal histories. Crime is a sensitive issue, and no matter what efforts the re-

searcher makes to assure subjects of confidentiality, people may be hesi-

tant to talk about crimes in their past. Accordingly, depending on the

degree of hesitancy of subjects, a researcher might gain different answers,

irrespective of a person’s actual criminal history. But even if a person is

willing to provide accurate responses to such questions, he or she may

not be able to. Some people have better memories than others, and the

reliability of this measure depends in part on a person’s ability to recall

events generally. Such issues of reliability have begun to be addressed di-

rectly by criminologists, who are trying to increase the reliability of self-

report methods by improving interview techniques and protocols.

Returning to the FBI rap sheets, we can also assess their reliability. In

general, not only is arrest assumed to be the most valid of official mea-

sures; it is also the measure most reliably recorded on the FBI rap sheets.

This is the case in good part because the rap sheets are built around fin-

gerprint records, which police agencies have come to routinely send to

the FBI. This helps the police agencies as well, because they often use

this information to check the identities of arrestees and to assess their

criminal histories. Other types of agencies are less consistent in their

transfer of information to the FBI, and as a result convictions and incar-

cerations are less reliably recorded.

The issues raised in connection with the validity and reliability of

criminal history information are good examples of the kinds of problems

you will encounter in assessing measures in criminal justice. You should

keep in mind that no variable is perfect. Some threat to validity is likely

velop or choose the best measure you can. The best measure is the one

that most closely reflects the concept you wish to study and assesses it in

a consistent and reliable way across subjects or events.

C h a p t e r S u m m a r y

In science, we use measurement to make accurate observations. All

measurement must begin with a classification process—a process

that in science is carried out according to systematic criteria. This

process implies that we can place units of scientific study in clearly

defined categories. The end result of classification is the development

of variables.

reliability is almost always present in measurement. Your task is to de-

to be encountered, no matter how careful you are. Some degree of un-

26 C H A P T E R T W O : M E A S U R E M E N T

There are four scales of measurement: nominal, ordinal, interval,

and ratio. With a nominal scale, information is organized by simple

classification. The aim is merely to distinguish between different phe-

nomena. There can be no overlap between categories nor can there be

cases that do not fit any one category. There is no theoretical limit to the

number of nominal categories possible. With an ordinal scale, not only

is information categorized, but these categories are then placed in order

of magnitude. An interval scale is one that, in addition to permitting the

processes of categorization and ordering, also defines the exact differ-

ence between objects, characteristics, or events. A ratio scale is an inter-

val scale for which a non-arbitrary, or true, zero value can be identified.

Data collected at a higher level of measurement may subsequently be

reduced to a lower level, but data collected at a lower level may not be

transformed to a higher one. For this reason, it is always advisable to

collect data at the highest level of measurement possible.

There are three separate factors that affect the quality of a measure.

The researcher should strive for a measure that has (1) a high scale of

measurement (one that uses the most information); (2) a high level of

validity (one that provides an accurate reflection of the concept being

studied); and (3) a high level of reliability (one that provides consistent

results across subjects or units of study).

K e y T e r m s

classification The process whereby data

are organized into categories or groups.

data Information used to answer a re-

search question.

interval scale A scale of measurement

that uses a common and standard unit and

enables the researcher to calculate exact

differences between scores, in addition to

categorizing and ordering data.

levels of measurement Types of mea-

surement that make use of progressively

larger amounts of information.

measurement The assignment of numeri-

cal values to objects, characteristics, or

events in a systematic manner.

nominal scale A scale of measurement

that assigns each piece of information to an

appropriate category without suggesting

any order for the categories created.

ordinal scale A scale of measurement that

categorizes information and assigns it an

order of magnitude without using a stan-

dard scale of equal intervals.

ratio scale A scale of measurement identi-

cal to an interval scale in every respect ex-

cept that, in addition, a value of zero on

the scale represents the absence of the

phenomenon.

reliability The extent to which a measure

provides consistent results across subjects

or units of study.

scale of measurement Type of catego-

rization used to arrange or assign values to

data.

E X E R C I S E S 27

validity The extent to which a variable

accurately reflects the concept being

measured.

variable A trait, characteristic, or attribute

of a person/object/event that can be mea-

sured at least at the nominal-scale level.

E x e r c i s e s

2.1 For each of the following examples of criminal justice studies, state whether the scale of measurement used is nominal, ordinal, or at least interval (i.e., interval or ratio). Explain your choice.

a. In a door-to-door survey, residents of a neighborhood are asked how many times over the past year they (or anyone in their house- hold) have been the victims of any type of crime.

b. Parole-board members rate inmate behavior on a scale with values ranging from 1 to 10; a score of 1 represents exemplary behavior.

c. One hundred college students are asked whether they have ever been arrested.

d. A researcher checks prison records to determine the racial back- ground of prisoners assigned to a particular cell block.

e. In a telephone survey, members of the public are asked which of the following phrases best matches how they feel about the perfor-

f. A criminologist measures the diameters (in centimeters) of the skulls of inmates who have died in prison, in an attempt to develop a biological theory of the causes of criminality.

g. Secretaries at a top legal firm are asked the following question: “Over the past year, have you been the victim of sexual harass-

2.2 You have been given access to a group of 12 jurors, with a mandate from your senior researcher to “go and find out about their prior jury experience.” Under each of the following three sets of restrictions, de- vise a question to ask the jurors about the number of experiences they have had with previous juries.

a. The information may be recorded only on a nominal scale of measurement.

indifferent, satisfied, or very satisfied. mance of their local police force: totally dissatisfied, dissatisfied,

follows: never, once, two or three times, more than three times, or ment—and if so, how many times?” Answers are categorized as

refused to answer.

28 C H A P T E R T W O : M E A S U R E M E N T

b. The information may be recorded on an ordinal scale but not on any higher scale of measurement.

c. The information may be recorded on an interval scale.

Your senior researcher subsequently informs you that she wishes to know the answers to the following five questions:

—How many of the jurors have served on a jury before?

—Who is the juror with the most prior experience?

—What is the sum total of previous jury experience?

—Is there anyone on the jury who has served more than three times?

—What is the average amount of prior jury experience for this group?

d. If you had collected data at the nominal level, which (if any) of the above questions would you be in a position to answer?

e. If you had collected data at the ordinal level, which (if any) of the above questions would you be in a position to answer?

f. If you had collected data at the interval level, which (if any) of the above questions would you be in a position to answer?

2.3 You have been asked to measure the public’s level of support for using the death penalty. Devise questions to gauge each of the following:

a. Overall support for using the death penalty.

b. Support for using the death penalty if there are other punishment options.

c. Support for using the death penalty if the chances of an innocent person being executed are

i. 1 in 1,000.

ii. 1 in 100.

iii. 1 in 10.

2.4 You are investigating the effects of a defendant’s prior record on vari- ous punishment decisions made by the court. One variable that you have access to in local court records is the total number of prior felony arrests for each defendant.

a. What kinds of questions would you be able to answer with prior record measured in this way?

b. Explain how you would recode this information on a nominal scale

c. Explain how you would recode this information on an ordinal scale

answer with prior record measured in this way?

answer with prior record measured in this way?

of measurement. What kinds of questions would you be able to

of measurement. What kinds of questions would you be able to

E X E R C I S E S 29

2.5 Because the Ministry of Transport (MOT) is concerned about the num- ber of road accidents caused by motorists driving too close together, it has, on an experimental 2-km stretch of road, painted “chevrons” (lane markings) every few meters in each lane. By the roadside it has erected a sign that reads: “KEEP YOUR DISTANCE: STAY AT LEAST 3 CHEVRONS FROM THE CAR IN FRONT!” The MOT has asked you to measure the extent to which this instruction is being followed. There are a number of possible measures at your disposal. Assess the relia- bility and validity of each approach suggested below. Which is the best measure?

a. Stand on a bridge over the experimental stretch of road and count how many of the cars passing below do not keep the required distance.

b. Compare police figures on how many accidents were recorded on that stretch of road over the periods before and after it was painted.

c. Study the film from a police camera situated 5 km farther down the same stretch of road (after the end of the experimental stretch) and count how many cars do not keep a safe distance.

2.6 The police are planning to introduce a pilot “community relations strategy” in a particular neighborhood and want you to evaluate whether it has an effect on the willingness of citizens to report crimes to the police. There are a number of possible measures at your dis- posal. Assess the reliability and validity of each approach suggested below. Which is the best measure?

a. Telephone every household and ask respondents to measure, on a scale of 1 to 10, how willing they are to report particular types of crime to the police. Repeat the experiment after the scheme has been in operation six months.

b. Compare a list of offenses reported by members of the neighbor- hood in the six months before introduction of the scheme with a similar list for the six months after introduction of the scheme. (It is standard procedure for the police to record the details of the com- plainant every time a crime is reported to them.)

2.7 You are comparing the psychological condition of three inmates serv- ing out long terms in different high-security prisons, and you are in- terested in the amount of contact each one has with the outside world. You wish to determine how many letters each one has sent over the past 12 months. No official records of this exist. There are a number of possible measures at your disposal. Assess the reliability and validity of each approach suggested below. Which is the best measure?

a. Ask each prisoner how many letters he or she sent over the past year.

30 C H A P T E R T W O : M E A S U R E M E N T

b. Check the rules in each of the prisons to see how many letters high security prisoners are allowed to send each year.

c. Check the records of the prison postal offices to see how many times each prisoner bought a stamp over the past year.

2.8 The government is interested in the link between employment and criminal behavior for persons released from prison. In a study de- signed to test for an effect of employment, a group of people released from prison are randomly assigned to a job training program, where they will receive counseling, training, and assistance with job place- ment. The other offenders released from prison will not receive any special assistance. There are a number of possible measures at your disposal. Assess the reliability and validity of each approach suggested below. Which is the best measure?

a. Eighteen months after their release from prison, interview all the of- fenders participating in the study and ask about their criminal activ- ity to determine how many have committed criminal acts.

2.9 In a recent issue of a criminology or criminal justice journal, locate a research article on a topic of interest to you. In this article, there should be a section that describes the data. A well-written article will describe how the variables were measured.

a. Make a list of the variables included in the article and how each was measured.

b. What is the level of measurement for each variable—nominal, ordi- nal, or at least interval? Explain why.

c. Consider the main variable of interest in the article. Assess its relia- bility and validity.

returned to prison within 18 months of release.

arrested for a new crime within 18 months of release.

b. Look at prison records to determine how many offenders were

c. Look at arrest records to determine how many offenders were

C o m p u t e r E x e r c i s e s

There are a number of statistical software packages available for data analysis.

Most spreadsheet programs will also perform the basic statistical analyses of the

kind described in this text through Chapter 17. The computer exercises included

in this text focus on the use of two different software programs: SPSS and Stata.

Most universities that we are aware of make at least one of these two statistical

programs available in student computer labs. There are also student versions of

each program that can be purchased separately, sometimes through a university

bookstore or other offices on campus—see each company’s Web site for details

(www.spss.com or www.stata.com). There are many excellent reference books

C O M P U T E R E X E R C I S E S 31

is not to repeat what is said in those books. Rather, our goal with the computer

exercises is to illustrate some of the power available to you in two widely used

packages. In real-world situations where you are perform some type of statistical

analysis, you rarely work through a problem by hand, especially if the number of

observations is large.

SPSS

To begin our exploration of SPSS, we will focus here on some of the data

management features available to users. After starting the SPSS program on

your computer, you will need to open the National Youth Survey data file from

the Web site (nys_1.sav). For those readers working with the Student Version

of SPSS, you are limited to data files with no more than 50 variables and 1,500

cases. We have also included a smaller version of the NYS data (nys_1_student.

sav) that contains a random sample of 1,000 cases (of the original 1,725).

After you start SPSS and open the data file, the raw data should appear in a

window that looks much like a spreadsheet. Each column represents a different

variable, while each row represents a different observation (individual, here). If

you scroll down to the end of the data file, you should see that there are 1,725

lines of data (or 1,000 if using the student version of the data file).

There are three direct ways to learn about the variables included in this data

file. First, notice the buttons in the lower center of the spreadsheet. One button

(which should appear as the darker shade of the two buttons) is labeled “Data

View,” and the other is labeled “Variable View.” The data view button presents

us with the spreadsheet of values for each observation and variable. If you click

on the button labeled “Variable View,” you should now see another spreadsheet,

in which variable names are listed in the first column and the other columns

contain additional information about each variable. For example, the first col-

umn provides the name of the variable, another column provides a label for the

Several SPSS and Stata files are available at the following Web address:

http://extras.springer.com. The data file we will use first represents a subset of

the data from the National Youth Survey, Wave 1. The sample of 1,725 youth

is representative of persons aged 11–17 years in the USA in 1976, when the first

wave of data was collected. While these data may seem old, researchers continue

to publish reports based on new findings and interpretations of these data. One

of the apparent strengths of this study was its design; the youth were inter-

viewed annually for 5 years from 1976 to 1980 and then were interviewed again

in 1983 and 1987. The data file on our Web site was constructed from the full

data source available at the Inter-University Consortium of Political and Social

Research, which is a national data archive. Data from studies funded by the

National Institute of Justice (NIJ) are freely available to anyone with an Internet

connection; go to http://www.icpsr.umich.edu/NACJD. All seven waves of data

from the National Youth Survey are available, for example.

on the use of either SPSS or Stata for statistical data analysis—our intent here

32 C H A P T E R T W O : M E A S U R E M E N T

variable (allowing us to add a more informative description of our variable), and

an additional column provides value labels. It is from this column that we will

be able to learn more about each variable. For example, click on the cell in this

column for the sex variable, and you should see a small gray box appear in the

cell. Now click on this small gray box and you will be presented with a new win-

dow that lists possible values for sex and the corresponding labels. Here, we see

that males have been coded as “1” and females as “2.” If you click on “OK” or

“Cancel,” the window disappears. You can then perform this same operation for

every other variable.

A second way of obtaining information about the variables in an SPSS data

file involves using the “Variables” command. To execute this command, click

on “Utilities” on the menu bar; then click on “Variables.” What you should see

is a list of variables on the left and another window on the right that presents

information about the highlighted variable. If you click on the sex variable, you

should see information on its coding and values in the window on the right. This

command is particularly useful if you are working with an SPSS data file and

simply need a reminder of how the variables are coded and what categories or

values are included. This feature is useful if you are working with a data set and

need to know what a particular variable refers to or how it is measured in order

to continue working.

A third way of obtaining information about the variables in an SPSS data file

involves the “Display Data File Information” command. To run this command,

click on “File” on the menu bar; then click on “Display Data File Information,”

and then select the option for “Working File” (the data you have already opened

up in SPSS). This command generates text for the output window in SPSS. This

output contains all the information SPSS has on every variable in a data file.

Executing this command is equivalent to executing the “Variables” command

for every variable in the data set and saving that information in another file. Be

aware that using this command on a data file with many variables will produce a

very large output file. This command is most useful when you are first working

with an SPSS data set that someone else has conveniently set up for you and you

need to verify the contents of the data set and the nature of the variables includ-

ed in the data set. Similar to what you are now doing with the NYS data file.

In subsequent chapters, the computer exercises will make use of syntax com-

mands and files. These are the command lines that each program uses to run a

particular procedure. To begin, open a syntax window in SPSS by clicking on

“File,” then “New” and “Syntax.” A blank window opens in which you may

enter command lines and save for future use. Alternatively, instead of open-

ing a blank syntax window, you could open one that had been saved, such as

Chapter_2.sps included on the text Web site.

Open the Chapter_2.sps syntax file. As a convention throughout the text, we

will use CAPS to denote required or really useful components to a command,

although you are free to use lowercase letters in any command that you run in

SPSS. The first line of this file begins with /*—this is a comment line. Any of

C O M P U T E R E X E R C I S E S 33

the SPSS syntax files that you open and begin with this line do not run com-

mands, but are included to help explain what it is you are reading in the file.

The first two command lines of Chapter_2.sps are:

The GET FILE command reads the NYS data file into SPSS. Note that you

will need to provide a directory name that contains your copy of the nys_1.sav

data file. The DATASET command gives you the option of naming the data

file something else internally, while you are working in SPSS. Here, the name

given is simply “nys.” The WINDOW = FRONT option is useful for those times

when you may have more than one data file opened in SPSS—this forces it to

keep this spreadsheet in the position where it is the first one you will see on your

computer screen.

It is important to note that each command line ends with a period—this

ensures that SPSS will know where the command ends and when to start execut-

ing the instruction.

To run these command lines—after you have edited the directory

information—highlight the two lines and do one of the following:

Click on the right-pointing triangle toward the top right of the syntax window.

Click on “Run” on the menu bar and then select the option for “Selection”—

The keystrokes for this same procedure will appear when you click on “Run.”

They are slightly different for Mac and Windows users, but may simplify your

To obtain the file information—the complete listing of variables and their char-

acteristics, run the third line of Chapter_2.sps:

DISPLAY DICTIONARY.

This will display all of the information we highlighted above in using the

point-and-click interactive method to display the file information.

Stata

Much like our discussion of SPSS, we begin our exploration of Stata by focusing

on some of the basic data management features available to users. After starting

the Stata program on your computer, you will see a window that has several dif-

ferent boxes. On the far left, the box labeled “Review” will contain a growing

list of commands that you run during a Stata session. The largest box (top and

center) is labeled “Results” and will present all of the output from various com-

mand that you have run as well as any error messages. Just below this box is the

“Command” box, where you will enter commands, if going one command at a

time. On the right side of the Stata window is a box labeled “Variables”—the

upper half will contain a complete list of all the variables in the data file, while

the lower half provides various details about the variable that is highlighted in the

upper half.

GET FILE = '[directory_name]nys_1.sav'.

DATASET NAME nys WINDOW = FRONT.

34 C H A P T E R T W O : M E A S U R E M E N T

Begin by opening the National Youth Survey data file obtained from the Web

site (nys_1.dta) using “File” from the menu bar and the “Open…” option listed.

Note that the “Review” box on the left now contains the Stata command that

was run to open the data file, the “Variables” box on the right contains the list-

ing of variables, and the “Results” window simply lists the Stata command that

was run.

If you would like to view the data in a spreadsheet format, click on the box at

the top center of the Stata window for “Data Browser” or “Data Editor”—the

“Browser” view will allow you to look at the data, while the “Editor” view will

allow you to make changes to the data. The window that opens will present each

variable as a separate column, each observation as a separate row, and the con-

tents of each cell will either be a numeric value or a value label that is associated

with a numeric value. If you click on a cell with a value label, the corresponding

numeric value appears in the white horizontal bar just above the spreadsheet.

There are several different ways of learning about the variables included in

this data file. As noted already, the “Variables” box on the right contains infor-

mation about each variable and is displayed as soon as the variable name is high-

lighted (i.e., clicked on) in the upper box.

A much more comprehensive method for obtaining information about the

variables in the data file is to use the “Codebook” command, which will list all

of the variables in the data file, their values, value labels, as well as other infor-

mation that will become useful in subsequent chapters of this text. To run the

“Codebook” command, click on “Data” on the menu bar, then the options for

“Describe data” and selecting the option for “Describe data contents (code-

book).” The “Results” window will generate a great deal of output that will list

each variable, the variable label, the possible (numeric) values when there are

fewer than 10 unique values, and any value labels. The other information pre-

sented will be discussed later.

Similar to the observation made in regard to the use of SPSS, in subsequent

chapters, the computer exercises will make use of syntax commands and files,

rather than the point-and-click options. Although Stata syntax is different from

SPSS syntax, how it works is very much the same: the command line(s) inform

Stata what procedure to run, whether it is opening a date file for use, or some

other statistical procedure. To open a new syntax window in Stata, where they

are called “do files,” start by clicking on “File,” then “New Do-file. A blank

window opens in which you may enter command lines and save for future use.

Alternatively, instead of opening a blank do file window, you could open one

that had been saved, such as Chapter_2.do included on the text Web site.

Stata commands must be in lowercase letters, unless otherwise noted in the

documentation for the command. As a convention throughout our text, we will

place Stata commands and useful or required options in boldface font.

Open the Chapter_2.do file. The first line of this file begins with /* and ends

with */—this is a comment line. In Stata, comments are bounded by these two

mirror sets of characters, so without the */, Stata would interpret everything

C O M P U T E R E X E R C I S E S 35

following the initial /* as a comment—so it is important to pay attention to your

use of these in any Stata do file.

The first command line of Chapter_2.do is:

The use command reads the NYS data file into Stata. Note that you will

need to provide a directory name that contains your copy of the nys_1.dta data

file. Also note that after the file name and location, there is a comma followed by

the clear option. Using this option tells Stata to clear out any existing data file

currently being used without saving any changes made to that data file.

To run this command line—after you have edited the directory information—

highlight the command line and do one of the following:

will run the command quietly, meaning there is no output in the “Results”

window, other than a line indicating a command was run.

will run the command and generate output in the “Results” window.

Keystroke options are available for both of these commands, to determine

what they are for your operating system (Mac or Windows), click on “View”

the keystroke shortcuts that can be used to run the command quietly or not.

To obtain the file information—the complete listing of variables and their char-

acteristics, run the second command line of Chapter_2.do:

codebook

This will display all of the information we highlighted above in using the point-

and-click interactive method to display the file information.

Problems

Using one of the ways described above, work through all the variables included

in the NYS data file:

1. Note the level of measurement for each variable and then briefly explain

why it is what it is. (You should not rely on the level of measurement

information given in the SPSS data file, especially if someone else has

constructed the SPSS data file for you.)

2. Describe the likely levels of reliability and validity for each variable and

explain why they are what they are.

use “[directory_name]nys_1.dta”, clear

Representing and Displaying Data

When Should a Pie Chart Be Used?

t i m e s e r i e s d a t a v i s u a l l y

How Can the Researcher Present Time Series Data Graphically?

C h a p t e r t h r e e

T h e f r e q u e n c y d i s t r i b u t i o n

U s i n g b a r c h a r t s t o r e p r e s e n t d a t a

U s i n g p i e c h a r t s t o r e p r e s e n t d a t a

R e p r e s e n t i n g h i s t o r i c a l o r

What is It?

When are Bar Charts Useful?

How are Bar Charts Created?

How are Pie Charts Created?

How are Data Represented Visually in a Histogram?

D. Weisburd and C. Britt, Statistics in Criminal Justice, DOI 10.1007/978-1-4614-9170-5_3,

© Springer Science+Business Media New York 2014

THE GRAPHICAL DISPLAY OF DATA is an important tool for presenting sta-

tistical results in such a way that the key features or characteristics of an

analysis are highlighted. There are many different ways the same data

might be displayed. Indeed, many books have been written that focus

entirely on graphical presentation of data. In this chapter, we introduce

some common ways of representing data in graphical form, along with

suggestions for effectively presenting information in an accurate way.

We begin by discussing the most basic way of summarizing—and then

graphing—data: frequency distributions and histograms. Building on the

discussion of histograms, we move on to more general bar charts, noting

W h a t A r e F r e q u e n c y D i s t r i b u t i o n s a n d H i s t o g r a m s ?

When we array scores according to their value and frequency, we con-

struct what is called a frequency distribution. Let’s take the following

data on previous arrests of 100 known offenders as an example:

14 0 34 8 7 22 12 12 2 8 6 1 8 1 18 8 1 10 10 2

12 26 8 7 9 9 3 2 7 16 8 65 8 2 4 2 4 0 7 2 1 2 11 2 1 1 5 7 4 10

11 3 41 15 1 23 10 5 2 10 20 0 7 6 9 0 3 1 15 5 27 8 26 8 1 1 11 2 4 4 8 41 29 18 8 5 2 10 1 0 5 36 3 4 9 5 10 8 0 7

We first group all of the cases with the same value together. Accordingly,

we group together the cases with no prior arrests, one prior arrest, two

Finally, we examine how graphs can be used to represent a series of

the variety of information that can be presented in bar and pie charts.

observations over time.

37

prior arrests, and so forth, until we have covered all of the potential

scores in the distribution. Then we arrange these scores in order of mag-

nitude, as is done in Table 3.1. Looking at the data in this way allows us

to get a sense of the nature of the distribution of scores.

In practice, creating a frequency distribution is usually the first step a

researcher takes in analyzing the results of a study. Looking at the distri-

bution of scores not only provides a first glance at the results of a study;

it also allows the researcher to see whether there are scores that do not

make sense. For example, coding errors in the data set may have given

rise to impossible scores. In our example, a result of thousands of arrests

would be very unlikely and would thus lead the researcher to take an-

other look at that particular case.

Constructing frequency distributions by hand can be very time and

labor intensive. Researchers today seldom construct frequency distribu-

tions by hand. This task can be done simply and easily with packaged

statistical software, such as SPSS.

Frequency Distribution of Prior Arrests for 100 Known Offenders

VALUE FREQUENCY

0 6 1 11 2 11 3 4 4 6 5 6 6 2 7 7 8 12 9 4

10 7 11 3 12 3 14 1 15 2 16 1 18 2 20 1 22 1 23 1 26 2 27 1 29 1 34 1 36 1 41 2 65 1 Total 100

Table 3.1

38 C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A

If you decide to present a frequency distribution of the results of a

study, you must choose what particular format to use. For example, the

distribution of prior arrests could simply be presented as in Table 3.1. Or

the same information could be presented graphically in what is called a

histogram. To make a histogram, we take the scores and values from a

frequency distribution and represent them in pictorial form. In this case,

we use a bar to represent each value in the frequency distribution. The

x-axis (the horizontal axis) of the histogram represents the values of the

variable we are analyzing—here, the number of arrests. The y-axis (the

vertical axis) captures the height of the bars and indicates the number of

scores—the frequency—found in each category. A histogram of the data

on prior arrests is provided in Figure 3.1. The information presented in

the histogram is identical to the information presented in the frequency

distribution in Table 3.1, but the histogram conveys to the reader an im-

mediate sense of the range of values, the location of clusters of cases,

and the overall shape of the distribution—information that is not as eas-

ily obtainable from a frequency distribution.

It is important to note that the x-axis in Figure 3.1 correctly represents

the full range of values for the variable. In particular, note that there is a

large gap in the distribution from 41 arrests to 65 arrests. Depending on

F re

q u

en cy

14

12

10

8

6

4

2

0

Number of Arrests

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Histogram of Frequency DistributionFigure 3.1

W H A T A R E F R E Q U E N C Y D I S T R I B U T I O N S A N D H I S T O G R A M S ? 39

the software used to generate a histogram, this information may or may

not be represented. Excluding this information and placing all the bars

adjacent to each other essentially means ignoring the level of measure-

ment of the variable and treating the variable as if it were nominal or or-

dinal. Why would you want to include information about those values

that have an observed frequency of zero? If you look at Figure 3.1, you

can see that there are relatively few observations for increasingly greater

numbers of arrests and that the distance between categories starts to in-

crease. If the bars were adjacent to each other and correctly labeled, it

would still be possible for the reader to discern the spread of observa-

tions. But when the values with an observed frequency of zero are por-

trayed as well, it is often easier to interpret the histogram.

Before we get into working with more complex graphs, it is worth

noting that frequency distributions of interval-level variables are often

dispersed across a large number of values. Thus, in presenting a fre-

people, we would likely not want to present the simple distribution of

income scores. If we did, we might end up with thousands of scores,

most of which would include only one or two cases. It would take pages

and pages to illustrate these data in the form of either a frequency distri-

bution or a histogram. The solution to this problem is to “group” data

together in larger categories—for example, by thousands or tens of thou-

sands of dollars in the case of incomes. Although there is no hard-and-

fast rule about how to create such larger groupings, it should be done in

a way that fairly represents the raw distribution of scores. Do not create

such a small group of categories that important variation in your data is

hidden.

A common source of confusion for students of statistics is the fact that

statisticians often represent distributions as “curves” rather than his-

tograms or frequency distributions. For example, Figure 3.2 uses a curve

to represent the 2001 SAT math scores of over 1.2 million college-bound

test takers.1 What is the relationship between a frequency distribution or

histogram and a distribution represented by a curve, such as the one in

Figure 3.2?

When we represent a distribution as a curve, it is usually a distribu-

tion with a very large number of cases, such as that of SAT math scores

of 2001 college-bound seniors. We can represent these distributions as

curves because, with a true interval-scale measure, as the number of

cases becomes very large, we can construct a histogram in such a way

1The College Board, 2001 College-Bound Seniors: A Profile of SAT Program Test Tak-

ers, accessed at http://www.collegeboard.com/sat/cbsenior/yr2001/pdf/NATL.pdf.

if we were looking at the incomes of a random sample of thousands of

together into categories that represent a range of values. For example,

quency distribution or histogram, it is often necessary to group scores

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A40

that it begins to approximate a curve. We do this by making the intervals

of scores smaller and smaller.

This process is illustrated in Figure 3.3. We begin with a histogram of

almost 10,000 cases, in which all of the scores are placed within 10

200 300 400 500 600 700 800

Smooth Curve Representation of a Histogram:

SAT Math Scores of 2001 College-Bound Seniors Figure 3.2

4,000

3,000

2,000

1,000

0

F re

q u

en cy

Std. dev. = 1.00 Mean = 0.00 N = 9,994.00

Values

Constructing a Distribution That Approximates a CurveFigure 3.3

(a) Distribution with 10 Intervals

W H A T A R E F R E Q U E N C Y D I S T R I B U T I O N S A N D H I S T O G R A M S ? 41

broad categories (see part a). Here each category is represented by one

large bar. When we increase the number of intervals, or categories, in

the histogram to 30 (part b), we can still see the individual bars but the

shape of the distribution is not as jagged. When the number of cate-

gories is increased to 650 (part c), the histogram looks more like a

smooth curve than a histogram, although if you look closely you will be

1,200

1,000

800

600

400

200

0

F re

q u

en cy

Std. dev. = 1.00 Mean = 0.00 N = 9,994.00

Values

(b) Distribution with 30 Intervals

50

40

30

20

10

0

F re

q u

en cy

Std. dev. = 1.00 Mean = 0.00 N = 9,994.00

Values

(c) Distribution with 650 Intervals

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A42

able to identify the bars that make up the curve. If our distribution had

included an even larger number of scores and categories, the curve

would have become even smoother.

E x t e n d i n g H i s t o g r a m s t o M u l t i p l e G r o u p s : U s i n g B a r C h a r t s

Although histograms provide a quick and easy way to present data from

a frequency distribution, they can be used to present the frequency dis-

tribution for only a single group. What happens when we have fre-

quency distributions for the same variable for more than one group?

There are many instances where an investigator will want to present re-

sults graphically for two or more groups, such as treatment and control

groups, males and females, or states. We could simply construct a his-

togram for each group we were interested in, but we would not be able

to make direct comparisons across the groups in the form and shape of

the distributions. A simple extension of the histogram to multiple groups

makes use of the bar chart. Bar charts allow us to present information

for multiple groups simultaneously. Bar charts are constructed in much

the same way as histograms. The x-axis generally represents the values

of the variable, and the y-axis the size of the bar.2 Most statistical soft-

ware and spreadsheet packages allow the user to construct bar charts to

depict patterns in the various groups. Two of the more common ap-

proaches involve placing the bars side by side and on top of each other.

Presenting data for each group in adjacent bars gives the reader a sense

of the distribution for each group and allows for immediate comparison

of distributions across groups.

Table 3.2 presents simulated frequency distributions for numbers of

prior convictions among 100 men and 100 women arrested for drug of-

fenses. The frequencies for male and female arrestees suggest that males

In addition to being able to incorporate data from more than one

group, a bar chart has other benefits. For example, bars may be pre-

2Most statistical software and spreadsheet packages allow for manipulation of many

characteristics of a bar chart, including color, shading, patterning, and dimensions

(two vs. three). While this allows for the construction of unique charts, the investiga-

tor should be wary of adding so much detail to a chart that the reader loses the point

the investigator is trying to make.

females. Figure 3.4 portrays the male and female frequency distributions

have had more prior contact with the criminal justice system than the

in a bar chart.

43E X T E N D I N G H I S T O G R A M S T O M U L T I P L E G R O U P S

sented either vertically, as in Figure 3.4, or horizontally, as in Figure 3.5.

The only difference between the bar charts in Figures 3.4 and 3.5 is that

in the latter the axes have been flipped: The y-axis now represents the

rules about which form of bar chart is better. Some people like vertical

bar charts because they can draw an imaginary horizontal line across the

graph to get a sense of which bars are larger and smaller. Alternatively,

other people prefer horizontal bars because looking at them mimics the

Frequency Distributions of Number of Prior Convictions

for Male and Female Drug Arrestees

FREQUENCY

Number of Prior Convictions Male Female

0 25 40 1 20 25 2 15 15 3 10 7 4 8 4 5 6 0 6 2 3 7 6 3 8 5 1 9 0 2

10 3 0 Total 100 100

Table 3.2

F re

q u

en cy

40

30

20

10

0

Number of Prior Convictions

0 1 2 3 4 5 6 7 8 9 10

Male

Female

Bar Chart for Male and Female Frequency DistributionsFigure 3.4

values of the variable (here, the number of prior convictions), and the

x-axis represents the size of the bar (the frequency). There are no specific

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A44

process of reading text from left to right. Since the research on visual

perception is mixed about whether vertical or horizontal bars are more

effective—there are benefits and drawbacks to both approaches—the

preferences of the investigator typically determine which approach is

used.3

A cautionary note is needed about the use of bar charts for comparing

multiple groups. In Figure 3.4, the number of individuals in each group

was equal, allowing us to make a direct comparison of each group’s fre-

quency distribution. An investigator would run into trouble, however, if

the two groups did not have the same number of cases. Say one group

had two or three times as many cases as a second group. If we simply

presented a bar chart of the observed frequencies, then we would be

limited to discussing the shape of each group’s distribution; we would

be unable to make valid comparisons across the two groups. For exam-

ple, in the frequency distributions of prior convictions in Table 3.2, sup-

Male

Female

10

9

8

7

6

5

4

3

2

1

0

0 10 20 30 40

Frequency

N u

m b er

o f

P ri

or C

on vi

ct io

n s

Horizontal Bar Chart for Male and Female Frequency DistributionsFigure 3.5

3For a good discussion of the benefits and drawbacks of various approaches, see Gary

T. Henry, Graphing Data: Techniques for Display and Analysis (Thousand Oaks, CA:

Sage, 1995).

4E X T E N D I N G H I S T O G R A M S T O M U L T I P L E G R O U P S 5

pose that we doubled the frequencies for the male arrestees, giving us a

total of 200 males in the sample. In Figure 3.6, the bars representing

male arrests are twice as large as they are in Figure 3.4, and one might

be tempted to say that more males have zero, one, or two arrests. Strictly

speaking, such an observation is correct—the frequency is larger for one

group than for the other group—but it misrepresents the relative sizes of

the two groups and the relative distribution of cases within each group,

which are unchanged by doubling the number of male arrests.

In research on crime and criminal justice, the groups we are inter-

ested in comparing rarely have an equal number of cases. For example,

Table 3.3 presents the frequency distributions of years of education for

capital offenders executed in the United States between 1977 and 1995,

distinguished by the recorded race of the offender. These data come

from a public data file archived by the Bureau of Justice Statistics.4 The

bar chart for these data is presented in Figure 3.7. It is important to note

F re

q u

en cy

50

40

30

20

10

0

Number of Prior Convictions

0 1 2 3 4 5 6 7 8 9 10

Male

Female

Bar Chart for 200 Male and 100 Female ArresteesFigure 3.6

4Capital Punishment in the United States, ICPSR Study #6956, available through the

National Archive of Criminal Justice Data (NACJD) at http://www.icpsr.umich.edu/

NACJD.

that whites outnumber African Americans at a rate of about 3 to 2. This tells

us that although we can evaluate the frequency distributions for whites and

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A46

Number of Years of Education Among Offenders Executed

in the United States, 1977 to 1995, by Race of Offender

RACE OF OFFENDER

Years of Education White African American

7 24 7 8 14 17 9 19 16

10 24 20 11 14 18 12 60 31 13 8 1 14 12 2 15 2 0 16 3 0 17 1 0 Total 181 112

Table 3.3

F re

q u

en cy

70

60

50

40

30

20

10

0

Years of Education

7 8 9 10 11 12 13 14 15 16 17

Whites

African Americans

Bar Chart for Number of Years of Education Among Offenders

Executed in the United States, 1977 to 1995, by Race of Offender Figure 3.7

4E X T E N D I N G H I S T O G R A M S T O M U L T I P L E G R O U P S 7

African Americans separately, we should avoid any direct comparisons

across the two groups.

How do we address the problem of unequal group sizes? The most

direct way is to convert the observed frequencies into proportions or

percentages. A proportion has a value between 0 and 1 and represents

the fraction of all observations that fall into a category. We calculate a

proportion as:

Equation 3.1

where Ncat refers to the number of observations in a given category and

Ntotal refers to the total number of observations. For example, to find the

proportion of whites who had 8 years of education, we would take the

number of white offenders who had 8 years of education (14) and divide

it by the total number of whites executed (181):

Proportion � Ncat

Ntotal

By convention, we generally round a proportion to the second decimal

place. (However, as we will discuss in more detail in the next chapter,

your decision as to how precisely to present a statistic should be based

on the specific research problem you are examining.) The proportion of

executed white offenders who had 8 years of education is thus about

0.08.

Sometimes researchers like to transform a proportion into a percent-

age, because percentages are more commonly used in our everyday

lives. We obtain the percentage simply by multiplying the proportion

by 100:

Equation 3.2Percentage � � Ncat

Ntotal � � 100

W orking It Out

� 0.077

� 14 181

Proportion � Ncat

Ntotal

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A48

For our example of executed white offenders with 8 years of education,

we multiply 14/181 by 100:

W orking It Out

� 7.7%

� � 14 181� � 100

Percentage � � Ncat

Ntotal � � 100

This tells us that about 8% of executed white offenders had 8 years of

education. Table 3.4 presents the observed frequencies and correspond-

ing percentages (calculated to the third decimal place) for every cell in

Table 3.3.

At this point, we can graph either the proportions or the percent-

ages in a bar chart. The selection of proportions or percentages does

not matter, as it will not have any bearing on the shape of the distribu-

tions. The bar chart in Figure 3.8 uses percentages to display years of

education among executed offenders in the United States, distinguished

by race.

Frequency Distributions and Percentages for Number of Years

of Education Among Executed Offenders, by Race

RACE OF OFFENDER

WHITE AFRICAN AMERICAN

Years of Education Freq. % Freq. %

7 24 13.260 7 6.250 8 14 7.735 17 15.179 9 19 10.497 16 14.286

10 24 13.260 20 17.857 11 14 7.735 18 16.071 12 60 33.149 31 27.679 13 8 4.420 1 0.893 14 12 6.630 2 1.786 15 2 1.105 0 0.000 16 3 1.657 0 0.000 17 1 0.552 0 0.000 Total 181 112

Table 3.4

4E X T E N D I N G H I S T O G R A M S T O M U L T I P L E G R O U P S 9

U s i n g B a r C h a r t s w i t h N o m i n a l o r O r d i n a l D a t a

The previous examples of bar charts and histograms have focused on

variables measured at the interval level. Bar charts are also quite useful

for visually representing nominally or ordinally measured variables. In

much of the research on public opinion about use of the death penalty in

the United States, there are sharp differences between the views of whites

and those of African Americans. Table 3.5 presents the level of support

for using the death penalty for first-degree murderers, distinguished by

P er

ce n

ta ge

35

30

25

20

15

10

5

0

Years of Education

7 8 9 10 11 12 13 14 15 16 17

Whites

African Americans

Percentages of Executed Offenders with Various Numbers of Years of Education, by RaceFigure 3.8

Level of Agreement with Use of the Death Penalty

for Convicted First-Degree Murderers

RACE OF RESPONDENT

WHITE AFRICAN AMERICAN

Level of Agreement Freq. % Freq. %

Strongly agree 567 51.266 57 37.748 Agree 290 26.221 32 21.192 Neutral/no opinion 76 6.872 24 15.894 Disagree 106 9.584 25 16.556 Strongly disagree 67 6.058 13 8.609 Total 1,106 151

Table 3.5

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A50

race of the respondent. The data come from the General Social Survey

(GSS), administered by the National Opinion Research Center at the Uni-

versity of Chicago. Respondents were asked: “How strongly do you agree

with the following statement: The death penalty should be used for per-

sons convicted of first-degree murder.” The responses were strongly

agree, agree, neutral/no opinion, disagree, and strongly disagree.5

To chart these data, we cannot graph the numbers of white and

African American respondents in each category and hope to make sense

of the patterns, since there are about seven times more white respon-

dents than African American respondents. Again, we construct a bar

chart using the proportion or percentage of respondents who fall into

each category. A bar chart presenting percentages of white and African

American respondents in each category appears in Figure 3.9.

P i e C h a r t s

Pie charts offer another way of displaying data graphically if the num-

ber of categories of a variable is relatively small. Each wedge in a pie

chart is a proportional representation of the number of cases in that cate-

gory. When we present data on the percentage or proportion of cases in

5The entire GSS database is publicly available and can be accessed at http://www.

icpsr.umich.edu/GSS. Data presented here are drawn from a 1991 study.

P er

ce n

ta ge

60

50

40

30

20

10

0

Use of Death Penalty

Whites

African Americans

Strongly Agree Agree Neutral Disagree Strongly Disagree

Bar Chart for Level of Agreement with Use of the Death Penalty

for Convicted First-Degree Murderers Figure 3.9

51P I E C H A R T S

a pie chart, the information contained in a pie chart is identical to that

We will use the death penalty opinion data in Table 3.5 to illustrate

The information presented in Figure 3.10 is identical to that in Figure

3.9. But the need for two separate pie charts to represent the data for the

two groups of respondents points out one of the limitations of using pie

charts for comparing multiple groups. In Figure 3.9, the bars for the two

groups of respondents are presented side by side. The reader is able to

quickly assess which group is larger, which group is smaller, and the mag-

nitude of the difference. With the two pie charts represented in parts a and

b of Figure 3.10, the reader has to go back and forth between the pies,

match up the category of interest, and then try to infer the magnitude of the

difference in size between the two wedges. Although pie charts are an easy

and effective way of representing the relative sizes of different categories of

a variable, we discourage the use of pie charts for any type of cross-group

comparison, to avoid the potential for confusing or misleading the reader.

T i m e S e r i e s D a t a

Study of many important issues in crime and criminal justice requires the

use of time series data. Time series data include measures on the same

variable for the same unit of analysis at more than one point in time. For

Strongly Disagree (6.1%)

Agree (26.2%)

Neutral (6.9%)

Disagree (9.6%) Strongly Agree (51.3%)

Strongly Disagree (8.6%)

Agree (21.2%)

Neutral (15.9%)

Disagree (16.6%)

Strongly Agree (37.7%)

Percentage of Agreement with Use of the Death Penalty for Convicted First-Degree MurderersFigure 3.10

(a) White Respondents (b) African American Respondents

aspects—size, shape, orientation, dimensions (two or three), position

of wedges, colors, shades, and so on.

and spreadsheet packages allow the researcher to manipulate various

presented in a bar chart. As with bar charts, most statistical software

responses for white and African American respondents, respectively.

the construction of pie charts. Parts a and b of Figure 3.10 present the

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A52

example, in a study of adolescents, the same individuals may be inter-

viewed two or more times. Or, in an examination of the effectiveness of

a treatment program, information on key characteristics of participants

may be collected before and after the treatment, to test for change.

An example of time series data more familiar to many students comes

from the Federal Bureau of Investigation’s annual Uniform Crime Re-

ports.6 Included in the reports are the “official” crime rates, which tell us

the total crime rate, the murder rate, the burglary rate, and so on. We

can easily locate a crime rate for the United States—as well as smaller

geopolitical units, such as states or counties—for some period of time.

Table 3.6 presents the total crime rate and murder rate for the United

States per 100,000 people for the years 1990 to 2000.

In a time series plot, data are graphed over time, with the measure

of time along the x-axis. For the data in Table 3.6, the measure of time is

years, but in other cases, we may have daily, weekly, monthly, or quar-

terly data. The y-axis then represents the value—here, the crime rate.

Figure 3.11 presents a plot of the total crime rate per 100,000 people in

the United States for the period 1990 to 2000. As you can see, the time

series plot provides a very powerful way to present data over time. The

“crime drop” in the United States during this period is clearly illustrated

in Figure 3.11.

It is also possible to plot more than one time series on the same

graph, but be careful about the values of the different series. If one se-

ries has values that are many times larger than those in the other series,

you run the risk of generating a plot in which the line of one group

looks misleadingly like a straight line. Figure 3.12, which shows total

Total Crime Rate and Murder Rate in the United States, 1990 to 2000

TOTAL CRIME RATE MURDER RATE

YEAR (PER 100,000) (PER 100,000)

1990 5,820.300 9.4000 1991 5,897.800 9.8000 1992 5,660.200 9.3000 1993 5,484.400 9.5000 1994 5,373.500 9.0000 1995 5,275.900 8.2000 1996 5,086.600 7.4000 1997 4,930.000 6.8000 1998 4,619.300 6.3000 1999 4,266.800 5.7000 2000 4,124.800 5.5000

Table 3.6

6Federal Bureau of Investigation, Crime in the United States, available at http://www.

fbi.gov/ucr/ucr.htm.

T I M E S E R I E S D A T A 53

crime rates and murder rates for the 1990 to 2000 period, illustrates this

problem. Since the total crime rate ranges between about 4,000 and

6,000 per 100,000 while the murder rate varies between about 5 and 10

per 100,000, the line representing the murder rate appears straight and

indeed can hardly be seen. In such a case, one solution is to construct a

R at

e

7,000

6,500

6,000

5,500

5,000

4,500

4,000

3,500

3,000

Year 1990

1991 1992

1993 1994

1995 1996

1997 1998

1999 2000

Total Crime Rate

Total Crime Rate (per 100,000 People) in the United States, 1990 to 2000Figure 3.11

R at

e

7,000

6,000

5,000

4,000

3,000

2,000

1,000

0

Year

1990 1991

1992 1993

1994 1995

1996 1997

1998 1999

2000

Total Crime Rate

Murder Rate

Total Crime Rate and Murder Rate (per 100,000 People)

in the United States, 1990 to 2000: Single Vertical Axis Figure 3.12

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A54

time series plot with a second y-axis on the right side of the chart. Figure

3.13 adds a second y-axis to account for the smaller values associated

with the murder rate, providing a better depiction of the annual variation

in both rates.

C h a p t e r S u m m a r y

A frequency distribution is an arrangement of data according to the

frequency with which each value occurs. The data may be represented

in a table or in graphic form in a histogram, which uses a bar to repre-

sent the frequency for each value.

Bar charts can be used to represent the frequencies, percentages,

or proportions of variables, regardless of whether they have been mea-

sured at the nominal, ordinal, or interval level. In a vertical bar chart, the

sizes of the bars are indicated along the y-axis and correspond to the fre-

quency, the percentage, or the proportion. The values, or categories, of

the variable are represented along the x-axis. When comparing two or

more groups in a bar chart, it is important to have the values represent

the percentage or proportion of cases, since the presentation of frequen-

cies could be misleading if the groups are of very different sizes.

Pie charts are another common way to represent variables with a

small number of categories or values. The size of each wedge in a pie

chart corresponds to the relative size of that category’s frequency count.

T ot

al C

ri m

e R

at e

M u

rd er

R at

e

7,000

6,000

5,000

4,000

3,000

2,000

1,000

0

Year

1990 1991

1992 1993

1994 1995

1996 1997

1998 1999

2000

Total Crime Rate

Murder Rate

12

10

8

6

4

2

0

Total Crime Rate and Murder Rate (per 100,000 People) in the United States,

1990 to 2000: Two Vertical Axes Figure 3.13

55C H A P T E R S U M M A R Y

Pie charts are better suited to describing data from a single sample or

group than data from multiple groups.

Time series data may be graphically displayed in a time series plot,

a line graph that portrays the values of a variable over some period of

time, such as days, weeks, months, or years. To allow for the compari-

son of multiple variables, additional lines can be easily incorporated into

the line graph. If the magnitudes of the variables differ by a large degree,

it is possible to add an additional y-axis so that the lines can be overlaid

and common or unique trends in the plotted variables discerned.

K e y T e r m s

bar chart A graph in which bars represent

frequencies, percentages, or proportions

for the categories or values of a variable.

frequency The number of times that a

score or value occurs.

frequency distribution An arrangement

of scores in order from the lowest to the

highest, accompanied by the number of

times each score occurs.

histogram A bar graph used to represent

a frequency distribution.

percentage A relation between two num-

bers in which the whole is accorded a

value of 100 and the other number is given

a numerical value corresponding to its

share of the whole.

pie chart A graph in which a circle (called

a pie) is cut into wedges to represent the

relative size of each category’s frequency

count.

proportion A relation between two num-

bers in which the whole is accorded a

value of 1 and the other number is given a

numerical value corresponding to its share

of the whole.

time series data Repeated measures of

the same variable over some regularly oc-

curring time period, such as days, months,

or years.

time series plot A line graph that con-

nects repeated measures of the same vari-

able over some regularly occurring time

period, such as days, months, or years.

S y m b o l s a n d F o r m u l a s

Ncat Number of cases in a category of a variable

Ntotal Total number of cases

To calculate the proportion of cases falling into a category:

Proportion � Ncat

Ntotal

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A56

To calculate the percentage of cases falling into a category:

E x e r c i s e s

3.1 A team of researchers observed the behavior of 20 children during breaktime on a school playground. The observers recorded how many times each child performed a “violent act”—be it a push, a kick, or a punch—against another child. The scores of the 20 children were as follows:

2 0 1 0 0 4 0 10 2 1 3 3 0 1 4 11 0 0 2 0

a. Construct a frequency distribution table for the above results.

b. Construct a histogram of the frequency distribution.

c. How might you interpret the results?

3.2 Workers at an inner-city rape crisis center asked all the callers on a given night for their ages. For a total of 50 callers, the results were as follows:

28 39 17 18 22 31 26 27 16 20

34 35 29 26 17 23 22 23 37 28

24 19 14 25 27 19 24 26 41 27

21 24 17 16 35 25 19 23 29 18

23 26 24 43 28 21 21 36 26 27

a. Construct a frequency distribution table for the above results.

b. Based on the frequency distribution, construct three different his- tograms, with the data grouped in 1-year intervals, 3-year intervals, and 10-year intervals.

3.3 A review of the records of 20 male and 20 female adjudicated delin- quents revealed the following numbers of prior arrests for violent crimes:

Male: 0 2 1 7 4 2 2 1 6 5 0 0 1 2 4 1 0 0 2 9

Female: 0 0 1 4 2 3 1 1 5 1 0 0 0 1 0 3 2 1 0 1

a. Construct three frequency distribution tables: one for males, one for females, and one for the male and female data combined.

b. Construct a histogram for each of the three frequency distributions.

c. How might you interpret the results?

Percentage � � Ncat

Ntotal � � 100

E X E R C I S E S 57

3.4 In a survey of 75 offenders convicted through guilty pleas, respon- dents were asked who had the greatest influence on their decision to plead guilty. Here are their responses: defense attorney (n � 35), spouse/partner (n � 25), prosecutor (n � 10), and judge (n � 5).

a. Graph these data on three bar charts using frequencies, propor- tions, and percentages.

b. Graph these data on a pie chart.

3.5 Suppose a researcher conducted a survey to assess the link between

researcher found that 150 thought the punishments were “too harsh,” 100 thought the punishments were “about right,” and 50 thought the punishments were “too lenient.”

b. Explain your choice of chart.

c. Describe the patterns that you find in this type of chart.

3.6 The burglary rates for two cities over a 10-year period are reported in the following table:

Year City A City B

1991 2,576 875

1992 2,675 966

1993 2,892 1,015

1994 3,014 1,325

1995 2,852 1,779

1996 2,651 1,954

1997 2,443 2,333

1998 2,519 2,121

1999 2,999 2,657

2000 2,840 2,005

a. Present this information in the way that you think is most informative.

b. Explain why you selected this type of chart.

c. Describe the patterns that you observe in this chart.

separately.

a. Present this information graphically in the way that you think is most informative—either for the full sample or for each gender

ments were “too harsh.” Among 300 female respondents, the

gender and attitudes about the criminal justice system. Among 850 male respondents, the researcher found that 500 thought the punishments given by the courts were “too lenient,” 200 thought the punish- ments were “about right,” and the remaining 150 thought the punish-

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A58

C o m p u t e r E x e r c i s e s

SPSS and Stata both contain a wide range of graphics tools that are comparable

to those found in many of the better spreadsheet packages. The following com-

puter exercises are intended to guide you through the basic steps in producing

graphs in the two different programs. In fact, the graphics commands we illus-

trate from SPSS are what are referred to as “legacy” commands in SPSS’s win-

dowed environment. The change in SPSS nomenclature is a consequence of the

graphics capabilities of the program having expanded greatly in recent years. The

graphics options in SPSS are now so complex that the best way to learn about

and work with the graphics commands is to do it interactively. The description

and discussion of each command is meant only as a brief overview of some of

the capabilities of either SPSS or Stata that will produce figures comparable to

those discussed in this chapter. We encourage you to experiment and to explore

the range of features available in the graphics commands. On the Web site for

the text, we have included syntax files that provide examples of each of the

graphics commands in either SPSS (Chapter_3.sps) or Stata (Chapter_3.do).

SPSS

Frequency Distributions and Histograms

Frequency distributions and histograms are produced quite easily in SPSS through

the use of the FREQUENCIES command (which can also be shortened as

FREQ):

Following VARIABLES = would be a list of the variables for which you wanted

SPSS to compute frequency distributions. The /HISTOGRAM option will gen-

erate a histogram for the corresponding variables included in the list. You should

note that the default in SPSS is to construct intervals for the values of the varia-

ble to be graphed. The size and the number of intervals can be modified by edit-

ing the chart in SPSS, which is accomplished by double-clicking on the chart in

the SPSS output window. This will open an additional window that allows you to

make numerous changes to your chart and customize its appearance. We encour-

age you to experiment with many of the options available to you.

The GRAPH command with the /HISTOGRAM option provides an

alternative way of obtaining just a histogram, without the corresponding frequen-

cy distribution. The simplest way of generating a histogram using SPSS syntax is

the following:

The histogram generated by this command will be identical to that produced by

the FREQUENCIES command.

C O M P U T E R E X E R C I S E S 59

FREQUENCIES VARIABLES = variable_names /HISTOGRAM.

GRAPH /HISTOGRAM = variable_name.

This will generate a bar chart where each bar reflects the number of cases with

that value, much like the histogram, but with one important difference. Bar

charts are not sensitive to the values of the variable being graphed—the values

are treated as ordinal and are all positioned along the x-axis at constant intervals.

Histograms make use of the level of measurement, so that a variable measured at

the interval level of measurement will be appropriately scaled on the x-axis.

To produce a bar chart that includes bars for multiple groups, there are sev-

eral different ways of doing this in SPSS. A simple bar chart for multiple groups

can be obtained by again executing the GRAPH command. To generate a mul-

tiple group bar chart that uses percentages as the point of comparison across

groups, use the following command:

To generate a multiple group bar chart that uses the number of cases in each

group, use the following command:

Note that the type of graph is now /BAR(GROUPED), the percentage of cases

by the PCT option, the number of cases by the COUNT option, the variable

we are interested in plotting is listed after the first BY and the grouping variable

after the second BY option.

For example, if we were interested in plotting a measure of delinquency by

sex of respondent in a self-report survey using percentages as the comparison,

we would use the following command:

The resulting graph would have two bars—one for each sex—at each level of the

delinquency measure.

Pie Charts

Pie charts are also very easy to produce in SPSS with the GRAPH command, but

changing the type of chart to /PIE=

Similar to the bar chart options, rather than the number of cases (here the

COUNT option), you can also ask for SPSS to use percentages (PCT option in

Bar Charts

To have SPSS produce bar charts, you will need to use the GRAPH command

again, but simply change the option to /BAR=

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A60

GRAPH /BAR = variable_name.

GRAPH /BAR(GROUPED) = PCT BY variable_name BY group_var_

name.

GRAPH /BAR(GROUPED) = COUNT BY variable_name BY

group_var_name.

GRAPH /BAR(GROUPED) = PCT BY delinquency BY sex.

GRAPH /PIE = COUNT BY variable_name.

the command line, replacing COUNT). As with every other chart in SPSS, you

can edit various dimensions of the chart after it has been created and presented

in the output window.

Line Graphs

The fourth type of chart we highlight here is the line graph. Again, the GRAPH

command is used and the type of chart is changed to /LINE

This command will generate a simple line graph that defaults to plotting the

number of cases at each value of the variable being plotted—equivalent to the

simple bar chart, but with a line connecting what would have been the top of

each bar.

To produce a time plot, such as that seen in Figure 3.11, we begin with the

same GRAPH command:

The VALUE option tells SPSS to plot the value of the variable included in the

parentheses. So, in Figure 3.11, we plotted a crime rate measure—we would

simply include the name of that variable here. After the BY option, we would

include the name of the time variable, such as month, quarter, year, etc.

Figure 3.12 includes two lines that represent two different crime rates. To

include additional lines in a time plot in SPSS, we would simply add the addi-

tional variable name to a list included in the parentheses following the VALUE

option:

NOTE: SPSS will not produce a line graph with two y-axes, such as that found

in Figure 3.13.

Stata

Frequency Distributions and Histograms

Frequency distributions are computed in Stata with the tab1 command and have

the following format:

If more than one variable name is listed, multiple frequency distributions will be

computed. Note that some of the other options for computing frequency dis-

tributions in Stata will produce more complicated tables should more than one

variable be listed (e.g., the tab command).

C O M P U T E R E X E R C I S E S 61

GRAPH /LINE(SIMPLE) = variable_name.

GRAPH /LINE(SIMPLE) = VALUE(variable_name)

BY time_variable.

GRAPH /LINE(SIMPLE) = VALUE(variable_name1

tab1 variable_name(s)

variable_name2) BY time_variable.

Histograms are also easy to produce in Stata with the histogram command:

Note that you are limited to one variable at a time in running the histogram

command.

Bar Charts

Although Stata has an impressive set of graphics commands, the ability to cre-

ate simple bar charts like those discussed in this chapter is limited. Rather than

rely on one of the built-in Stata commands, we will use a user-written command

called fbar that is available for installation on your local computer running Stata.

The command to type and run (one time only) is:

This command will install fbar from an archive maintained by Stata and will make it

available to you while you’re running Stata.1 Once the command has been installed,

the structure to the fbar command is:

Where everything appearing after the comma is an option. The default for fbar

is to produce a bar chart using frequencies. To have it compute percentages, use

the percent option (which can be abbreviated perc). If there is an interest in

comparing bar charts for different groups, then use the by option and include the

grouping variable’s name in the parentheses and use the option totalpc, which

will calculate percentages in the multiple group bar chart using all of the catego-

ries within each group as the reference, as we have in this chapter (see Figure 3.8).

Similar to our discussion of multiple bar charts in SPSS, if we were to

compare delinquency counts by sex, we would run the following command:

Pie Charts

Pie charts are very easy to produce in Stata using the graph command:

Where the over(…) option tells Stata how to classify the information into

categories.

1 Throughout the text, we will make use of user-written procedures for Stata. The process will

be very much the same every time we execute one of these commands to make a procedure

available within Stata. If you are working on a computer in a student computer lab setting, you

will need to install these kinds of procedures every time you move to a different machine and

run Stata.

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A62

histogram variable_name

ssc install fbar

fbar variable_name, by(group_variable_name)

percent totalpc

fbar delinquency, by(sex) totalpc

graph pie, over(variable_name)

Where the y_variable_name and x_variable_name refer to the placement of the

variable on the two axes of the plot.

To produce a time plot, such as that seen in Figure 3.11, we begin with the

same twoway command, and just make explicit reference to the x-axis being

the indicator of time:

When we want to add an additional line to the time series plot, we just include

another line graph and pair of variable names in parentheses:

Note that the time variable should be the same, if the comparison is between two

(or more) measures taken at the same points in time.

One of the options available in Stata, but not SPSS, is the ability to add a

second y-axis, which as we saw in Figure 3.13 can be useful for comparing trends

measured on different scales. The key is to include a reference to the axis within

each set of parentheses in the twoway command:

Where the axis(#) is inserted following a comma that follows the name of the

time variable. The Stata do file for Chapter 3 (Chapter_3.do) illustrates how to

use this command to reproduce Figures 3.11 through 3.13.

Problems

1. Enter the data from Exercise 3.1.

a. Produce a frequency distribution table and a histogram.

b. How does the histogram produced by the statistical software differ

from the one you constructed?

c. Experiment with the interval length to see if you can reproduce in the

statistical software the chart you constructed by hand.

2. Enter the data from Exercise 3.3.

a. Produce two frequency distribution tables and two histograms, one

each for males and for females.

Line Graphs

The final chart we highlight here is the line graph, many of which can be

produced with the twoway command in Stata, which for simple examples takes

on the following form:

C O M P U T E R E X E R C I S E S 63

twoway (line y_variable_name x_variable_name)

twoway (line y_variable_name time_variable_name)

twoway (line y_var_name1 time_variable_name)

(line y_var_name2 time_variable_name)

twoway (line y_var_name1 time_variable_name, axis(1)) (line

y_var_name2 time_variable_name, axis(2))

b. Produce a bar chart that presents the information for males and

females simultaneously, in terms of

i. Number of cases

ii. Percentage of cases.

3. Enter the data from Exercise 3.6. Produce three time plots:

a. One for City A

b. One for City B

c. One that contains lines representing both City A and City B.

4. Open the NYS data file (nys_1.sav, nys_1_student.sav, or nys_1.dta).

a. Produce a frequency distribution table and a histogram for each of the

delinquency variables.

b. Describe the shapes of the distributions produced in part a.

c. Choose one of the delinquency variables:

i. Produce a bar chart using percentages for each gender. Describe how

the patterns are similar and different across gender.

ii. Produce a bar chart using percentages for each race. Describe how

the patterns are similar and different across race.

C H A P T E R T H R E E : R E P R E S E N T I N G A N D D I S P L A Y I N G D A T A64

d. Select two of the nominal or ordinal variables (excluding gender and

race).

i. Produce pie charts for both variables.

ii. Produce pie charts for both variables, distinguished by gender.

iii. Produce pie charts for both variables, distinguished by race.

iv. Describe how the patterns are similar and different across gender and

race.

Describing the Typical Case:

Measures of Central Tendency

What Information Does It Use?

What Information Does It Use?

C h a p t e r f o u r

T h e m o d e

T h e m e d i a n

T h e m e a n

How is It Calculated?

What are Its Advantages and Disadvantages?

How is It Calculated?

What are Its Advantages and Disadvantages?

How is It Calculated?

What are Its Advantages and Disadvantages?

What are Its Other Unique Properties?

What Information Does It Use?

D. Weisburd and C. Britt, Statistics in Criminal Justice, DOI 10.1007/978-1-4614-9170-5_4,

© Springer Science+Business Media New York 2014

THE NATURAL FIRST STEP in summarizing research is to provide a basic

portrait of the characteristics of a sample or population. What is the typ-

ical case? If the researcher could choose one case to represent all oth-

ers, which would it be? When a sample is very small, it is possible

merely to show the array of cases and let the reader decide. However,

as the number of cases grows, it becomes difficult to make a decision

about typicality from the distribution as a whole. This is the function of

measures of central tendency in statistics. They provide us with a simple

snapshot of our data that can be used to gain a picture of the average

case.

In this chapter, three commonly used measures of central tendency

are discussed and compared. The first, the mode, is used primarily with

nominal-level data. It is the simplest measure of central tendency, draw-

ing information only about the frequency of events in each category. The

second measure, the median, takes into account not only frequency but

also the order or ranking of study subjects. Finally, the mean adds the

additional factor of the exact scores associated with each subject studied.

As in the discussion of levels of measurement, we emphasize in this

chapter the benefits gained from statistics that use more information. But

we also illustrate the importance of looking carefully at the distribution

of cases in your study before deciding which measure of central ten-

dency is most appropriate.

T h e M o d e : C e n t r a l T e n d e n c y i n N o m i n a l S c a l e s

Faced with a nominal-scale measure, how would you define a typical

case? Take as an example Table 4.1. Here you have a nominal scale of

legal representation for a sample of offenders convicted of white-collar

crimes in U.S. federal courts. Offenders were placed into one of five cat-

egories, indicating the type of legal representation they had: no attorney

66

T H E M O D E : C E N T R A L T E N D E N C Y I N N O M I N A L S C A L E S 67

present, legal-aid attorney, court-appointed attorney, public defender,

and privately retained attorney. The number of individuals who fall in

each category—or, in statistical language, the N of cases—is reported.

Clearly, you have very limited information in this example on which

to base a choice about typicality. Here, as with other nominal-scale mea-

sures, you simply know how many cases fall into one category or an-

other. You would probably choose the category “private attorney” as

most representative of this sample, because it contains by far the most

cases (380). And indeed, this is precisely how statisticians define typical-

ity for nominal-level variables. We call the category with the largest N, or

number of cases, the mode. In this sample of white-collar offenders, the

modal category for type of representation is “private attorney.”

By defining one category as the modal category, we are able to pro-

vide a summary of the type of case that is typical of our sample or popu-

lation. Such statements are common in criminal justice research. We

often are interested in the racial category that appears most often in our

data or the type of offense that is most common. The modal category

can also provide a basis for making comparisons among samples. For

example, let’s say that a sample of offenders convicted of nonviolent

property crimes that would not ordinarily be defined as white collar was

compared to this larger sample of offenders convicted of white-collar

crimes. For the former group, as is apparent from Table 4.2, the modal

category is not “private attorney” but rather “court-appointed attorney.”

Although this comparison of the two samples is not a complex one, it

Legal Representation for White-Collar Crime

CATEGORY FREQUENCY (N)

No Attorney 20 Legal Aid 26 Court Appointed 92 Public Defender 153 Private Attorney 380 Total (�) 671

Table 4.1

Legal Representation for Common Crime

CATEGORY FREQUENCY (N)

No Attorney 40 Legal Aid 7 Court Appointed 91 Public Defender 22 Private Attorney 70 Total (�) 230

Table 4.2

68 C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

In general, we do not use the mode to describe central tendency with

ordinal or interval scales. The reason, in good part, is that the mode does

not take advantage of the additional information that such scales pro-

vide. The average case should not be chosen simply on the basis of the

frequency of events in a particular category, because higher-level scales

also provide information on the order or nature of the differences be-

tween categories.

Nonetheless, there are cases where researchers choose to use the

mode to describe ordinal- or interval-level measures. Generally this oc-

curs when there is a very large group of cases in one particular category.

Table 4.3, for example, provides an ordinal-level measure of the financial

harm caused by a sample of convicted offenders. Because almost two-

thirds of the individuals studied fall in the category “$101–$2,500,” you

might want to describe typicality in this case by saying that this category

is the modal category. Similarly, if you were examining prior arrests and

two-thirds of the offenders in your sample had no prior arrests, you

might want to report no arrests as the modal category. Even though this

measure is an interval measure, the mode provides a fairly good sum-

mary of the typical case in your sample.

T h e M e d i a n : T a k i n g i n t o A c c o u n t P o s i t i o n

In constructing the median, we utilize information not only on the num-

ber of cases found in a particular category, but also on the positions of

the categories. The median may be defined simply as the middle score in

a distribution. For ordinal scales, it is the category in which the middle

score lies. For interval scales, the median is the value that splits the dis-

tribution of scores in half.

There are two general steps in determining the median for a distribu-

tion of scores. First, the values need to be arranged from low to high

Financial Harm for a Sample of Convicted Offenders

CATEGORY FREQUENCY (N)

Less than $100 15 $101–$2,500 92 $2,501–$10,000 20 More than $10,000 19 Total (�) 146

Table 4.3

illustrates the different backgrounds of the two groups. White-collar

offenders are much more likely than common criminals to have the

resources to pay for private legal representation.

T H E M E D I A N : T A K I N G I N T O A C C O U N T P O S I T I O N 69

scores. As we saw in Chapter 3, a frequency distribution allows us to

represent our data in this way. Table 4.4 presents a frequency distribu-

tion of views of public drunkenness, drawn from a survey of students.

The students were presented with an ordinal-scale measure that allowed

them to rate the seriousness of a series of crimes. The ratings ranged

from “not serious at all” to “most serious.”

Second, we need to determine which observation splits the distribu-

tion. A simple formula, Equation 4.1, allows us to define which observa-

tion is the median when the number of observations in the distribution is

odd, as is the case with our example of views of public drunkenness.

Equation 4.1

In this case, we add 1 to the total number of observations in the sample

or population we are studying and then divide by 2. For the frequency

distribution in Table 4.4, the median observation is the 141st score:

Median observation � N � 1

2

Student Views on Public Drunkenness

CATEGORY FREQUENCY (N) CUMULATIVE N

Not serious at all 73 73 A bit serious 47 120 Somewhat serious 47 167 Serious 27 194 Very serious 26 220 Extremely serious 39 259 Most serious 22 281 Total (�) 281

Table 4.4

W orking It Out

� 141

� 281 � 1

2

Median observation � N � 1

2

However, because our variable, student views on drunkenness, is mea-

sured on an ordinal scale, it does not make sense to simply state that the

141st observation is the median score. To give a substantive meaning to

the median, it is important to define which category the median score

C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

falls in. The 141st observation in our distribution of ordered scores falls

in the category labeled “somewhat serious.”

The advantage of the median over the mode for describing ordinal

scales is well illustrated by our example of views of public drunkenness.

If we used the mode to describe typicality in student assessments of the

seriousness of public drunkenness, we would conclude that the typical

student did not see drunkenness as at all serious. But even though the

“not serious at all” category includes the largest number of cases, almost

three-quarters of the students rate this behavior more seriously. The me-

dian takes this fact into consideration by placing the typical case in the

middle of a distribution. It is concerned with not only the number of

cases in the categories, but also their position.

If the number of observations or cases in your distribution is even, then

you cannot identify a single observation as the median. While statisticians

recognize that the median is ambiguously defined in this case, by conven-

tion they continue to use Equation 4.1 to identify the median for an ordi-

nal-level distribution. In practice, this places the median score between two

observations. For example, consider the distribution of 146 scores in Table

4.3, representing financial harm in a sample of offenders. Here the number

1

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73rd observation: $101–$2,500

74th observation: $101–$2,500

� 73.5

� 146 � 1

2

Median observation � N � 1

2

1With this method, it is possible that the defined median value will fall between two

categories of an ordinally measured variable. In that case, you simply note that the

median falls between these two categories.

The median is sometimes used for defining typicality with interval

scales. For example, Table 4.5 presents the average number of minutes

of public disorder (per 70-minute period) observed in a sample of 31

of scores is even, and thus there is not a single observation that can be

defined as the median. Using Equation 4.1, we can see that the median is

defined as the halfway point between the 73rd and the 74th observation.

This means that the median falls in the category defined as $101 to $2,500.

70

T H E M E D I A N : T A K I N G I N T O A C C O U N T P O S I T I O N

“hot spots of crime,” or city blocks with high levels of crime. The hot

spots are arranged in ascending order on the basis of the number of

minutes of disorder observed. In this case, the distribution has an odd

number of observations, and thus the median is the score in the middle

of the distribution, or the 16th observation, which has a value of 2.12.

Hot Spots: Minutes of Public Disorder (A)

HOT SPOT

SCORE FREQUENCY (N) CUMULATIVE (N)

0.35 1 1 0.42 1 2 0.46 1 3 0.47 1 4 0.52 1 5 0.67 1 6 1.00 1 7 1.06 1 8 1.15 1 9 1.19 2 11 1.48 1 12 1.60 1 13 1.63 1 14 2.02 1 15 2.12 1 16 2.21 1 17 2.34 1 18 2.45 1 19 2.66 1 20 3.04 1 21 3.19 1 22 3.23 1 23 3.46 1 24 3.51 1 25 3.72 1 26 4.09 1 27 4.47 1 28 4.64 1 29 4.65 1 30 6.57 1 31 Total (�) 31 31

Table 4.5

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� 16

� 31 � 1

2

Median observation � N � 1

2

71

C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

Accordingly, using the median, we would describe the average hot spot as

having a little more than two minutes of disorder in each 70-minute period.

As noted above, when the number of observations in a distribution is

even, the median is ambiguously defined. Let’s, for example, delete the hot

spot with a score of 6.57 from Table 4.5. In this case, there is no single mid-

dle value for the array of cases in the table. If we use Equation 4.1 to define

the median observation, we get a value of 15.5.

2Sometimes the median for ordinal-level variables is also calculated using this method.

In such cases, the researcher should realize that he or she is treating the variable

under consideration as an interval-level measure. Only for an interval-level measure

can we assume that the units of measurement are constant across observations.

W orking It Out

� 15.5

� 30 � 1

2

Median observation � N � 1

2

But what is the value or score associated with an observation that lies

between two scores in an interval-level scale? If both the 15th and the

16th observation are in the same category, then the solution is easy. You

simply define the median as the score associated with both the 15th and

the 16th observation. However, it will sometimes be the case with an

interval-level variable that each of these observations will have a differ-

ent value on the scale, as we find here. There is no true median value

for this example. By convention, however, we define the median with

interval-level measures as the midpoint between the observation directly

below and the observation directly above the median observation. In our

example, this is the midpoint on our scale between the scores 2.02 and

2.12. The median in this case is defined as 2.07.2

W orking It Out

� 2.07

Median � 2.02 � 2.12

2

16th case � 2.12

15th case � 2.02

72

T H E M E D I A N : T A K I N G I N T O A C C O U N T P O S I T I O N

The median is generally more appropriate than the mode for assess-

ing central tendency for both ordinal- and interval-level measures. How-

ever, the median does not take advantage of all the information included

in interval-level scales. Although it recognizes the positions of the values

of a measure, it does not take into account the exact differences among

these values. In many cases, this can provide for a misleading estimate of

typicality for interval-level measures.

For example, let’s say that the distribution of disorder in hot spots is

that represented in Table 4.6. In this case, the median is 1.83. But is 1.83

a good estimate of central tendency for this distribution? The 17th score

is 3.34, which is not very close to 1.83 at all. The score of 1.83 is not an

the values of the 16th and 17th cases. This is because it looks only at

Hot Spots: Minutes of Public Disorder (B)

HOT SPOT

SCORE FREQUENCY (N) CUMULATIVE (N) CUMULATIVE %

0.35 1 1 3.2 0.42 1 2 6.5 0.46 1 3 9.7 0.47 1 4 12.9 0.52 1 5 16.1 0.67 1 6 19.4 1.00 1 7 22.6 1.06 1 8 25.8 1.15 1 9 29.0 1.19 2 11 35.5 1.48 1 12 38.7 1.60 1 13 41.9 1.63 1 14 45.2 1.73 1 15 48.4 1.83 1 16 51.6 3.34 1 17 54.9 3.44 1 18 58.1 3.45 1 19 61.3 3.66 1 20 64.5 4.04 1 21 67.7 4.19 1 22 71.0 4.23 1 23 74.2 4.46 1 24 77.4 4.51 1 25 80.6 4.72 1 26 83.9 5.09 1 27 87.1 5.47 1 28 90.3 5.64 1 29 93.5 5.65 1 30 96.8 5.57 1 31 100.0 Total (�) 31 31 100.0

Table 4.6

bution. The median is not sensitive to the gap in our measure between

ideal estimate of typicality, as it is far below half the scores in the distri-

73

C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

position and not at the size of the differences between cases. The me-

dian does not take advantage of all the information provided by interval-

level measures.

Another way to describe the median in interval-level measures is to

say that it is the 50th percentile score. A percentile score is the point or

score below which a specific proportion of the cases is found. The 50th

percentile score is the score below which 50% of the cases in a study lie.

For the data in Table 4.6, if we defined the median in this way, we again

choose 1.83 as the median minutes of disorder observed in the hot spots.

In this case, if we add the percentage of cases for all of the scores up

until the middle, or 16th, score, we come to a total (or cumulative per-

centage) of 51.6. At the 15th score, or 1.73, the cumulative percentage is

only 48.4, less than 50%.

T h e M e a n : A d d i n g V a l u e t o P o s i t i o n

The mean takes into account not only the frequency of cases in a cate-

gory and the positions of scores on a measure, but also the values of

these scores. To calculate the mean, we add up the scores for all of the

subjects in our study and then divide the total by the total number of

subjects. In mathematical language, the mean can be written as a short

equation:

Equation 4.2

Even though equations sometimes put students off, they are an im-

portant part of statistics. Indeed, equations are the language of statistics.

They show how a statistic is constructed and the method we use to cal-

culate it. Equations provide a short way of writing out what would often

take a number of sentences to describe in English. One of our tasks in

this text is to help you to translate such equations and to become more

comfortable with them.

In the case of the mean, we introduce what are for most students of

criminal justice some new symbols and concepts. First, to express the

mean, statisticians provide us with a shorthand symbol, —in English,

“X bar.” The equation also includes the summation symbol, �. Under the

symbol is i � 1, and above it is N. What this means is that you should

start summing your cases with the first subject in the sample and end

X

X � � N

i�1

Xi

N

74

T H E M E A N : A D D I N G V A L U E T O P O S I T I O N

with the last one (represented by N because, as we have already dis-

cussed, N is the number of cases in your sample). But what should you

sum? X represents the measure of interest—in the case of our example,

minutes of disorder. We use the subscript i to denote each of the obser-

vations of the variable X. If, for example, we wrote X3, we would be re-

ferring only to the 3rd observation of the variable. So Equation 4.2 says

that you should sum the scores for minutes of disorder from the first to

the last case in your study. Then you should divide this number by the

total number of cases.

Table 4.7 presents information about the total number of prior arrests

for a sample of 20 individuals arrested for felony offenses. To calculate

the mean, we first sum all of the scores, as shown in the numerator of

Equation 4.2:

Total Number of Prior Arrests

TOTAL NUMBER

OF ARRESTS FREQUENCY (N) CUMULATIVE (N)

0 4 4 1 1 5 2 2 7 4 3 10 5 3 13 7 4 17 8 2 19

10 1 20

Table 4.7

W orking It Out

� 86

� 5 � 5 � 5 � 7 � 7 � 7 � 7 � 8 � 8 � 10

� 0 � 0 � 0 � 0 � 1 � 2 � 2 � 4 � 4 � 4

� N

i�1

Xi � � 20

i�1

Xi

We then take the sum of the values, 86, and divide by the number of ob-

servations in the sample.

75

76 C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

The result, 4.3, tells us that in this sample the typical person arrested for

a felony has, on average, 4.3 prior arrests.

As another example, let’s take the data from Table 4.5 on minutes of

disorder in crime hot spots. According to Equation 4.2, the first step is to

sum all of the scores:

W orking It Out

� 4.3

� 86 20

X � � N

i�1

Xi

N

W orking It Out

� 71.56

� 3.51 � 3.72 � 4.09 � 4.47 � 4.64 � 4.65 � 6.57

� 2.21 � 2.34 � 2.45 � 2.66 � 3.04 � 3.19 � 3.23 � 3.46

� 1.15 � 1.19 � 1.19 � 1.48 � 1.60 � 1.63 � 2.02 � 2.12

� 0.35 � 0.42 � 0.46 � 0.47 � 0.52 � 0.67 � 1.00 � 1.06

� N

i�1

Xi � � 31

i�1

Xi

We then take this number, 71.56, and divide it by N, or 31, the number

of cases in our sample.

W orking It Out

� 2.308387097

� 71.56

31

X � � N

i�1

Xi

N

T H E M E A N : A D D I N G V A L U E T O P O S I T I O N 77

The result, 2.308387097 (rounded to the ninth decimal place), brings up

an issue that often arises in reporting statistics. Do you really need to

provide your audience with the level of precision that is given by your

statistic? In this case, for example, at what level of precision should min-

utes of disorder be presented?

A basic rule of thumb is to use your common sense in answering such

questions. Don’t provide statistics developed out to a large number of

decimal places just to impress others. In making this decision, you

should ask: What is the simplest presentation of my results that will pro-

vide the reader or listener with enough information to understand and

evaluate my work? Overall, criminal justice researchers seldom report the

mean to more than two decimal places. This is a good choice in our ex-

ample. Rounding to the second decimal place gives a mean of 2.31. Pro-

viding a more precise representation of the mean here would not add

important information for the reader.

In some cases, it is useful to develop estimates with much greater pre-

cision. In particular, if the values for the cases you are examining are

very small in the first place, you will want to present a more precise

mean. For example, Lawrence Sherman and his colleagues looked at the

mean daily rate of reported domestic violence in a study that compared

the impact of arrests versus warnings as a strategy for controlling spouse

abusers.3 Had they reported their findings only to the second decimal

place, as recommended above, they would have ended up with a mean

daily rate over the longest follow-up period (361–540 days) of 0.00 for

short arrest and 0.00 for warning. The difficulty here is that individuals

are unlikely to report cases of domestic violence on a very frequent

basis. Sherman et al. needed a much higher degree of precision to exam-

ine the differences between the two groups they studied. Accordingly,

they reported their results to the fourth decimal place. For arrests, the

rate was 0.0019, and for warnings it was 0.0009. These differences,

though small, were found to be meaningful in their research.

Comparing Results Gained Using the Mean and Median

Returning to the example from Table 4.5, we see that the mean for min-

utes of disorder, 2.31, is very similar to the median of 2.12 calculated

earlier. In this case, adding knowledge about value does not change our

portrait of the typical hot spot very much. However, we get a very differ-

ent sense of the average case if we use the data from Table 4.6. Here,

the median provided a less than satisfying representation of the average

3L. Sherman, J. D. Schmidt, D. Rogan, P. Gartin, E. G. Cohn, D. J. Collins, and A. R.

Bacich, “From Initial Deterrence to Long-Term Escalation: Short-Custody Arrest for

Poverty Ghetto Domestic Violence,” Criminology 29 (1991): 821–850.

78 C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

case. It was not sensitive to the fact that there was a large gap in the

scores between the 16th and 17th cases. Accordingly, the median, 1.83,

was very close in value to the first half of the cases in the sample, but

very far from those hot spots with higher values. The mean should pro-

vide a better estimate of typicality here, because it recognizes the actual

values of the categories and not just their positions. Let’s see what hap-

pens when we calculate the mean for Table 4.6.

Following our equation, we first sum the individual cases:

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� 84.41

� 4.51 � 4.72 � 5.09 � 5.47 � 5.64 � 5.65 � 5.77

� 3.34 � 3.44 � 3.45 � 3.66 � 4.04 � 4.19 � 4.23 � 4.46

� 1.15 � 1.19 � 1.19 � 1.48 � 1.60 � 1.63 � 1.73 � 1.83

� 0.35 � 0.42 � 0.46 � 0.47 � 0.52 � 0.67 � 1.00 � 1.06

� N

i�1

Xi � � 31

i�1

Xi

We then divide this number by the total number of cases:

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� 2.7229

� 84.41

31

X � � N

i�1

Xi

N

Here, we gain an estimate of typicality of 2.72 (rounding to the second

decimal place). As you can see, this score is much better centered in

our distribution than is the median. The reason is simple. The median

does not take into account the values of the categories. The mean does

take value into account and thus is able to adjust for the gap in the

distribution.

T H E M E A N : A D D I N G V A L U E T O P O S I T I O N 79

There are cases in which the sensitivity of the mean to the values of

the categories in a measure can give misleading results. For example,

let’s say that one case in your study is very different from the others. As

noted in Chapter 1, researchers call such a case an outlier, because it is

very much outside the range of the other cases you studied. Taking the

example of minutes of disorder from Table 4.5, let’s say that the last case

had 70 minutes of disorder (the maximum amount possible) rather than

6.57 minutes. When we calculate the mean now, the sum of the cases is

much larger than before:

W orking It Out

� 134.99

� 3.51 � 3.72 � 4.09 � 4.47 � 4.64 � 4.65 � 70.0

� 2.21 � 2.34 � 2.45 � 2.66 � 3.04 � 3.19 � 3.23 � 3.46

� 1.15 � 1.19 � 1.19 � 1.48 � 1.60 � 1.63 � 2.02 � 2.12

� 0.35 � 0.42 � 0.46 � 0.47 � 0.52 � 0.67 � 1.00 � 1.06

� N

i�1

Xi � � 31

i�1

Xi

Dividing this sum by the total number of cases provides us with a mean

of 4.35 (rounded to the second decimal place):

W orking It Out

� 4.3545

� 134.99

31

X � � N

i�1

Xi

N

The mean we calculated with the original score was 2.31 (see page 76).

Accordingly, merely by changing one score to an outlier, we have almost

doubled our estimate of typicality. In this case, the sensitivity of the

mean to an extreme value in the distribution led it to overestimate the

C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

average case. This illustrates the general principle that the mean is sensi-

tive to outliers. Because the mean is used to develop many other more

complex statistics, this principle is relevant not only to the mean itself

but also to a number of other important statistical techniques used by re-

searchers.

So what should you do if outliers lead to a misleading conclusion re-

garding typicality in your study? One solution is simply to exclude the

outliers from specific analyses and let your readers or audience know

that some cases have been excluded and why. If the number of extreme

cases is large enough, you may want to analyze these cases separately.

Other Characteristics of the Mean

Two other traits of the mean are important because they play a role in

how we develop other statistics. The first concerns what happens when

we look at deviations (or differences) from the mean. This will be-

come an issue in the next chapter, when we discuss measures of disper-

sion. The second, often termed the least squares property of the

mean, will become important to us in Chapter 15, when we discuss re-

gression.

If we take each score in a distribution, subtract the mean from it, and

sum these differences, we will always get a result of 0. In equation form,

this principle is represented as follows:

Equation 4.3

In English, this equation says that if we sum the deviations from the

mean, from the first to the last case, we will always get a result of 0. This

principle is illustrated in Table 4.8, using the data on minutes of public

disorder from Table 4.5. Here we have taken the 31 scores and sub-

tracted the mean from each one. We then added these differences. Be-

cause the positive scores balance out the negative ones, the result is 0.

This will always happen when we use the mean.

The second trait, the least squares property, is very important for un-

derstanding regression analysis (introduced in Chapter 15), a technique

commonly used for describing relationships among variables in criminal

justice. For the moment, it is enough to note this fact and that the issues

� N

i�1

(Xi � X ) � 0

Another solution is to transform the outliers. That is, you may want to

highest value that is not an outlier). In this way, you can include the

replace them with values closer to the rest of the distribution (e.g., the

cases, but minimize the extent to which they affect your estimate of

typicality. However, you should be cautious in developing such trans-

formations of your scores, keeping in mind that you are changing the

character of the distribution examined in your study.

80

T H E M E A N : A D D I N G V A L U E T O P O S I T I O N

we address early on in statistics are often the bases for much more com-

plex types of analysis. “Don’t forget the basics” is a good rule. Many mis-

takes that researchers make in developing more complex statistics come

from a failure to think about the basic issues raised in the first few chap-

ters of this text.

The least squares property is written in equation form as follows:

Equation 4.4

What this says in English is that if we sum the squared deviations from

the mean for all of our cases, we will get the minimum possible result.

That is, suppose we take each individual’s score on a measure, subtract

� N

i�1

(Xi � X )2 � minimum

Deviations from the Mean for Minutes of Public Disorder (A)

SCORE (X) DEVIATION FROM THE MEAN (Xi � )

0.35 0.35 � 2.31 � �1.96 0.42 0.42 � 2.31 � �1.89 0.46 0.46 � 2.31 � �1.85 0.47 0.47 � 2.31 � �1.84 0.52 0.52 � 2.31 � �1.79 0.67 0.67 � 2.31 � �1.64 1.00 1.00 � 2.31 � �1.31 1.06 1.06 � 2.31 � �1.25 1.15 1.15 � 2.31 � �1.16 1.19 1.19 � 2.31 � �1.12 1.19 1.19 � 2.31 � �1.12 1.48 1.48 � 2.31 � �0.83 1.60 1.60 � 2.31 � �0.71 1.63 1.63 � 2.31 � �0.68 2.02 2.02 � 2.31 � �0.29 2.12 2.12 � 2.31 � �0.19 2.21 2.21 � 2.31 � �0.10 2.34 2.34 � 2.31 � �0.03 2.45 2.45 � 2.31 � �0.14 2.66 2.66 � 2.31 � �0.35 3.04 3.04 � 2.31 � �0.73 3.19 3.19 � 2.31 � �0.88 3.23 3.23 � 2.31 � �0.92 3.46 3.46 � 2.31 � �1.15 3.51 3.51 � 2.31 � �1.20 3.72 3.72 � 2.31 � �1.41 4.09 4.09 � 2.31 � �1.78 4.47 4.47 � 2.31 � �2.16 4.64 4.64 � 2.31 � �2.33 4.65 4.65 � 2.31 � �2.34 6.57 6.57 � 2.31 � �4.26 Total (�) 0*

*Because of rounding error, the actual column total is slightly less than zero.

X

Table 4.8

81

C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

the mean from that score, and then square the difference. If we then

sum all of these values, the result we get will be smaller than the result

we would have gotten if we had subtracted any other number besides

the mean. You might try this by calculating the result for minutes of dis-

order using the mean. Then try other values and see if you can find

some other number of minutes that will give you a smaller result. The

least squares property says you won’t.

Using the Mean for Noninterval Scales

The mean is ordinarily used for measuring central tendency only with in-

terval scales. However, in practice, researchers sometimes use the mean

with ordinal scales as well. Is this wrong? In a pure statistical sense, it is.

However, some ordinal scales have a large number of categories and

thus begin to mimic some of the characteristics of interval-level mea-

sures.

This is particularly true in cases where the movements from one cate-

gory to another in an ordinal scale can be looked at as equivalent, no

matter which category you move from. Taking our example of student

attitudes toward public drunkenness in Table 4.4, a researcher might

argue that the difference between “somewhat serious” and “a bit serious”

is about equivalent to that between “very serious” and “extremely seri-

ous,” and so forth. Thus, the difference between these categories is not

just a difference of position; it is also a movement of equal units up the

scale. Taking this approach, we can say that this measure takes into ac-

count both position and value, although the values here are not as

straightforward as those gained from true interval scales such as number

of crimes or dollar amount stolen.

A researcher might argue that the mean is appropriate for presenting

findings on views of public drunkenness because this ordinal-scale mea-

sure of attitudes is like an interval-scale measure. Although it is easy to

see the logic behind this decision, it is important to note that such a de-

cision takes a good deal of justification. In general, you should be very

cautious about using the mean for ordinal-level scales, even when the

above criteria are met.

S t a t i s t i c s i n P r a c t i c e : C o m p a r i n g t h e M e d i a n a n d t h e M e a n

The general rule is that the mean provides the best measure of central

tendency for an interval scale. This follows a principle stated in Chap-

ter 1: In statistics, as in other decision-making areas, more information

is better than less information. When we use the mean, we take into

82

S T A T I S T I C S I N P R A C T I C E

tion, but also the values or scores of those categories. Because more in-

formation is used, the mean is less likely than other measures of central

tendency to be affected by changes in the nature of the sample that a re-

searcher examines. It is useful to note as well that the mean has some al-

gebraic characteristics that make it more easily used in developing other

types of statistics.

The mean is generally to be preferred, but when the distribution of a

variable is strongly skewed, the median provides a better estimate of

central tendency than the mean. “Skewed” means that the scores on the

cases. A distribution that has extreme values lower than the main cluster

of observations (i.e., there is a “tail” to the left in the distribution) is said

to be negatively skewed, while a distribution that has extreme values

greater than the main cluster of observations (i.e., there is a “tail” to the

right in the distribution) is said to be positively skewed.4

A good example of a skewed distribution in criminal justice is crimi-

nal history as measured by self-reports of prisoners. Horney and Mar-

shall, for example, reported results on the frequency of offending for a

sample of prisoners.5 As is apparent from Figure 4.1, most of the offend-

ers in their sample had a relatively low offending rate—between 1 and

20 offenses in the previous year. But a number of offenders had rates of

more than 100, and a fairly large group had more than 200. The mean

for this distribution is 175.

Clearly, 175 offenses provides a misleading view of typical rates of of-

fending for their sample. Because the mean is sensitive to value, it is in-

flated by the very high frequency scores of a relatively small proportion

of the sample. One solution suggested earlier to the problem of outliers

4A formal statistic for measuring the degree of skewness of a distribution is given by

the following equation:

In words, this equation tells us to take the deviation between a value and the mean

and cube it, then sum these values over all observations; the sum of the cubed devia-

tions is then divided by the sample size (N) multiplied by the standard deviation

cubed. The measure of skewness will have a value of 0 if the distribution is symmetri-

cal, a negative value if the distribution is negatively skewed, and a positive value if

the distribution is positively skewed. The greater the value of the measure, the greater

the degree of positive or negative skewness. 5J. Horney and I. H. Marshall, “An Experimental Comparison of Two Self-Report Meth-

ods for Measuring Lambda,” Journal of Research in Crime and Delinquency 29 (1992):

102–121.

skewness � (Xi � X )3

Ns3

account not only the frequency of events in each category and their posi-

variable are very much weighted to one side and that frequencies of

extreme values trail off in one direction away from the main cluster of

83

� N

i�1

C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

was to exclude such cases. But here, this would mean excluding almost

30% of the sample. Thus, these are not outliers in the traditional sense.

Another option mentioned earlier is to analyze the “outliers” separately.

But again, there is quite a spread of scores even if we look at those

above 50 or 100 separately, and the analysis of the outliers might in itself

provide a misleading view of central tendency. A common solution used

for describing this type of skewed interval-level distribution is to use the

median rather than the mean to describe central tendency. The median

for this distribution is 4, which is certainly more representative of the av-

erage case than is the mean. But even if you choose this solution, it is

very important to note to your audience that the distribution is skewed

and to tell them a bit about the nature of the distribution.

How should you decide when a distribution is so skewed that it is

preferable to use the median as opposed to the mean? You should begin

by comparing the mean and the median. When there is a very large dif-

ference between them, it may be the result of skewness. In such cases,

you should look at the distribution of the scores to see what is causing

the mean and median to differ widely. But there is no solid boundary

line to guide your choice.

cases where the mean and median provide relatively close estimates,

P er

ce n

t of

R es

p on

d en

ts

Frequency of Offending

1–20

0

20

40

60

80

21–40 41–60 61–80 81–100 101–120 121–140 141–160 161–180181–200 200+

Individual Frequency of Offending for a Sample of Offenders:

A Case Where the Mean Is a Misleading Measure of Central Tendency Figure 4.1

your choice will be clear. In the former case you would choose the

84

In extreme cases (such as that of criminal history in our example) or

C H A P T E R S U M M A R Y

where in between, you will have to use common sense and the experi-

ences of other researchers working with similar data as guidelines. What

seems to make sense? What have other researchers chosen to do? One

way of being fair to your audience is to provide results for both the

mean and the median, irrespective of which you choose as the best mea-

sure of typicality.

C h a p t e r S u m m a r y

The mode is calculated by identifying the category that contains the

greatest number of cases. It may be applied to any scale of measure-

ment. Because the mode uses very little information, it is rarely used

with scales of measurement higher than the nominal scale. It can occa-

sionally serve as a useful summary tool for higher-level scales, however,

when a large number of cases are concentrated in one particular cate-

gory.

The median is calculated by locating the middle score in a distribu-

tion and identifying in which category it falls. It is also known as the

50th percentile score, or the score below which 50% of the cases lie. The

information used includes both the number of cases in a particular cate-

gory and the positions of the categories. The median uses more informa-

tion than does the mode and requires a scale of measurement that is at

least ordinal in magnitude.

The mean is calculated by dividing the sum of the scores by the

number of cases. The information used includes not only the number of

cases in a category and the relative positions of the categories, but also

the actual value of each category. Such information normally requires at

least an interval scale of measurement. For this reason, the researcher

should be cautious about using the mean to describe an ordinal scale.

The mean uses more information than the mode and the median. It is,

however, sensitive to extreme cases—outliers. Faced with the distorting

effect of outliers, the researcher may choose to keep them, to transform

them to other values, or to delete them altogether. If a distribution of

scores is substantially skewed, then it may be more appropriate to use

the median than to use the mean.

The sum derived by adding each score’s deviation from the mean

will always be 0. If the deviation of each score from the mean is

squared, then the sum of these squares will be less than it would be if

any number other than the mean were used. This is called the least

squares property.

median, and in the latter the mean. However, when your results fall some-

85

86 C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

K e y T e r m s

deviation from the mean The extent to

which each individual score differs from

the mean of all the scores.

least squares property A characteristic

of the mean whereby the sum of all the

squared deviations from the mean is a

minimum—it is lower than the sum of the

squared deviations from any other fixed

point.

mean A measure of central tendency cal-

culated by dividing the sum of the scores

by the number of cases.

median A measure of central tendency cal-

culated by identifying the value or category

of the score that occupies the middle posi-

tion in the distribution of scores.

mode A measure of central tendency cal-

culated by identifying the score or category

that occurs most frequently.

outlier(s) A single or small number of ex-

ceptional cases that substantially deviate

from the general pattern of scores.

skewed Describing a spread of scores that

is clearly weighted to one side.

S y m b o l s a n d F o r m u l a s

X Individual score

Mean

N Number of cases

� Sum

To calculate the median observation:

To calculate the mean:

To show how the sum of the deviations from the mean equals 0:

To express the least squares property:

� N

i�1

(Xi � X)2 � minimum

� N

i�1

(Xi � X) � 0

X � � N

i�1

Xi

N

Median observation � N � 1

2

X

E X E R C I S E S 87

E x e r c i s e s

4.1 Drivers cited for moving violations are required by a state’s laws to take a driving safety course taught by the local police department. The sign-in sheet asks individuals to note why they received a ticket. The 14 participants at a recent class noted the following:

Speeding, Running a red light, Running a stop sign, Speeding, Speeding, Running a red light, Tailgating, Speeding, Running a red light, Recklessness, Speeding, Running a red light, Speeding, Running a stop sign

a. Categorize these data and calculate the mode.

b. Explain why the median would not be an appropriate measure of central tendency for these data.

4.2 Calculate the mode, median, and mean for the following data:

a. Number of previous employments held by 25 convicts:

3 3 1 1 0 1 0 2 1 0 8 4 3

1 2 1 9 0 1 7 0 7 2 0 1

b. Weeks of training undergone by 20 prison guards:

10 16 12 16 16 16 10 8 10 12

16 18 12 16 16 8 0 12 10 16

c. Height (in meters) of 30 convicts:

1.72 1.78 1.73 1.70 1.81 1.64 1.76 1.72 1.75 1.74

1.88 1.79 2.01 1.80 1.77 1.79 1.69 1.74 1.75 1.66

1.77 1.73 1.72 1.91 1.80 1.74 1.72 1.82 1.86 1.79

4.3 A researcher checked the response times of police to ten emergency telephone calls. The data below record the number of minutes that elapsed from when the telephone call ended to when the police arrived:

24 26 14 27 198 22 27 17 19 29

a. Calculate the mode, the median, and the mean.

b. Which of these measures is the most suitable for this particular case? Explain your choice.

4.4 Airport officials wished to check the alertness of their security officers over the two busiest weeks of the summer. During this period, they sent out 50 undercover staff carrying suspicious items of hand lug- gage. Five of them were stopped at the entrance to the airport. Six made it into the airport, but were stopped at check-in. Thirteen more got into the airport and through check-in, only to be stopped at the

88 C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

hand-luggage inspection point. Two passed the airport entrance, check-in, and hand-luggage inspection, but were stopped when presenting their boarding cards at the gate. Four people made it past every one of these stages, only to be stopped when boarding the plane. Twenty of the undercover staff were not detected at all.

a. Categorize the data and calculate the median category.

b. Is the median a good measure of central tendency in this case? Ex- plain your answer. If you think it is not, suggest an alternative and explain why.

4.5 On the first day of the term in a statistics course, the professor admin- istered a brief questionnaire to the students, asking how many statis- tics courses they had ever taken before the current term. Of the 33 students who answered the question, 17 said none, 9 said one, 3 said two, 2 said three, 1 said four, and 1 said five.

a. Calculate the mode, median, and mean for number of prior statis- tics classes.

b. Which one of these measures of central tendency best measures the typicality of these data?

4.6 As part of her undergraduate thesis, a criminal justice student asked ten other criminal justice majors to rate the fairness of the criminal jus- tice system. The students were asked to say whether they strongly agreed, agreed, were uncertain, disagreed, or strongly disagreed with the following statement: “The criminal justice system in our country treats all defendants fairly.” The ten responses were

Strongly agree, Strongly agree, Strongly disagree, Strongly disagree, Uncertain, Disagree, Disagree, Agree, Strongly disagree, Uncertain

a. Categorize these data and calculate an appropriate measure of cen- tral tendency.

b. Explain why this measure of central tendency best represents the typicality of these data.

4.7 There are five prisoners in the high-security wing in a prison—Albert, Harry, Charlie, Dave, and Eddie. Only Eddie’s biographical details have been lost. The information available is as follows:

Age Previous Convictions

Albert 23 1

Harry 28 4

Charlie 18 1

Dave 41 1

Eddie ? ?

E X E R C I S E S 89

a. Can we compute any of the following for previous convictions of the five prisoners: the mode, the median, or the mean? If any (or all) of these three measures may be calculated, what are their values?

b. If the mean number of previous convictions is 2.0, how many con- victions does Eddie have?

c. If we know that the median age for the five prisoners is 28, what does this tell us about Eddie’s age? Explain why.

d. If the mean age for the five prisoners is 28.2, how old is Eddie?

4.8 A researcher sat on a bench along the main shopping street of a city center on ten successive Saturdays from 11:00 A.M. to 2:00 P.M.—the three busiest shopping hours of the day—and recorded the number of times a police officer passed by. The results for the ten weeks are as follows:

Week No.: 1 2 3 4 5 6 7 8 9 10

No. of Police Officers Observed: 4 3 6 4 4 5 4 35 3 5

On week 8, local unionists held a demonstration in the city center, and the high number of observations for that week can be explained by the extra officers called in to police the rally.

b. Which measure of central tendency best represents typicality for these data? Discuss the issues involved in choosing the most appro- priate means of describing the data.

c. Imagine that the unionists had decided to hold a regular demon- stration in the city center on alternating weeks. The results recorded for the same study would be as follows:

Week No.: 1 2 3 4 5 6 7 8 9 10

No. of Police Officers Observed: 4 30 6 31 6 52 4 35 4 34

4.9 On a recent evening, a police crackdown on prostitution solicitation resulted in 19 arrests. The ages of the persons arrested were

17 18 24 37 32 49 61 20 21 21

25 24 24 26 30 33 35 22 19

a. Calculate an appropriate measure of central tendency.

b. Explain why this measure of central tendency best represents typi- cality for these data.

officers observed. a. Calculate the mode, median, and mean for the number of police

Would the measure of central tendency you recommended in part b still be the best measure of typicality? Explain why.

C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E

4.10 Using your answers from part a of Exercise 4.5, calculate

a. The sum of the deviations from the mean.

b. The sum of the squared deviations from the mean.

c. The sum of the squared deviations from the median.

d. The sum of the squared deviations from the mode.

e. Which of these sums of squared deviations has the smallest value?

90

C o m p u t e r E x e r c i s e s

Measures of central tendency are reported along with many other statistics in

many of the software programs you might encounter. The commands that we

describe below will be the same as those that we highlight at the end of

Chapter 5.

SPSS

There are two primary ways of obtaining measures of central tendency in SPSS.

The quickest way to obtain information on the mean for one or more variables

measured at the interval level of measurement is to use the DESCRIPTIVES

command

The output window will contain quite a bit of information for each variable

that you have named—much of it will make more sense after reading Chapter

5. It is important to note that in regard to measures of central tendency, the

DESCRIPTIVES command will report only the mean, not the mode or the

median. Since the only measure of central tendency this command will calculate

is the mean, this command is generally useful only for interval-level data. This

command is most useful when you are working with a data set that contains

almost exclusively interval-level variables.

An alternative to obtaining measures of central tendency is to again use the

FREQUENCIES command discussed at the end of Chapter 3. As you may recall

from the previous chapter’s computer exercises, this command will produce

frequency distributions for the variables whose names are included in the list of

variables. To obtain measures of central tendency for the variables of interest, we

simply need to add an option requesting these values:

DESCRIPTIVES VARIABLES = variable_names.

FREQUENCIES VARIABLES = variable_names

/FORMAT = NOTABLE

/STATISTICS = MEAN MEDIAN MODE.

C O M P U T E R E X E R C I S E S 91

Where the /STATISTICS = option lists the three measures of central tendency

we are interested in. The /FORMAT = NOTABLE option suppresses the print-

ing of the frequency distributions for all the variables included in the list. Should

you want the frequency distributions, just omit this line. Also, the separation of

the command across three lines is simply for ease of reading—all of this material

can be included on a single line in a syntax file. The formatting on this page of

the text would have made it difficult to read what was being done within SPSS.

After running this command, the output window will contain a box labeled

“Statistics.” Each column of this table refers to a separate variable. As you move

down the rows, you should see the reported values for the mode, the median,

and the mean for each variable.

Caution: The mode and the median are listed as numbers, even though the data

may be nominal or ordinal and you have entered value labels. To report correctly

the value of the mode or median, you need to report the category represented

by that number. For example, suppose you had analyzed the variable labeled

“gender,” where males were coded as 1 and females as 2, the mode would be

reported by SPSS as either 1 or 2, but it would be up to you to report correctly

whether the modal category was male or female–not a 1 or a 2.

Specific examples of the use of each of these commands are provided in the

accompanying SPSS syntax file for Chapter 4 (Chapter_4.sps).

Stata

In Stata, there are also two primary ways of obtaining median and mean—both

methods are fairly straightforward. The summarize command is

The basic output will include the mean and other measures on the variables

included in the list. To obtain the median, you will need to add the detail option:

You should note that the output will label the median as the 50th percentile.

Alternatively, if we want to avoid looking at a variety of other statistics that

may not be of interest, we can use the tabstat command and explicitly ask for

only the median and the mean:

The output will list the variables named across column and the median and the

mean will appear in separate rows of the table.

The mode is not reported in any of the standard Stata output, but is easily

determined by running the tab1 command to obtain a frequency distribution

that was described in the Computer Exercises at the end of Chapter 3.

Specific examples of the use of each of these commands are provided in the

accompanying Stata do file for Chapter 4 (Chapter_4.do).

summarize variable_names

summarize variable_names, detail

tabstat variable_names, statistics ( median mean)

C H A P T E R F O U R : D E S C R I B I N G T H E T Y P I C A L C A S E92

Recoding Variables

A common situation in statistical analysis is the need to recode the values for

some variable included in our data file. There are many reasons for why we may

need to recode, such as collapsing categories to simplify the categories of a vari-

able or defining some values as “missing” or inappropriate for our analysis.

The recode commands in SPSS and Stata are quite similar. In SPSS,

RECODE takes the following form:

The structure of RECODE will have a series of old and new values listed in

parentheses and can refer to specific values, ranges of values as well as missing

values. For the values that are not going to be changed, the (ELSE = COPY)

ensures that they are copied to the new variable. We would also like to empha-

size the importance of using new variable names to contain the recoded

values—it is good practice and helps to protect the data file that you are working

with. More than one researcher has made the mistake of sending the recodes

back into the original variable, realizing a mistake was made, and just damaged

the data file that was being used and the need to start over.

The EXECUTE command following the RECODE command is necessary to

force SPSS to perform the recodes now, rather than waiting for a call to another

procedure and performing the recodes at that time.

In Stata, the recode command takes the form

The recoding of old to new values occurs in however many parentheses are

required, just as in SPSS. There is no need to refer to the other values in the

original variable—they are automatically carried over to the new variable.

Of special use is the option of coding one or more values as missing. For

example, a person responding to a survey writes down an incorrect number that

is beyond the range of acceptable responses. In SPSS, system missing values are

noted by SYSMIS in the RECODE command. In Stata, a period (.) denotes a

missing value.

Problems

1. Open the NYS data file (nys_1.sav, nys_1_student.sav, or nys_1.dta).

Consider the level of measurement for each variable, and then compute

and report appropriate measures of central tendency for each variable.

RECODE variable_name (old = new)(old = new) (ELSE = Copy) INTO

new_variable_name.

EXECUTE.

recode variable_name (old = new), gen(new_variable_name)

Following all of the recodes in parentheses, the creation of a new variable is noted by

adding a comma to the command and then the gen(new_variable_name)

command to generate a new variable that contains the recoded values.

93

2. Do any of the variables included in the data file appear to have potential

outliers? (You may want to consult your histograms from the Chapter 3

computer exercises or create histograms now for the interval-level vari-

ables included in the data file.)

3. If you find one or more potential outliers, take the following steps to

investigate their effect on the measures of central tendency.

a. Use the recode command to create two new variables: one that recodes

the outliers as “System Missing” values and one that recodes the

outliers as the next smaller value.

Hints:

b. Using your variable that recodes the outliers as missing, report how

the values of the mean and the median change when potential outliers

are removed from the analysis.

c. Using your variable that recodes the outliers as the next smaller value,

report how the values of the mean and the median change when

potential outliers are made less extreme.

d. Which approach for handling potential outliers do you think is more

appropriate for analyzing these data? Explain why. Faced with potential

outliers, which measure of central tendency would you report?

C O M P U T E R E X E R C I S E S

� You will need to look at frequency distributions to determine the

maximum and next to maximum values.

� Keep in mind that SPSS will look for SYSMIS in the RECODE

command, while Stata will look for a period (.) in the recode command.

How Typical Is the Typical Case?:

Measuring Dispersion

What Do They Tell Us About Our Data?

a n d o r d i n a l s c a l e s : p r o p o r t i o n s ,

p e r c e n t a g e s , a n d t h e v a r i a t i o n r a t i o

r a n g e , v a r i a n c e , a n d s t a n d a r d d e v i a t i o n

C h a p t e r f i v e

M e a s u r e s o f d i s p e r s i o n

M e a s u r i n g d i s p e r s i o n i n n o m i n a l

M e a s u r i n g d i s p e r s i o n i n i n t e r v a l s c a l e s :

How are They Calculated?

What are Their Characteristics?

How are They Calculated?

What are Their Characteristics?

D. Weisburd and C. Britt, Statistics in Criminal Justice, DOI 10.1007/978-1-4614-9170-5_5,

© Springer Science+Business Media New York 2014

MEASURES OF CENTRAL TENDENCY provide a snapshot of the typical

case; however, the same statistic may be obtained from samples or pop-

ulations that are in fact quite dissimilar. For example, a sample of police

recruits with a mean or median age of 23 is not likely to include people

younger than 18 or older than 30, because most police departments have

age requirements for incoming officers. A sample of offenders with a

mean or median age of 23, however, will include offenders younger than

18 and much older than 30. In both these samples, the average person

studied is 23 years old. But the sample of offenders will include more

younger and older people than the sample of police recruits. The ages of

the offenders are dispersed more widely around the average age.

Measures of dispersion allow us to fill a gap in our description of the

samples or populations we study. They ask the question: How typical is

the typical case? They tell us to what extent the subjects we studied are

similar to the case we have chosen to represent them. Are most cases

clustered closely around the average case? Or, as with the sample of of-

fenders above, is there a good deal of dispersion of cases both above

and below the average?

M e a s u r e s o f D i s p e r s i o n f o r N o m i n a l - a n d O r d i n a l - L e v e l D a t a

With nominal scales, we define the typical case as the category with

the largest number of subjects. Accordingly, in Chapter 4 we chose

“private attorney” as the modal category for legal representation for a

sample of white-collar offenders. But how would we describe to what

extent the use of a private attorney is typical of the sample as a whole?

Put another way, to what degree are the cases concentrated in the

modal category?

95

C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

The Proportion in the Modal Category

The most straightforward way to answer this question is to describe the

proportion of cases that fall in the modal category. Recall from Chapter 3

that a proportion is represented by the following equation:

Accordingly, we can represent the proportion of cases in the modal cate-

gory using Equation 5.1:

Equation 5.1

That is, we take the number of cases in the modal category and divide it

by the total number of cases in the sample.

Taking the example of legal representation, we divide the N of cases

in the modal category (private attorney) by the total N of cases in the

sample (see Table 5.1):

Proportion � Nmodal cat.

Ntotal

Proportion � Ncat

Ntotal

Legal Representation for White-Collar Crime

CATEGORY FREQUENCY (N )

No Attorney 20 Legal Aid 26 Court Appointed 92 Public Defender 153 Private Attorney 380 Total (�) 671

Table 5.1

W orking It Out

� 0.5663

� 380

671

Proportion � Nmodal cat.

Ntotal

Following our earlier suggestions regarding rounding to the second deci-

mal place, we say that the proportion of white-collar offenders in the

modal category was about 0.57.

96

M E A S U R E S O F D I S P E R S I O N F O R N O M I N A L A N D O R D I N A L D A T A

Table 5.2 presents information about the method of execution used

on the 683 persons executed in the United States from 1977 to 2000. The

modal category is lethal injection, so the proportion in the modal cate-

gory is found by dividing the N of cases in that category by the total N of

cases:

Method of Execution in the United States, 1977–2000

CATEGORY FREQUENCY (N )

Lethal Injection 518 Electrocution 149 Lethal Gas 11 Hanging 3 Firing Squad 2 Total (�) 683

Source: Tracy L. Snell, “Capital Punishment 2000,” Bureau of Justice Statistics Bulletin, 2001, p. 12.

Table 5.2

W orking It Out

� 0.7584

� 518

683

Proportion � Nmodal cat.

Ntotal

Of persons executed in the United States from 1977 to 2000, the propor-

tion killed through lethal injection was about 0.76.

The Percentage in the Modal Category

Alternatively, we may refer to the percentage in the modal category.

Most people find percentages easier to understand than proportions. Re-

call that a percentage is obtained by taking a proportion and multiplying

it by 100. Accordingly, we can take Equation 5.1 and multiply the result

by 100 to get the percentage of cases in the modal category.

Percentage � Nmodal cat.

Ntotal

� 100

97

C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

For our legal representation example,

W orking It Out

� 56.6319

� 380

671 � 100

Percentage � Nmodal cat.

Ntotal

� 100

That is, about 57% of the cases in the sample fall in the modal category.

Similarly, for the method of execution example, the percentage in the

modal category is

The Variation Ratio

Another way to describe the degree to which the modal category repre-

sents the cases in a sample is to use a statistic called the variation ratio

(VR). The variation ratio is based on the same logic as a proportion, but

it examines the extent to which the cases are spread outside the modal

category, rather than concentrated within it. The proportion of cases in

the modal category is subtracted from 1:

Equation 5.2

For the legal representation example, the variation ratio is

VR � 1 � �Nmodal cat.

Ntotal �

� 75.8419

Percentage � 518

683 � 100

W orking It Out

� 0.4337

� 1 � �380

671� VR � 1 � �Nmodal cat.

Ntotal �

involved the use of lethal injection.

About 76% of all executions in the United States from 1977 to 2000

98

M E A S U R E S O F D I S P E R S I O N F O R N O M I N A L A N D O R D I N A L D A T A 99

The variation ratio for legal representation in this sample of white-collar

offenders is about 0.43. But what does this say about the extent to

which cases in the sample are clustered around the typical case? Is a

variation ratio of 0.43 large or small? What rule can we use for deciding

more generally whether the distribution we are examining is strongly

clustered?

One approach is to define at the outset the upper and lower limits for

the variation ratio or proportion for a particular measure. Obviously, the

largest proportion, regardless of the study, is 1.0, which would mean that

all of the cases were in the modal category. Having all of the cases in

the modal category would lead to a variation ratio of 0, indicating no

dispersion.

The smallest proportion (or largest VR) depends, however, on the

number of categories in your measure. The mode is defined as the cate-

gory in your measure with the most cases, so it must have at least one

more case than any other category. If you have only two categories, then

the modal category must include one more than half of the cases in your

study. So, in the instance of two categories, the least possible concentra-

tion is just over 0.50 of the cases. The least possible dispersion, as mea-

sured by the variation ratio, would be 1 minus this proportion, or just

under 0.50. If you have four categories, the modal category must have

more than one-quarter of the cases. Accordingly, the smallest variation

ratio would be a bit smaller than 0.75.

What about our example of legal representation? We have five cate-

gories and 671 cases. The smallest number of cases the modal cate-

gory could have with these numbers is 135. In this instance, each of

the other four categories would have 134 cases. This is the maximum

amount of dispersion that could exist in this sample, and it amounts to

about 20.12% of the total number of cases in the sample, or a variation

ratio of 0.7988. As noted earlier, the greatest degree of concentration

in the modal category would yield a proportion of 1 and a variation

ratio of 0. The estimates we calculated for legal representation

(proportion � 0.57; VR � 0.43) lie somewhere between these two

extremes.

Is this dispersion large or small? As with many of the statistics we will

examine, the answer depends on the context in which you are working.

“Large” or “small” describes a value, not a statistical concept. Statistically,

you know that your estimate falls somewhere between the largest possi-

ble degree of concentration and the largest possible degree of disper-

sion. But whether this is important or meaningful depends on the prob-

lem you are examining and the results that others have obtained in prior

research.

For example, if, in a study of legal representation for white-collar

crime in England, it had been found that 90% of the cases were concen-

trated in the private attorney category, then we might conclude that our

C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

results reflected a relatively high degree of dispersion of legal represen-

tation in the United States. If, in England, only 25% of the cases had

been in the modal category, we might conclude that there was a rela-

tively low degree of dispersion of legal representation in the United

States.

The proportion and the variation ratio are useful primarily for describ-

ing dispersion with nominal-level measures. In some circumstances,

however, they can be useful for describing ordinal-level variables as

well. This is true primarily when there are just a few categories in a mea-

sure or when there is a very high degree of concentration of cases in

one category. The problem in using a simple proportion or variation

ratio for ordinal-level measures is that the mode, upon which these sta-

tistics are based, is often a misleading measure for ordinal scales. As dis-

cussed in Chapter 4, the mode does not take into account the positions

of scores in a measure, and thus it may provide a misleading view of the

average case.

Index of Qualitative Variation

One measure of dispersion that is not based on the mode—and that

can be used for both nominal and ordinal scales—is the index of

qualitative variation (IQV). The IQV compares the amount of varia-

tion observed in a sample to the total amount of variation possible,

given the number of cases and categories in a study. It is a standard-

ized measure. This means that whatever the number of cases or cate-

gories, the IQV can vary only between 0 and 100. An IQV of 0 means

that there is no variation in the measure, or all of the cases lie in one

category. An IQV of 100 means that the cases are evenly dispersed

across the categories.

Equation 5.3

Equation 5.3 provides a guide for how to compute the IQV. You are

already familiar with the summation symbols within the parentheses.

Here we are summing not across cases, but across products of distinct

categories. Nobs represents the number of cases we observe within a cate-

gory in our study. Nexp represents the number of cases we would expect

in a category if the measure were distributed equally across the cate-

gories. That is, it is the N we would expect if there were the maximum

amount of dispersion of our cases. We use the subscripts i, j, and k as a

IQV � �� k�1

i�1

� k

j�i�1

Nobsi Nobsj

� k�1

i�1

� k

j�i�1 Nexpi

Nexpj � � 100

100

M E A S U R E S O F D I S P E R S I O N F O R N O M I N A L A N D O R D I N A L D A T A

shorthand way to say that we should multiply all of the potential pairs of

categories. Here’s how this works: k represents the total number of cate-

gories of a variable. In the legal representation example, k � 5. Sub-

scripts i and j index the categories of the variable. Use of the subscripts i

and j provides us with a way of keeping track and making sure that we

have multiplied all possible pairs of observed frequencies from each of

the categories.

For example, if a variable had three categories, then the numerator

(the measure of observed variation) would be equal to

If a variable had four categories, then the numerator would be equal to

A concrete example will make it much easier to develop this statistic

in practice. Let’s say that we wanted to describe dispersion of an ordinal-

scale measure of fear of crime in a college class of 20 students. The stu-

dents were asked whether they were personally concerned about crime

on campus. The potential responses were “very concerned,” “quite con-

cerned,” “a little concerned,” and “not concerned at all.” The responses

of the students are reported under the “N observed” column in Table 5.3.

As you can see, the cases are fairly spread out, although there are more

students in the “very concerned” and “quite concerned” categories than

in the “a little concerned” and “not concerned at all” categories. The ex-

pected number of cases in each category under the assumption of maxi-

mum dispersion is 5. That is, if the cases were equally spread across the

categories, we would expect the same number in each. Following Equa-

tion 5.3, we first multiply the number of cases observed in each category

by the number observed in every other category and then sum. We then

divide this total by the sum of the number expected in each category

Nobs1 Nobs2

� Nobs1 Nobs3

� Nobs1 Nobs4

� Nobs2 Nobs3

� Nobs2 Nobs4

� Nobs3 Nobs4

Nobs1 Nobs2

� Nobs1 Nobs3

� Nobs2 Nobs3

Fear of Crime Among Students

CATEGORY N OBSERVED N EXPECTED

Not Concerned at All 3 20/4 � 5 A Little Concerned 4 20/4 � 5 Quite Concerned 6 20/4 � 5 Very Concerned 7 20/4 � 5 Total (�) 20 20

Table 5.3

101

C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

multiplied by the number expected in every other category. This amount

is then multiplied by 100:

W orking It Out

� 96.6667

� �145 150� � 100

� �(3 � 4) � (3 � 6) � (3 � 7) � (4 � 6) � (4 � 7) � (6 � 7)

(5 � 5) � (5 � 5) � (5 � 5) � (5 � 5) � (5 � 5) � (5 � 5)� � 100

IQV � � k�1

i�1

� k

j�i�1

Nobsi Nobsj

� k�1

i�1

� k

j�i�1

Nexpi Nexpj

� 100

The observed variation is 145. The expected variation is 150, represent-

ing the maximum amount of dispersion possible for the measure. The

IQV for this measure is 96.67, meaning that the cases studied are very

dispersed among the categories of the measure.

M e a s u r i n g D i s p e r s i o n i n I n t e r v a l S c a l e s : T h e R a n g e , V a r i a n c e , a n d S t a n d a r d D e v i a t i o n

A common method of describing the spread of scores on interval or

higher scales is to examine the range between the highest and lowest

scores. Take, for example, the distribution of cases in Table 5.4. Let’s say

that this was a distribution of crime calls at hot spots over a one-year pe-

riod. In describing typicality in this distribution, we would report the

mean number of calls for the 12 places, which is 21.50. In describing

how dispersed the scores are, we would report that the scores range be-

tween 2 and 52, or that the range of scores is 50.

The range is very simple and easy to present. Its attraction lies pre-

cisely in the fact that everyone understands what a range represents.

However, the range is an unstable statistic because it uses very little of

the information available in interval-level scales. It bases its estimate of

dispersion on just two observations, the highest and lowest scores. This

means that a change in just one case in a distribution can completely

102

M E A S U R I N G D I S P E R S I O N I N I N T E R V A L S C A L E S

alter your description of dispersion. For example, if we changed the case

with the most calls in Table 5.4 from 52 to 502, the range would change

from 50 to 500.

One method for reducing the instability of the range is to examine

cases that are not at the extremes of your distribution. In this way, you

are likely to avoid the problem of having the range magnified by a few

very large or small numbers. For example, you might choose to look at

the range between the 5th and 95th percentile scores, rather than that

between the lowest and highest scores. It is also common to look at the

range between the 25th and 75th percentile scores or between the 20th

and 80th percentile scores. But however you change the points at which

the range is calculated, you still rely on just two scores in determining

the spread of cases in your distribution. The range provides no insight

into whether the scores below or above these cases are clustered to-

gether tightly or dispersed widely. Its portrait of dispersion for interval

scales is thus very limited.

How can we gain a fuller view of dispersion for interval scales? Re-

member that we became interested in the problem of dispersion because

we wanted to provide an estimate of how well the average case repre-

sented the distribution of cases as a whole. Are scores clustered tightly

around the average case or dispersed widely from it? Given that we have

already described the mean as the most appropriate measure of central

tendency for such scales, this is the natural place to begin our assess-

ment. Why not simply examine how much the average scores differ from

the mean?

In fact, this is the logic that statisticians have used to develop the

main measures of dispersion for interval scales. However, they are faced

with a basic problem in taking this approach. As we discussed in Chap-

ter 4, if we add up all of the deviations from the mean, we will always

Crime Calls at Hot Spots in a Year

HOT SPOT NUMBER NUMBER OF CALLS

1 2 2 9 3 11 4 13 5 20 6 20 7 20 8 24 9 27

10 29 11 31 12 52

Table 5.4

103

C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

come up with a value of 0. You can see this again by looking at the data

on crime calls at hot spots in Table 5.5. If we take the sum of the differ-

ences between each score and the mean, written in equation form as

the total, as expected, is 0.

As discussed in Chapter 4, when we add up the deviations above and

below the mean, the positive and negative scores cancel each other out.

In order to use deviations from the mean as a basis for a measure of dis-

persion, we must develop a method for taking the sign, or direction, out

of our statistic. One solution is to square each deviation from the mean.

Squaring will always yield a positive result because multiplying a posi-

tive number or a negative number by itself will result in a positive out-

come. This is the method that statisticians have used in developing the

measures of dispersion most commonly used for interval scales.

The Variance

When we take this approach, the variance (s 2) provides an estimate of

the dispersion around the mean. It is the sum of the squared deviations

from the mean divided by the number of cases. Written in equation

form, it is

Equation 5.4s 2 � � N

i�1

(Xi � X )2

N

� N

i�1

(Xi � X )

Deviations from the Mean for Crime Calls at Hot Spots in a Year

HOT SPOT NUMBER DEVIATIONS FROM

NUMBER OF CALLS THE MEAN (Xi � )

1 2 2 � 21.5 � �19.5 2 9 9 � 21.5 � �12.5 3 11 11 � 21.5 � �10.5 4 13 13 � 21.5 � �8.5 5 20 20 � 21.5 � �1.5 6 20 20 � 21.5 � �1.5 7 20 20 � 21.5 � �1.5 8 24 24 � 21.5 � 2.5 9 27 27 � 21.5 � 5.5

10 29 29 � 21.5 � 7.5 11 31 31 � 21.5 � 9.5 12 52 52 � 21.5 � 30.5

Total (�) � 0.0

X

Table 5.5

104

M E A S U R I N G D I S P E R S I O N I N I N T E R V A L S C A L E S

In practice, you must take the following steps to compute the variance

(as we do for our example in Table 5.6):

1. Take each case and subtract the mean from it, to get the deviation

from the mean. For our example of crime calls at hot spots, we first

take the case with 2 calls and subtract the mean of 21.5 from it, to

get a score of �19.5.

2. Square each of these scores. For the first case, our result is 380.25.

3. Sum the results obtained in step 2. For our example of hot spots of

crime, this yields a total of 1,839.

4. Finally, divide this result by the number of cases in the study. For our

12 cases, this leads to a variance of 153.25.1

Variance for Crime Calls at Hot Spots in a Year

HOT SPOT NUMBER

NUMBER OF CALLS (Xi � ) (Xi � )2

1 2 2 � 21.5 � �19.5 380.25 2 9 9 � 21.5 � �12.5 156.25 3 11 11 � 21.5 � �10.5 110.25 4 13 13 � 21.5 � �8.5 72.25 5 20 20 � 21.5 � �1.5 2.25 6 20 20 � 21.5 � �1.5 2.25 7 20 20 � 21.5 � �1.5 2.25 8 24 24 � 21.5 � 2.5 6.25 9 27 27 � 21.5 � 5.5 30.25

10 29 29 � 21.5 � 7.5 56.25 11 31 31 � 21.5 � 9.5 90.25 12 52 52 � 21.5 � 30.5 930.25

Total (�) � 0.0 Total (�) � 1,839.00

XX

Table 5.6

1If you are working with SPSS or another computer package, you will notice that the

result you get computing the variance by hand using this formula and the result pro-

vided by the computer package are slightly different. For example, SPSS computes a

variance of 167.18 for the distribution provided in Table 5.6. The difference develops

from the computer’s use of a correction for the bias of sample variances: 1 is sub-

tracted from the N in the denominator of Equation 5.4. The correction is used primar-

ily as a tool in inferential statistics and is discussed in Chapter 10. Though it is our

view that the uncorrected variance should be used in describing sample statistics,

many researchers report variances with the correction factor for sample estimates.

When samples are larger, the estimates obtained with and without the correction are

very similar, and thus it generally makes very little substantive difference which ap-

proach is used.

105

C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

amounts required for a group of 15 defendants. The mean bail amount is

$3,263.33. Following the same procedure as before, we subtract the

mean from each of the individual observations. These values are pre-

cases, we gain a variance of $6,984,155.56 for the dollar amount of bail.

W orking It Out

� 153.25

� 1,839

12

� � 12

i�1

(Xi � 21.5)2

12

s

2 � � N

i�1

(Xi � X )2

N

Variance for Bail Amounts for a Sample of Persons Arrested for Felonies

BAIL

DEFENDANT AMOUNT (Xi � ) (Xi � )2

1 500 �2,763.33 7,635,992.69 2 1,000 �2,263.33 5,122,662.69 3 1,000 �2,263.33 5,122,662.69 4 1,000 �2,263.33 5,122,662.69 5 1,200 �2,063.33 4,257,330.69 6 1,500 �1,763.33 3,109,332.69 7 2,500 �763.33 582,672.69 8 2,500 �763.33 582,672.69 9 2,500 �763.33 582,672.69

10 2,750 �513.33 263,507.69 11 5,000 1,736.67 3,016,022.69 12 5,000 1,736.67 3,016,022.69 13 5,000 1,736.67 3,016,022.69 14 7,500 4,236.67 17,949,372.69 15 10,000 6,736.67 45,382,722.69

Total (�) � 0.05 Total (�) � 104,762,333.33

XX

Table 5.7

sented in the third column. The squared deviations from the mean

appears at the bottom of the column. When we divide the total by the N of

appear in the fourth column, and the sum of the squared deviations

As another example, consider the data presented in Table 5.7 on bail

106

M E A S U R I N G D I S P E R S I O N I N I N T E R V A L S C A L E S

With the variance, we now have a statistic for computing dispersion

based on deviations from the mean. However, how can we interpret

whether the variance for a distribution is large or small? If you are having

trouble making sense of this from our two examples, you are not alone.

While squaring solves one problem (the fact that the raw deviations from

the mean sum to 0), it creates another. By squaring, we generally obtain

numbers that are much larger than the actual units in the distributions

we are examining.2

The Standard Deviation

Another measure of dispersion based on the variance provides a solution

to the problem of interpretation. This measure, the standard deviation,

is calculated by taking the square root of the variance. Accordingly, it re-

duces our estimate of dispersion, using a method similar to the one we

employed to solve the problem of positive and negative differences from

the mean adding to 0. The standard deviation (s) provides an estimate of

dispersion in units similar to those of our original scores. It is described

in equation form as

Equation 5.5s � �� N

i�1

(Xi � X )2

N

W orking It Out

� 6,984,155.56

� 104,762,333.33

15

� � 15

i�1

(Xi � 3,263.33)2

15

s

2 � � N

i�1

(Xi � X )2

N

2In the special case of a fraction, the result will be smaller numbers.

107

Although Equations 5.4 and 5.5 provide a useful way of conceptualizing and

measuring the variance and standard deviation, you can also use a com-

puting formula that has fewer steps and is less likely to result in computa-

tional error. In Table 5.6, we rounded the mean and then calculated squared

deviations based on values that were rounded at each step. In an attempt

to limit the amount of rounding, and consequently decrease the chances of

a mistake, an alternative equation that can be used for the variance is

And an alternative for the standard deviation is

Let’s reconsider the data in Table 5.6 on hot spots. The following table

illustrates the key calculations:

HOT SPOT NUMBER NUMBER OF CALLS (Xi)

1 2 4

2 9 81

3 11 121

4 13 169

5 20 400

6 20 400

7 20 400

8 24 576

9 27 729

10 29 841

11 31 961

12 52 2,704

Total (�) 258 7,386

X

2 i

s � �� N

i�1

X

2 i �

��N i�1

Xi�2

N

N

s

2 � � N

i�1

X

2 i �

�� N

i�1

Xi�2

N

N

Computational Equations for the Variance and Standard Deviation

The variance (s 2) is then calculated with the computational equation as

And the standard deviation is simply the square root of the variance:

Similarly, let’s revisit the bail data in Table 5.7 and compute the vari-

ance with the computational formula. The following table illustrates the

key calculations.

DEFENDANT BAIL AMOUNT (Xi )

1 500 250,000

2 1,000 1,000,000

3 1,000 1,000,000

4 1,000 1,000,000

5 1,200 1,440,000

6 1,500 2,250,000

7 2,500 6,250,000

8 2,500 6,250,000

9 2,500 6,250,000

10 2,750 7,562,500

11 5,000 25,000,000

12 5,000 25,000,000

13 5,000 25,000,000

14 7,500 56,250,000

15 10,000 100,000,000

Total (�) 48,950 264,502,500

The variance is

And the standard deviation is

s � �6,984,155.56 � 2,642.76

s

2 �

264,502,500 � (48,950)2

15 15

� 6,984,155.56

X

2 i

s � �153.25 � 12.38

s 2 �

7,386 � (258)2

12 12

� 153.25

110 C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

In calculating the standard deviation, we add one step to our calcula-

tion of variance: We take the square root of our result. For the example

of crime calls at 12 hot spots (where the variance equaled 153.25), we

obtain a standard deviation of .3 If you were to define,

on average, how much the scores differed from the mean just by looking

at these 12 cases, you would probably come to a conclusion close to that

provided by the standard deviation. Similarly, if we take the square root

of the variance for the bail example above, we come up with a figure

that makes much more intuitive sense than the variance. In this case, the

standard deviation is , or $2,642.76.

The standard deviation has some basic characteristics, which relate

generally to its use:

1. A standard deviation of 0 means that a measure has no variability.

For this to happen, all of the scores on a measure have to be the

same. For example, if you examine a group of first-time offenders,

there will be no variation in the number of offenses in their criminal

records. By definition, because they are all first-time offenders, the

standard deviation (and the variance) will be 0.

2. The size of the standard deviation (and the variance) is dependent

on both the amount of dispersion in the measure and the units of

analysis that are used. When cases are spread widely from the mean,

there is more dispersion and the standard deviation will be larger.

When cases are tightly clustered around the mean, the standard

deviation will be smaller.

Similarly, when the units of analysis in the measure are large, the

standard deviation will reflect the large units. For example, if you

report the standard deviation of police salaries in a particular city in

dollars, your standard deviation will be larger than if you reported

those salaries in units of thousands of dollars. If the standard

deviation is 3,350 in dollars, the standard deviation would be 3.35

using the unit of thousands of dollars.

3. Extreme deviations from the mean have the greatest weight in

constructing the standard deviation. What this means is that here, as

with the mean, you should be concerned with the problem of

outliers. In this case, the effect of outliers is compounded because

they affect not only the mean itself, which is used in computing the

standard deviation, but also the individual deviations that are

obtained by subtracting the mean from individual cases.

�6,984,155.56

�153.25 � 12.38

3As discussed in footnote 1, SPSS and many other computer packages would provide

a slightly different result, based on the use of a correction of �1 in the denominator.

M E A S U R I N G D I S P E R S I O N I N I N T E R V A L S C A L E S 111

The standard deviation is a useful statistic for comparing the extent to

which characteristics are clustered or dispersed around the mean in dif-

4

samples are very similar (61.05 for antitrust violators; 59.27 for bribery

offenders), but the standard deviation for those convicted of bribery is

about twice that of those convicted of antitrust violations.

Figure 5.1 illustrates why these two samples yield similar means but

very different standard deviations. The scores for most antitrust offend-

the distribution, including many more cases between 75 and 90 and

below 50. What this tells us is that the antitrust sample includes a fairly

homogeneous group of offenders, ranking on average relatively high

on the Duncan socioeconomic index. Bribery is a much more diverse

category. Although the means are similar, the bribery category includes

many more lower- and higher-status individuals than does the antitrust

category.

The Coefficient of Relative Variation

For the data on bribery and antitrust offenders in Table 5.8, in which the

means of the two groups are fairly similar, a direct comparison of stan-

dard deviations provides a good view of the differences in dispersion.

When the means of two groups are very different, however, this compar-

ison may not be a fair one. If the mean Duncan score for one group was

10 and for the other was 50, we might expect a larger standard deviation

in the latter group simply because the mean was larger and there was

Duncan SEI for Bribery and Antitrust Offenders

CATEGORY N s

Bribery 83 59.27 19.45 Antitrust 112 61.05 11.13 Total (�) 195

X

Table 5.8

4See Albert J. Reiss, Occupations and Social Status (New York: Free Press, 1961).

victed of antitrust violations is compared to a sample of offenders convicted

ferent samples. For example, in Table 5.8, a sample of offenders con-

of bribery. The characteristic examined is social status, as measured

by the interval-scale Duncan socioeconomic index (SEI). The index

is based on the average income, education, and prestige associated

with different occupations. The mean Duncan scores for these two

offenders, in contrast, the scores are much more widely spread across

ers are clustered closely within the range of 55 to 75. For bribery

112 C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

N u

m b er

o f

O ff

en d

er s

Std. dev. = 11.13 Mean = 61.05 N = 112.00

0 20 40 60 80

100

80

60

40

20

0

SEI

Std. dev. = 19.45 Mean = 59.27 N = 83.00

N u

m b er

o f

O ff

en d

er s

SEI

30

20

10

0 0 20 40 60 80

Socioeconomic IndicesFigure 5.1

(a) Antitrust Offenders

(b) Bribery Offenders

M E A S U R I N G D I S P E R S I O N I N I N T E R V A L S C A L E S 113

greater potential for dispersion. Similarly, if two measures use different

units of analysis—for example, dollars and number of offenses—a direct

comparison of standard deviations does not make sense.

One solution to this problem is to use the coefficient of relative

variation (CRV). The coefficient of relative variation looks at the size of

the standard deviation of a measure relative to the size of its mean:

Equation 5.6

In the example of the SEI for antitrust offenders, we divide the standard

deviation (11.13) by the mean (61.05) to obtain a CRV of 0.18, meaning

that the standard deviation is about one-fifth the size of the mean. Be-

cause the CRV expresses dispersion in a measure in a standardized form

relative to the mean, we can compare the CRV across measures that have

widely different means and standard deviations.

CRV � s

X

W orking It Out

� 0.1823

� 11.13

61.05

CRV � s

X

A Note on the Mean Deviation

The standard deviation allows us to measure dispersion in interval

scales, taking into account the deviation from the mean of each case in

our sample or population. But it is not the only measure that allows us

to do this. The mean deviation takes a similar approach, but relies on

absolute values, rather than squaring, to overcome the fact that the sum

of the deviations from the mean equals 0. When you take the absolute

value of a number, you ignore its sign. Accordingly, �8 and 8 both have

an absolute value of 8; in mathematical notation, � �8 � � � 8 � � 8.

The equation for the mean deviation is similar to that for the vari-

ance. The only difference is that we take the absolute value of the

antitrust offenders.

A measure that has a CRV of 1, for example, may be considered to

include much greater relative variation than is found in our sample of

114 C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

Mean Deviation for Crime Calls at Hot Spots in a Year

HOT SPOT NUMBER DEVIATIONS FROM

NUMBER OF CALLS THE MEAN �Xi � �

1 2 � 2 � 21.5 � � 19.5

2 9 � 9 � 21.5 � � 12.5

3 11 � 11 � 21.5 � � 10.5

4 13 � 13 � 21.5 � � 8.5

5 20 � 20 � 21.5 � � 1.5

6 20 � 20 � 21.5 � � 1.5

7 20 � 20 � 21.5 � � 1.5

8 24 � 24 � 21.5 � � 2.5

9 27 � 27 � 21.5 � � 5.5

10 29 � 29 � 21.5 � � 7.5

11 31 � 31 � 21.5 � � 9.5

12 52 � 52 � 21.5 � � 30.5

Total (�) � 111.0

X

Table 5.9

difference between each score and the mean, rather than the square of

the difference:

Using the data on crime calls in hot spots from Table 5.4, we take the

following steps to obtain the mean deviation. We first take the absolute

value of the difference between each score and the mean (see Table

5.9). We then sum up the 12 scores. Notice that we obtain a positive

number now (111), and not 0, because we are taking the absolute values

of the differences. Dividing this sum by the number of cases, N, we get a

mean deviation of 9.25.

Mean deviation � � N

i�1 �Xi � X �

N

W orking It Out

� 9.25

� 111 12

� � 12

i�1 �Xi � 21.5 �

N

Mean deviation � � N

i�1 �Xi � X �

N

Equation 5.7

C H A P T E R S U M M A R Y 115

The mean deviation and the standard deviation provide similar esti-

mates of dispersion, but the mean deviation here is a bit smaller than the

standard deviation of 12.38 that we calculated earlier. Which is the better

estimate of dispersion? In some sense, the mean deviation is more

straightforward. It simply looks at the average deviation from the mean.

In obtaining the standard deviation, we first must square the deviations;

then later, to return our result to units similar to those of the original dis-

tribution, we must take the square root of the variance.

Given our rule that we should use the least complex presentation

that is appropriate to answering our research question, you may won-

der why the standard deviation is almost always preferred over the

mean deviation in criminal justice research. As you will see in the next

few chapters, the answer is that the standard deviation is relevant to a

number of other statistics that we use in analyzing and describing

data.

C h a p t e r S u m m a r y

Measures of dispersion describe to what extent cases are distributed

around the measure of central tendency. They tell us just how typical the

typical case is.

There are several measures of dispersion for nominal and ordinal

scales. Proportions and percentages describe the extent to which cases

are concentrated in the modal category. The variation ratio (VR) de-

scribes the extent to which cases are spread outside the modal category.

A proportion of 1 (VR of 0) means that all the cases are in the modal cat-

egory. This represents the least possible amount of dispersion. The value

for the greatest possible dispersion can be determined by calculating the

minimum possible value of the modal category and then translating that

into a proportion or VR value. These measures can, in principle, be used

with ordinal-level data, but the results may be misleading, as they take

into account only the value of the mode. As an alternative, the index of

qualitative variation (IQV) is a standardized measure that takes into ac-

count variability across all the categories of a nominal- or ordinal-level

variable. An IQV of 0 means that there is no variation; an IQV of 100

means that there is maximum variation across the categories.

A different set of measures is used to measure dispersion for inter-

val and ratio scales. The range measures the difference between the

highest and lowest scores. It has the advantage of simplicity, but it

uses very little information (only two scores) and the scores used are

taken from the two extremes. It is also very sensitive to outliers. A

116 C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

95th and the 5th percentile. Such measures, however, are still based

on minimal information and thus are generally considered unstable

statistics. A more stable statistic for measuring dispersion in interval-

level scales is the variance. The variance is the sum of the squared

deviations of each score from the mean divided by the number of

cases. The standard deviation (s) is the square root of the variance.

The advantage of the standard deviation over the variance is that the

results are more easily interpreted. If all the scores in a sample are the

same, s will be 0. The more widely the scores are spread around the

mean, the greater will be the value of s. Outliers have a considerable

impact on the standard deviation.

Comparing the standard deviations of means is problematic when

the means are very different or when their units of measurement are

different. An alternative measure, the coefficient of relative varia-

tion (CRV), enables comparisons among samples with different

means. A less often used measure of dispersion for interval scales is

the mean deviation. The mean deviation is computed by taking the

sum of the absolute values of the deviations from the mean divided by

the number of cases.

K e y T e r m s

coefficient of relative variation A mea-

sure of dispersion calculated by dividing

the standard deviation by the mean.

index of qualitative variation A measure

of dispersion calculated by dividing the

sum of the possible pairs of observed

scores by the sum of the possible pairs of

expected scores (when cases are equally

distributed across categories).

mean deviation A measure of dispersion

calculated by adding the absolute deviation

of each score from the mean and then di-

viding the sum by the number of cases.

range A measure of dispersion calculated

by subtracting the smallest score from the

largest score. The range may also be calcu-

lated from specific points in a distribution,

such as the 5th and 95th percentile scores.

standard deviation A measure of disper-

sion calculated by taking the square root of

the variance.

variance (s2) A measure of dispersion cal-

culated by adding together the squared de-

viation of each score from the mean and

then dividing the sum by the number of

cases.

variation ratio A measure of dispersion

calculated by subtracting the proportion of

cases in the modal category from 1.

researcher may instead choose to measure the range between, say, the

S Y M B O L S A N D F O R M U L A S 117

S y m b o l s a n d F o r m u l a s

Nmodal cat. Number of cases in the modal category

Ntotal Total number of cases

Nobs Number of cases observed in each category

Nexp Number of cases expected in each category

s Standard deviation

s 2 Variance

To calculate the proportion of cases falling in the modal category:

To calculate the percentage of cases falling in the modal category:

To calculate the variation ratio:

To calculate the index of qualitative variation:

To calculate the variance:

To calculate the standard deviation:

s � �� N

i�1

(Xi � X )2

N

s

2 � � N

i�1

(Xi � X )2

N

IQV � �� k�1

i�1

� k

j�i�1

Nobsi Nobsj

� k�1

i�1

� k

j�i�1

Nexpi Nexpj

� � 100

VR � 1 � �Nmodal cat.

Ntotal �

Percentage � Nmodal cat.

Ntotal � 100

Proportion � Nmodal cat.

Ntotal

118 C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

To calculate the coefficient of relative variation:

To calculate the mean deviation:

E x e r c i s e s

5.1 Police records for 105 rape victims were analyzed to determine whether any prior relationship existed between the victim and the offender. The results were as follows:

Spouse 41

Family member other than spouse 14

Acquaintance 22

No prior relationship 28

a. Calculate the modal proportion and the variation ratio.

b. What are the minimum and maximum possible values for the varia- tion ratio?

c. Calculate the index of qualitative variation.

5.2 As part of a larger study on the influence of delinquent peers, a sam- ple of high school youth were asked how much they wanted to be like their best friend. The responses were coded as follows: in every way, 26; in most ways, 36; in some ways, 41; and not at all, 8.

a. Calculate the variation ratio for these data.

b. Calculate the index of qualitative variation for these data.

5.3 People convicted of minor traffic offenses who appeared in the magistrate’s court of a given locality on a given day were sentenced as follows: conditional discharge, 14; fine, 35; and license dis- qualification, 11.

a. Calculate the variation ratio.

b. Calculate the index of qualitative variation.

c. Why do these two results differ?

5.4 A sample of women was drawn from town A, and another sample was drawn from town B. All the women were asked how safe or unsafe

Mean deviation � � N

i�1 �Xi � X �

N

CRV � s

X

E X E R C I S E S 119

they felt walking alone at night in their neighborhoods. The results were recorded on a scale as follows: totally unsafe (town A: 40; town B: 25), quite unsafe (town A: 29; town B: 23), quite safe (town A: 10; town B: 15), and totally safe (town A: 21; town B: 17).

a. For each town, describe the typical case, using an appropriate mea- sure of central tendency. Explain why this is the best measure of central tendency for these data.

b. For each town, describe how typical the typical case is, using an appropriate measure of dispersion. Explain why this is the best measure of dispersion for these data.

c. In comparing the measures of central tendency and dispersion for the two towns, what conclusions may be drawn about the attitudes of the women?

5.5 For a sample of 12 offenders convicted of weapons violations, the length of prison sentence in months was recorded as:

6 6 2 12 36 48 60 24 24 20 18 15

a. Calculate the range for these data.

b. Calculate the mean and the variance for these data.

5.6 A group of 20 prisoners in a particular cell block were tested on their knowledge of the rules of the institution. The marks (out of a possible 70) were as follows:

31 28 27 19 18 18 41 0 30 27

27 36 41 64 27 39 20 28 35 30

a. Calculate the range.

b. Remove the largest and smallest scores. Calculate the range for the remaining cases.

c. How do you account for the difference between the values of the above two measures of dispersion?

5.7 Police crack a drug ring of 18 suppliers and discover that of the 18, only 4 have no previous convictions for drug- or theft-related offenses. Eight of those arrested have 1 previous conviction, and the others have 2, 3, 4, 5, 6, and 8, respectively.

a. Calculate the mean and the standard deviation of the 18 cases.

b. If each of the drug suppliers is convicted this time around, does the extra conviction on each of their criminal records affect the mean or the standard deviation in any way? Explain your answer.

120 C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

5.8 Use the data collected from tests of prisoners’ knowledge of institution rules in Exercise 5.6.

a. Calculate the mean and the standard deviation for these data.

b. If you remove the two most extreme scores, 0 and 64, what are the new mean and standard deviation?

c. How do you account for this effect?

5.9 When asked about how often in the last year they drank more than four beers in one evening, a sample of college students reported the following:

Number

of Times Frequency

0 187

1 213

2 162

3 94

4 71

5 55

6 39

7 12

8 9

9 5

10 13

a. Calculate an appropriate measure of dispersion for these data. Explain why this measure is most appropriate for these data.

b. Describe one way these data could be recoded to reduce the number of categories. Calculate an appropriate measure of dispersion for the recoded data and explain why this measure is most appropriate.

5.10 A researcher takes a sample of shop owners in Tranquiltown and a sample of shop owners in Violenceville and asks them to estimate the value of goods stolen from their shops in the past 12 months. The mean figure is $11.50 (s � $2.50) for Tranquiltown and $4,754.50 (s � $1,026.00) for Violenceville. When the study is published, the mayor of Violenceville protests, claiming that the mean sum for his town is a misleading figure. Because the standard deviation for Vio- lenceville is much bigger than that for Tranquiltown, he argues, it is clear that the mean from Violenceville is a much less typical descrip- tion of the sample than the mean from Tranquiltown.

a. What statistic might help the researcher to refute this criticism? Why?

C O M P U T E R E X E R C I S E S 121

b. Calculate this statistic for each town. What should the researcher conclude?

5.11 A researcher investigating differences in violence among preschool- age boys and girls found that the average number of violent acts per week was 7.6 (s � 4.8) for boys and 3.1 (s � 1.9) for girls.

a. Calculate the coefficient of relative variation for boys and for girls.

b. How can the coefficient of relative variation be used to compare these two groups? What does it tell you?

C o m p u t e r E x e r c i s e s

Similar to the measures of central tendency discussed in Chapter 4, there are

several ways to obtain measures of dispersion for interval-level variables in SPSS

and Stata (Neither program computes measures of dispersion, such as the index

of qualitative variation, for nominal and ordinal variables.).

SPSS

The same two commands that we used to obtain measures of central tendency—

DESCRIPTIVES and FREQUENCIES—will be used to obtain measures of

dispersion. Since you have already used these commands, our discussion here is

focused on the default measures of dispersion and additional options available

for measuring dispersion.

The DESCRIPTIVES command allows you to compute the standard devia-

tion and variance. The default is to compute the standard deviation, as well as

minimum and maximum values. To obtain the variance and/or range, add the

option /STATISTICS = to the command line:

The abbreviations should be clear, but STDDEV is the standard deviation, MIN

is the minimum value, and MAX is the maximum value.

In computing the variance, SPSS uses a correction for the bias of sample

measures of variance and dispersion: 1 is subtracted from the N in the denomi-

nator of Equations 5.4 and 5.5. The correction is used primarily as a tool in

inferential statistics and is discussed in greater detail in Chapter 10. Though it is

our view that the uncorrected variance and standard deviation should be used

in describing sample statistics, many researchers report these statistics with the

correction factor for sample estimates. When samples are larger, the estimates

obtained with and without the correction are very similar, and thus it generally

makes very little substantive difference which approach is used.

DESCRIPTIVES VARIABLES = variable_names

/STATISTICS = MEAN STDDEV VARIANCE RANGE MIN MAX.

122 C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

The FREQUENCIES command provides similar measures of disper-

sion through use of the /STATISTICS = option. You may request the stand-

ard deviation, variance, range, minimum, and maximum. In addition to these

measures of dispersion, the FREQUENCIES command has the option of com-

puting percentiles. Since calculation of percentiles by hand can be very difficult

(and error prone), this is a nice feature. There are three options for calculating

percentiles in SPSS: quartiles (25th, 50th, and 75th percentiles) or cut points for

some number of equally spaced groups (e.g., 10th, 20th, …, 90th percentiles)

by using the NTILES = option, as well as specific percentiles that you may be

interested in (e.g., 5th, 95th, 99th) by using the /PERCENTILES = option. The

general form, including all the various options, is:

If you are not interested in percentiles or equally spaced groups (e.g., quartiles,

deciles, etc.), then omit those lines from the command.

Specific examples of the use of each of these commands are provided in the

accompanying SPSS syntax file for Chapter 5 (Chapter_5.sps).

Stata

To obtain measures of dispersion in Stata, the same two commands used in

obtaining measures of central tendency will again be used. To obtain detailed

measures of dispersion, it is best to simply add the detail option to the

command line:

This will produce the variance, standard deviation, and numerous percentiles.

Stata makes the correction to the computation of the variance in the same way

that SPSS does by subtracting 1 from the total sample size.

The tabstat command is also useful in trimming out the unnecessary pieces

of information and printing only those items we want:

Where min is the minimum, max is the maximum, sd is the standard deviation,

var is the variance, cv is the coefficient of variation (not available in SPSS), and

FREQUENCIES VARIABLES = variable_names

/PERCENTILES = list_of_percentiles

/NTILES = number_of_equally_spaced_groups

/STATISTICS = STDDEV VARIANCE RANGE MINIMUM MAXIMUM

MEAN MEDIAN MODE.

summarize variable_names, detail

tabstat variable_names, statistics (median mean min max

range sd var cv p#)

C O M P U T E R E X E R C I S E S 123

in the 5th, 10th, 90th, and 95th percentiles, we would include p5 p10 p90 p95

on the command line (and within the parentheses).

Specific examples of the use of each of these commands are provided in the

accompanying Stata do file for Chapter 5 (Chapter_5.do).

Problems

1. Enter the data from Exercise 5.6 on the 20 prisoners’ test scores.

a. What is the range?

b. What are the 5th and 95th percentiles? What is the range between the

5th and 95th percentiles?

c. How does your answer to part b compare to your answer to part b in

Exercise 5.6?

2. Enter the data from Exercise 5.7. (Be sure that you have 18 lines of data,

since there are 18 observations listed in the question.)

a. What are the mean and the standard deviation? How does the standard

deviation differ from the value you calculated in Exercise 5.7?

b. To add 1 to each person’s number of prior convictions, you will need to

create a new variable.

In SPSS:

In Stata:

3. Open the NYS data file (nys_1.sav, nys_1_ student.sav, or nys_1.dta).

a. Choose five of the delinquency measures. What are the quartiles for

b. What is the range between the 25th and 75th percentiles for each

delinquency measure? (This difference is known as the “inter- quartile

range.”) What do the differences in inter-quartile ranges appear to

indicate about the dispersion of these different measures self-reported

delinquency?

p# refers to a specific percentile we may be interested in and included for as

many percentiles as we would like reported. For example, if we were interested

COMPUTE new_var_name = old_var_name + 1.

EXECUTE.

gen new_var_name = old_var_name +1

What are the mean and standard deviation for this new

variable? What has changed? What has remained the same?

124 C H A P T E R F I V E : H O W T Y P I C A L I S T H E T Y P I C A L C A S E ?

-

able (Remember, this measure is available in the tabstat command in

Stata, but will need to be calculated by hand if using SPSS.). What do

variable indicate about the relative dispersion of these variables?

c. What number of delinquent acts would mark the 15 % least delinquent

youth? The 20 % most delinquent youth?

The Logic of Statistical Inference:

Making Statements About Populations

from Sample Statistics

How Can a Sample Teach Us About a Population?

C h a p t e r s i x

S a m p l e d i s t r i b u t i o n s a n d p o p u l a t i o n d i s t r i b u t i o n s

A s k i n g t h e r e s e a r c h q u e s t i o n

A n s w e r i n g t h e r e s e a r c h q u e s t i o n

How are They Defined?

What Symbols are Used?

How are the Two Interrelated?

What Types of Error are Possible?

What is an “Acceptable” Risk of Error?

When Might It be Necessary to Accept a Different Level of Risk?

What are the Research and Null Hypotheses?

How are They Set Up?

D. Weisburd and C. Britt, Statistics in Criminal Justice, DOI 10.1007/978-1-4614-9170-5_6,

© Springer Science+Business Media New York 2014

IN THIS CHAPTER, we look at an important dilemma that researchers face

in conducting criminal justice research. Although they seek to make

statements about populations, generally they collect data on samples

drawn from such populations. Statistical inference provides a solution to

about the characteristics of a population from data collected from a sam-

ple drawn from the population. We begin our discussion of statistical in-

ference by explaining the dilemma researchers face in making statements

about populations from samples. We then examine the logic of statistical

inference and the statistical risks associated with using this logic. You

will be introduced to how null and research hypotheses are set up, how

risks of error are assessed, and how levels of statistical significance are

used to limit this error.

T h e D i l e m m a : M a k i n g S t a t e m e n t s A b o u t P o p u l a t i o n s f r o m S a m p l e S t a t i s t i c s

In descriptive statistics, we are concerned with two basic types of distrib-

utions. One is the distribution of scores in the sample, or the sample

distribution. The second is the distribution of scores in the population

from which the sample is drawn. This is referred to as the population

distribution. One of the fundamental problems in research in criminal

justice, as in other fields, is that we want to make statements about the

characteristics of the population distribution, but we generally have in-

formation only about the distribution of sample scores. For example,

when we draw a sample of 2,000 voters in an election survey, we are

not interested per se in how those people will vote. Rather, we examine

their voting preference to learn something about how all people will

vote in the election. In statistical terms, we want to use information on

characteristics of the distribution of sample scores to make statements

about characteristics of the distribution of population scores.

126

this dilemma: it allows the researcher to make statements, or inferences,

T H E D I L E M M A 127

It is important to note at the outset that populations can be defined in

a number of different ways. There is, for example, the population of the

entire United States or the population of a particular state. There is the

population of all prisoners in the United States or the population of pris-

oners only in a specific state. Although the population of cases is fixed at

any particular time, actual populations are constantly changing across

time. For example, we can speak of the population of prisoners on any

particular day. But every day new people enter prisons and some prison-

ers are freed. Since the population of prisoners changes every day, the

population of prisoners at any one time is only a sample of the popula-

tion of prisoners across a longer period of time—for example, a year or

two.

Statisticians use different symbols to distinguish statistics on a popula-

tion from statistics on a sample (see Table 6.1). Population statistics, or

parameters, are defined using Greek letters. For example, the parame-

ter for the mean in a distribution of population scores is represented by

� and the standard deviation by �. Sample statistics are represented by

roman letters. We denote the mean in a distribution of sample scores as

and the standard deviation as s.

Why do we study sample statistics if we really want to say something

about population parameters? It certainly makes more sense to collect in-

formation on the population if that is what is of interest in the long run.

In practice, however, it is usually very difficult to gain information on the

universe, or total group of cases in the population. One reason is sim-

ply financial. As was pointed out in Chapter 1, to carry out just one sur-

vey of the U.S. population regarding their attitudes toward crime would

exhaust the budget of the National Institute of Justice (the major funder

of research in criminal justice in the United States) for many years. But

beyond the costs of such studies, there is the problem of their manage-

ment. A study of an entire population will often demand contact with

hundreds of thousands or even millions of people. Such an effort is

likely to be not just expensive, but difficult to manage and time consum-

ing to complete.

Because of the difficulty of gaining information on the characteristics

of an entire population, such parameters are generally unknown. How-

ever, when a parameter is available, there is no point in drawing statis-

tics from a sample. In recent years, advances in computer technology

X

Representing Population Parameters and Sample Statistics

MEAN VARIANCE STANDARD DEVIATION

Sample distribution s 2 s Population distribution � �2 �

X

Table 6.1

128 C H A P T E R S I X : T H E L O G I C O F S T A T I S T I C A L I N F E R E N C E

and recognition by public officials of the importance of data in making

policy decisions about the criminal justice system have led to the devel-

opment of a number of databases that include information on the popu-

lation of cases. For example, we now have population parameters on

characteristics of sentencing in the federal courts and basic demographic

characteristics of offenders held in jails and prisons. Information on the

population of arrests and emergency calls to the police is routinely col-

lected and computerized in most cities. One scholar suggests that this

trend means that criminologists in the future will have to pay less and

less attention to samples and the problems they create for researchers.1

However, whatever the future will bring, at present criminal justice re-

searchers must rely primarily on sample statistics in trying to say some-

thing about the characteristics of a population.

Given our reliance on sample statistics, it is important at the outset to

define how they might differ from population parameters. One obvious

difference is that sample statistics are generally known—put differently,

they can be defined by the researcher in the context of a research study.

In contrast, parameters are generally unknown, although, as we noted

above, there is a trend toward development of parameters about major

issues in criminal justice.

Even though parameters are often unknown, they are assumed to be

fixed. By that we mean that there is one true parameter for any measure.

For example, there is a true mean age at first arrest for the population of

all criminals in the United States at a specific time. In contrast, sample

statistics vary from sample to sample. For example, if you were to draw

10 samples from a population, using exactly the same method each time,

each sample would likely provide different sample statistics.

This is illustrated in Table 6.2. Ten random samples of 100 offenders

were drawn from a population of 1,940 offenders. Sample statistics are

presented for mean age and number of prior arrests. Although the sample

statistics obtained are generally similar to the population parameters, each

sample provides a somewhat different group of estimates, and in some

cases the differences are relatively large. In the case of sample 10, for ex-

ample, the average number of arrests for the sample is more than a third

lower than the population score. In sample 4, the average age is more

than two years older than the population parameter. This occurs despite

the fact that we drew each of the samples using the same technique and

from the same population of scores. You might want to try this yourself

by drawing a series of samples from your class or dormitory. Using the

same method, you will almost always obtain different sample statistics.

1M. Maltz, “Deviating from the Mean: The Declining Significance of Significance,”

Journal of Research in Crime and Delinquency 31 (1994): 434–463.

T H E R E S E A R C H H Y P O T H E S I S 129

This fact is one of the fundamental problems we face in statistics. We

want to make statements about populations, but we generally must rely

on sample statistics to do so. If sample statistics vary from sample to

sample, how can we use them to make reliable statements about the pa-

rameters associated with a population? Put differently, what is the use of

most studies in criminal justice, if they are based on samples rather than

populations? Fortunately, there is an area of statistics that provides us

with a systematic way to make decisions about population parameters

based on sample statistics. This area is called statistical inference, and

in the remaining sections of this chapter we focus on the logic that un-

derlies statistical inference.

T h e R e s e a r c h H y p o t h e s i s

Statistical inference begins with the definition of the questions that the

researcher seeks to answer in a research project. Sometimes research

questions in criminal justice are focused on specific agencies in the

criminal justice system. For example, we may want to learn more about

the police, the courts, or probation services. Other times research ques-

tions revolve around broad theoretical concerns that can be applied

across criminal justice agencies. We may, for example, seek to define

common features of criminal justice programs that lead to a reduction in

recidivism (reoffending). Sometimes our questions relate to offenders,

other times to victims of crime or criminal justice agents.

To answer a research question, we have to set up at least one and

sometimes several research hypotheses related to it. A hypothesis is a

proposed answer to our research question that we can then test in the

Ten Random Samples of 100 Offenders Drawn

from a Population of 1,940 Offenders

MEAN AGE MEAN ARRESTS

Population 39.7 2.72 Sample 1 41.4 2.55 Sample 2 41.2 2.19 Sample 3 38.8 2.09 Sample 4 42.1 3.45 Sample 5 37.9 2.58 Sample 6 41.1 2.62 Sample 7 39.2 2.79 Sample 8 39.2 2.48 Sample 9 37.8 2.55 Sample 10 37.7 1.72

Table 6.2

130 C H A P T E R S I X : T H E L O G I C O F S T A T I S T I C A L I N F E R E N C E

context of a study. Stating a research hypothesis does not mean that we

assume that the hypothesis is true. Rather, it focuses our research ques-

tion in such a way that it can be directly examined in the context of a

study. When the research hypothesis does not indicate a specific type of

outcome, stating only that there is a relationship or a difference, we say

that it is a nondirectional hypothesis. However, in those cases where

a researcher has a very clear idea of what to expect—based on prior re-

search evidence and/or theory—the research hypothesis may be more

Suppose we are interested in comparing the arrest records of drug-

involved offenders with those of offenders who do not use drugs. Our

research hypothesis might be simply that the arrest records of drug-

behavior among drug-involved offenders, we might want to state a direc-

tional hypothesis—that drug-involved offenders have more serious arrest

records than do non–drug-involved offenders. One problem with choos-

ing the latter option is that if we state our research hypothesis as a direc-

tional hypothesis, we are stating that we are not interested in outcomes

that fall in the opposite direction. In criminal justice research, we can

often be surprised by what we learn in a study. Accordingly, researchers

generally are cautious in defining a directional research hypothesis.

Having defined a research hypothesis, we want to examine whether it

is true for the population in which we are interested. For our example of

drug-involved offenders, if we could collect information about all offend-

ers, we could simply look at the parameters drawn from our study to see

whether they support the research hypothesis and, if so, to what degree.

In this case, we would not need to use the logic of statistical inference.

We would collect data directly on the population parameters. But ordi-

narily we cannot collect information on the population parameters and

must rely on the statistics drawn from a sample in making our decision.

Our problem is that we cannot come to an absolute conclusion regard-

ing the research hypothesis because we know that statistics vary from

sample to sample.

On the basis of a sample, we can never be sure of the true value of a

population parameter. Accordingly, we can never be absolutely certain

as to whether the research hypothesis is true. But does the fact that we

cannot be sure mean that we cannot come to a reasonable conclusion

regarding our hypotheses?

In fact, we often make decisions about hypotheses on the basis of

samples in our daily lives. For example, let’s say that you are deciding

involved offenders and offenders who do not use drugs are different

(a nondirectional hypothesis). But based on prior knowledge of criminal

precise. In this case, the researcher may specify the nature of the relation-

ship that is expected. Such a research hypothesis is called a directional

hypothesis. When a directional hypothesis is used, the researcher

states at the outset that he or she is interested in a specific type of

outcome—for example, that one group has more arrests than another.

T H E N U L L H Y P O T H E S I S 131

whether to sign up for a course taught by an instructor named Professor

Justice. One issue that you are particularly concerned about is the impact

the course will have on your grade point average. To make an informed

decision about the course, you might decide to ask friends of yours who

took the course last year how Professor Justice grades in comparison to

others at your college. Although you might not think of them quite in

this way, your friends represent your sample. In turn, the hypothesis that

the professor grades differently from other faculty members in your col-

lege is similar to a research hypothesis.

If your friends gave a mixed view or generally were unable to say

whether Professor Justice grades more harshly or more easily than other

professors, you would likely conclude that the course would not have

much impact on your grade point average. Put differently, you would

decide that the research hypothesis is probably false. If most of your

friends said that the professor is an easy grader or, conversely, that she

was a hard grader, you would take this as evidence that the research hy-

pothesis is most likely correct—that the professor grades differently and

that the course is likely to have an impact on your grade point average.

Once you have made the decision that Professor Justice is different

from others, you are likely to assess how she is different. If your friends

define the professor as a hard grader, you might decide to avoid the

course because you fear you would get a lower grade than is usual with

other professors. If they define the professor as an easy grader, you

might be encouraged to take the course, with the expectation that your

grade will be higher than usual.

In effect, you make a decision about the research hypothesis based

on information that you draw from your “sample” of friends. Your confi-

dence in making a decision will depend greatly on how reliable you be-

lieve your friends’ observations to be and to what degree they represent

other students in the class. This is very similar to the logic we use in

making statistical inferences from samples to populations. However, in

statistical inference, we test hypotheses not in reference to the research

hypothesis but in reference to a type of hypothesis that statisticians call

the null hypothesis.

T h e N u l l H y p o t h e s i s

The null hypothesis—or H0—gains its name from the fact that it usually

states that there is no relationship, or no difference. It is the flip side of

the research hypothesis (H1), which usually posits that there is a relation-

ship. In the example of the professor’s grading, the null hypothesis

would simply be that “there is no difference between the grading of Pro-

fessor Justice and that of others in the university.”

132 C H A P T E R S I X : T H E L O G I C O F S T A T I S T I C A L I N F E R E N C E

In practice, in statistics, we make decisions about hypotheses in rela-

tion to the null hypothesis rather than the research hypothesis. This is

because the null hypothesis states that the parameter in which we are in-

terested is a particular value. For example, returning to the comparison

of drug-involved and other offenders, your null hypothesis (H0) might be

that there is no difference between the two groups in the average num-

ber of crimes committed, or, put differently, that the difference is equal

to zero. In the case of Professor Justice’s grading, the null hypothesis

also states that there is no difference, or again that the difference be-

tween the average grade given by Professor Justice and the average

grade given by her colleagues is equal to zero.

In contrast, the research hypothesis is ordinarily not stated in exact

terms. A number of potential outcomes can satisfy the research hypothe-

sis. In the example of the professor’s grading, any average grade that is

different from that of other professors in the college is consistent with

the research hypothesis. But only one result, that the professor’s grading

is the same as that of others, is consistent with the null hypothesis. The

null hypothesis, accordingly, has the advantage of defining a specific

value for the population parameter.2

By stating the null and research hypotheses, we have taken a first

very important step in making statistical inferences. However, we still

have the problem of how to make decisions about these hypotheses on

the basis of sample statistics.

Whenever we rely on sample statistics to make statements about popu-

This means that when we test hypotheses in research, we generally

do not ask whether a hypothesis is true or false. To make such a state-

ment would require knowledge about the population parameters.

Rather, we ask whether we can make an inference, or draw a conclu-

sion, about our hypotheses based on what we know from a sample. In

statistical inference, we use sample statistics to infer to, or draw conclu-

sions about, population parameters.

2Some statisticians prefer to call the research hypothesis the “alternative” hypothesis,

because we can, in theory, choose any value as the null hypothesis, and not just the

value of zero or no difference. The alternative hypothesis, in this case, can be defined

as all other possible outcomes or values. For example, you could state in your null

hypothesis that the professor’s grades are, on average, five points higher than those of

other professors in the college. The alternative hypothesis would be that the profes-

sor’s grades are not, on average, five points higher than those of other professors.

lation parameters, we must always accept that our conclusions are

tentative. The only way to come to a definitive conclusion regarding the

population parameter is to actually examine the entire population. This

happens more and more with criminal justice data today. However, in most

research, we are still able to collect information only about sample statistics.

R I S K S O F E R R O R I N H Y P O T H E S I S T E S T I N G 133

In order to understand the logic of making inferences, it will help to

return to our example of drug-involved offenders. Let’s say we are inter-

ested in the number of crimes that offenders commit in a given year. We

decide to use arrests as our measure of criminal behavior. We might state

our null hypothesis as follows: “Drug-involved offenders and offenders

who do not use drugs have, on average, the same number of arrests in a

given year.” To test our hypothesis, we take a sample of drug-involved

As illustrated earlier in regard to Professor Justice’s grading, in every-

day life we make such decisions through a combination of intuition,

prior experience, and guesswork. In statistical inference, we take a sys-

tematic approach to this decision making, which begins with the recog-

nition that whatever decision we make has a risk of error.

R i s k s o f E r r o r i n H y p o t h e s i s T e s t i n g

What types of error do we risk when making a decision about a popula-

pothesis concerning arrests among drug-involved and non–drug-involved

offenders, there are only two possible scenarios for the population. In

the first case, the null hypothesis is true, meaning that there is no differ-

ence in average number of arrests between offenders in the population

who use drugs and those who do not. Alternatively, the null hypothesis

may be false, meaning that there is a difference in the population be-

Based on our sample statistic, we can, as well, come to only two pos-

sible conclusions regarding the null hypothesis. We can reject the null

hypothesis and infer that there is a difference in the average numbers of

arrests of drug-involved and other offenders in the population. Alterna-

tively, we can fail to reject the null hypothesis and state that our sample

does not provide sufficient evidence to conclude that there is a differ-

ence in the average numbers of arrests of drug-involved offenders and

offenders in the population who do not use drugs.

tion parameter from a sample? A simple way to examine this question

is to compare the potential decisions that can be made about a null

hypothesis with the value of the population parameter. For our null hy-

do offenders who do not use drugs.

offenders have, on average, either fewer or more arrests in a year than

tween drug-involved and other offenders. In this case, drug-involved

offenders and another sample of offenders who do not use drugs. We find

that drug-involved offenders in our sample have a mean of five arrests

per year, whereas offenders who do not use drugs have a mean of three

arrests per year. Should we reject the null hypothesis? Should we conclude

that there is a difference in the numbers of arrests in the population

based on results from our sample?

134 C H A P T E R S I X : T H E L O G I C O F S T A T I S T I C A L I N F E R E N C E

If we cross these two sets of possibilities, we define four possible situ-

ations, as represented in Figure 6.1. Two of these are desirable, because

they suggest that our decision about the null hypothesis is consistent

with the population parameter. In one case (box 1), we fail to reject the

null hypothesis, and it is in fact true. In the second (box 4), we reject the

null hypothesis on the basis of our sample results, and the null hypothe-

sis is false.

In the remaining situations, however, our decisions are not consistent

with the population parameter. In one (box 2), we fail to reject the null

hypothesis on the basis of our sample statistic, but it is in fact false. In

this case, we have made what statisticians call a Type II (or beta)

error. A Type II error would occur in our example of arrests among

offenders if we did not reject the null hypothesis on the basis of our

sample results when in fact the average numbers of arrests for drug-

involved and other offenders were different in the population to which

we want to infer. We make a Type I (or alpha) error (see box 3)

when we reject the null hypothesis on the basis of sample statistics but

H0 is true. In this case, we infer from our sample that drug offenders are

different from offenders who do not use drugs, when in fact they are

similar in the population.

Whenever we make a decision about a population parameter from a

sample statistic, we risk one of these two types of statistical error. If we

Box 1 ✔

Box 3 �

Type I error

Box 2 �

Type II error

Box 4 ✔

H0 = True H0 = False

Population

Reject H0

Decision

Fail to Reject H0

Types of Error in a Statistical TestFigure 6.1

S T A T I S T I C A L L E V E L S O F S I G N I F I C A N C E 135

fail to reject the null hypothesis, there is always the possibility that we

have failed to reject it when it was false (Type II error). If we reject the

null hypothesis, there is always the possibility that we have rejected it

when it was true (Type I error). Although we cannot avoid the possibil-

ity of error when we study samples, we can decide at the outset how

R i s k s o f E r r o r a n d S t a t i s t i c a l L e v e l s o f S i g n i f i c a n c e

In statistical inference, we assess the risk of making a wrong decision

about the population parameter in reference to Type I error. This is why

it is very important in statistical inference that we use the phrase “reject”

or “fail to reject” the null hypothesis, rather than the simpler statement

that we “reject” or “accept” the null hypothesis. Type I error is concerned

with rejection of the null hypothesis when it is true. It does not refer di-

rectly to the risk of accepting the null hypothesis when it is false (Type II

error). This latter problem will be the focus of a discussion in Chapter

make a decision either to “reject” or to “fail to reject” the null hypothesis

on the basis of the amount of Type I error we are willing to risk.

We define the amount of Type I error we are willing to risk as the

significance level of a test of statistical significance. In a test of statis-

tical significance, we make a decision to reject or to fail to reject the null

hypothesis on the basis of a sample statistic. The significance criterion,

or level, of a test of statistical significance is ordinarily represented by the

symbol �. The estimate of the risk of Type I error that is associated with

rejecting the null hypothesis in a test of statistical significance (based on

a sample statistic) is called the observed significance level and is ordi-

narily represented by the symbol p. In statistical inference, we first iden-

tify the amount of Type I error we are willing to risk, or the significance

level of a test. We then estimate the observed significance level from our

sample statistics. Finally, we compare the observed significance level

gained from our study with the criterion significance level we set at the

outset of our test of statistical significance. If the observed significance

level is less than the significance criterion, or level, that we set at the

outset of the test, we reject the null hypothesis. In the next chapter, we

examine how statisticians estimate the observed significance level of a

test. At this juncture, it is important to consider how we decide on the

amount of Type I error we are willing to risk. How do we choose the

significance level in a test of statistical significance?

hypotheses.

much risk or error we are willing to take in making decisions about

21. For now, it is important to remember that in statistical inference we

136 C H A P T E R S I X : T H E L O G I C O F S T A T I S T I C A L I N F E R E N C E

If we are willing to take a good deal of risk of Type I error, we set a

very lenient significance level. This means that we are willing to reject

the null hypothesis on the basis of our sample statistic, even if the risk of

a Type I error is fairly large. If we set a very strict significance level, this

means that we are unwilling to reject the null hypothesis unless we are

fairly certain that our decision is correct.

If we return to the example of Professor Justice’s grading, the impor-

tance of Type I error in making statistical inferences will become clearer.

Let’s say that every one of the friends you ask reports that the professor

is a much easier grader than other professors at your college. In this

case, you would probably conclude that the observed significance level,

or risk of a Type I error, in your study was very small. It is unlikely that

all of your friends would say that the professor was an easy grader if she

was in reality very similar to others in the college. Of course, your

friends could provide a mistaken view of the professor’s grading habits.

But you would probably assume that this is not very likely.

But what if your friends provided you with a mixed view of Professor

Justice’s grading? What if 40% of your friends say that Professor Justice

grades similarly to other professors and 60% say that she is easier? Would

you be so confident in rejecting the null hypothesis with these results?

Overall, the majority of your friends still say the professor is a relatively

easy grader. But in this case there is a substantial group reporting that

she grades much as other professors do. If you set a strict significance

level, you might not be willing to reject the null hypothesis based on

your observations (your observed significance level), and you might sim-

ply conclude that there is not enough evidence, on the basis of your

sample of friends, to say that Professor Justice grades differently.

It might seem at first that we would want to be fairly lenient in setting

our significance level. Often, in defining the research hypothesis, we are

expressing what we believe to be true. Why, then, would we want to

make it difficult to reject the null hypothesis? By rejecting the null hy-

pothesis that there is no difference, we are led to infer that the research

hypothesis is correct. This would seem in our best interest. In fact, by

convention we set a fairly strict significance level. In order to reject the

null hypothesis, we are expected to provide convincing evidence that

our conclusions reflect the true population parameters.

What is “convincing evidence”? How much risk of a Type I error

should we be willing to take in a test of statistical significance? In

criminal justice, and in most of the social sciences, a 5% level of statis-

tical significance is generally considered rigorous enough for tests of

hypotheses. This means that if the observed significance level of our

test is greater than 0.05, we will fail to reject the null hypothesis. If the

observed significance level is less than 0.05, we will reject the null

hypothesis.

D E P A R T I N G F R O M C O N V E N T I O N A L C R I T E R I A 137

D e p a r t i n g f r o m C o n v e n t i o n a l S i g n i f i c a n c e C r i t e r i a

In statistics, as in everyday life, it is simplest to follow the accepted con-

research may be better served by departing from established norms.

Sometimes this is the case because our criterion is not cautious enough

for the issue we are examining.

For example, a criminal justice agency making a decision about its

policies may want you to use a stricter standard than you would ordinar-

ily apply to your research. This may be the case if your conclusions

could lead to expensive or time-consuming changes in the structure or

activities of the agency. The agency will likely not want to make such

changes unless you are very confident that your conclusions are correct.

In this situation, you might decide to set your significance level at 0.01 or

0.001, meaning that you are willing to reject the null hypothesis only if

the observed significance level of a test is less than 1% or less than 0.1%.

Sometimes a researcher may decide to use a more lenient significance

level than 5%. This ordinarily occurs when the researcher is particularly

concerned with a Type II error rather than a Type I error. In the Min-

neapolis hot spots experiment, for example, which evaluated the impact

of police patrol on street blocks with high crime activity, the principal in-

vestigators discussed the dangers of a Type II error at the outset of their

study.3 They argued that conventional significance criteria might be too

strict for assessing the effectiveness of new police initiatives. Failure to

reject the null hypothesis of no program impact in this case, if it were

false, would lead the police not to pursue a potentially effective new

method of police patrol. The principal investigators, accordingly, de-

cided to use a 0.10, rather than a 0.05, significance level.

Why would a concern with a Type II error lead us to change the sig-

nificance level of our test of hypotheses? As we noted earlier, the signifi-

cance level is based on a Type I error, not a Type II error. However, the

two types of statistical error are related. When we increase the risk of a

Type I error, we reduce the risk of a Type II error. When we decrease

the risk of a Type I error, we increase the risk of a Type II error. This

3See Lawrence Sherman and David Weisburd, “General Deterrent Effects of Police

Patrol in Crime ‘Hot Spots’: A Randomized Study,” Justice Quarterly 12:4 (1995):

625–648.

ventions. Accordingly, most criminal justice researchers apply the 5%

significance level fairly automatically to the research questions they

consider. The problem with this approach is that the purposes of the

reject the null hypothesis in your study.

Whenever you use a significance level more stringent than 5%, it is

important to explain clearly why you have chosen to make it harder to

138 C H A P T E R S I X : T H E L O G I C O F S T A T I S T I C A L I N F E R E N C E

relationship is an obvious one, because setting a stricter significance

level (by decreasing the risk of a Type I error) naturally makes it less

likely that you will reject the null hypothesis. Similarly, when we make it

easier to reject the null hypothesis by increasing the risk of a Type I

error, it is that much harder in practice to fail to reject the null hypothe-

sis (and therefore to make a Type II error). The relationship between

Type I and Type II error is easy to understand, but it is not directly pro-

cept of “statistical power,” other factors in addition to the significance

level of a study affect the risk of a Type II error.

You should always consider carefully the implications of risks of error in

your research before setting the significance level of a test of statistical sig-

nificance. Even though you are likely in the end to rely on the conventional

norm of 5%, there are, as discussed above, some cases for which you might

want to consider stricter or more lenient levels of significance. If you do

choose a level other than 0.05, you must explain to your audience the fac-

tors that led you to depart from common practice in criminal justice.

The level of significance in a study should be defined at the outset,

and not after your results are in. If you wait until you have your sample

data, there will always be the temptation to adjust your significance level

to fit your sample statistics. This is particularly important if you decide to

use more stringent or more lenient criteria. If you use a more lenient cri-

terion, others might argue that you have made that decision in order to

allow rejection of the null hypothesis (and thus support for the research

hypothesis). If you use a stricter criterion, others might argue that you

are trying to avoid rejecting the null hypothesis. In many funded re-

search studies, researchers specify in their original proposals the signifi-

cance levels that they intend to apply to tests of hypotheses in order to

prevent such criticism later on.

In this chapter, we discussed the logic underlying statistical inference.

In the coming chapters, we will examine how statisticians define the risk

of Type I error associated with a specific outcome in a study, or the ob-

served significance level of a test. We also detail how such tests are ap-

plied to different types of statistics. You should not expect to have a full

understanding of statistical inference at this point. We have only begun

to develop these ideas and will return to them again and again.

C h a p t e r S u m m a r y

In descriptive statistics we are concerned with two types of distribu-

tions. The sample distribution is the distribution of scores in the sam-

ple. The population distribution is the distribution of scores in the

portional. As we will discuss in Chapter 21 when we examine the con-

C H A P T E R S U M M A R Y 139

population from which the sample is drawn. Population statistics are

known as parameters, and they have symbols different from those

which you have encountered so far. The mean in a population distribu-

tion is represented by � and the standard deviation by �. Parameters

are assumed to be fixed and are generally unknown. Sample statistics

are by definition known, but they vary from sample to sample. Statisti-

cians are faced with a fundamental dilemma in that they are usually in-

terested in making statements about population parameters, but they

generally study sample statistics. Statistical inference provides a solu-

tion to this dilemma.

Statistical inference begins with the definition of the research hy-

pothesis and the null hypothesis. These hypotheses are set up by the

researcher to answer the broader research question. The research hy-

pothesis is the proposed answer to a specific research question. When

the research hypothesis does not indicate a specific type of outcome,

stating only that there is a relationship or a difference, we say that it is a

nondirectional hypothesis. When the research hypothesis identifies

the nature of the relationship that is expected, it is called a directional

hypothesis. The null hypothesis generally posits that there is no such

relationship or no difference. The null hypothesis is stated in exact

terms. The research hypothesis encompasses a range of possible alterna-

Given the difficulties involved in collecting information on an entire

population, we are forced to work with samples. The tools of statistical

inference enable us to infer from a sample to a population by identifying

the risk of making a mistaken decision and determining the amount of

risk we are prepared to take. Two possible errors can be made when

making decisions about a population from a sample. A researcher who

rejects the null hypothesis when it is in fact true has made a Type I

error. A researcher who fails to reject the null hypothesis when it is in

fact false has made a Type II error.

In a test of statistical significance, we make a decision to reject or

to fail to reject the null hypothesis on the basis of a sample statistic. The

significance level defines the risk of Type I error that a researcher is

willing to take in a test of statistical significance. The risk of Type I error

associated with a specific sample statistic is the observed significance

level of a test. A commonly accepted standard significance level is 5%,

but researchers may choose to set a lower level if they want it to be

more difficult to reject the null hypothesis—or a higher level if they want

to make it easier. A researcher who wishes to depart from the accepted

standard should explain the reasons for such a decision at the outset.

Decreasing the risk of a Type I error increases the risk of a Type II error,

and vice versa.

hypothesis.

tives. It is thus easier to focus decisions on whether to reject the null

140 C H A P T E R S I X : T H E L O G I C O F S T A T I S T I C A L I N F E R E N C E

K e y T e r m s

directional hypothesis A research hy-

pothesis that indicates a specific type of

outcome by specifying the nature of the re-

lationship that is expected.

nondirectional hypothesis A research

hypothesis that does not indicate a specific

type of outcome, stating only that there is a

relationship or a difference.

null hypothesis A statement that reduces

the research question to a simple assertion

to be tested by the researcher. The null hy-

pothesis normally suggests that there is no

relationship or no difference.

observed significance level The risk of

Type I error associated with a specific sam-

ple statistic in a test. When the observed

significance level is less than the criterion

significance level in a test of statistical sig-

nificance, the researcher will reject the null

hypothesis.

parameter A characteristic of the pop-

ulation—for example, the mean number

of previous convictions for all U.S.

prisoners.

population The universe of cases that the

researcher seeks to study. The population

of cases is fixed at a particular time (e.g.,

the population of the United States). How-

ever, populations usually change across

time.

population distribution The frequency

distribution of a particular variable within a

population.

research hypothesis The antithesis of the

null hypothesis. The statement normally

answers the initial research question by

suggesting that there is a relationship or a

difference.

sample A set of actual observations or

cases drawn from a population.

sample distribution The frequency distri-

bution of a particular variable within a

sample drawn from a population.

sample statistic A characteristic of a sam-

ple—for example, the mean number of

previous convictions in a random sample

of 1,000 prisoners.

significance level The level of Type I

error a researcher is willing to risk in a sta-

tistical test of significance.

statistical inference The process of mak-

ing generalizations from sample statistics to

population parameters.

test of statistical significance A test in

which a researcher makes a decision to re-

ject or to fail to reject the null hypothesis

on the basis of a sample statistic.

Type I error Also known as alpha error.

The mistake made when a researcher re-

jects the null hypothesis on the basis of a

sample statistic (i.e., claiming that there is a

relationship) when in fact the null hypothe-

Type II error Also known as beta error.

The mistake made when a researcher fails

to reject the null hypothesis on the basis of

a sample statistic (i.e., failing to claim that

there is a relationship) when in fact the

null hypothesis is false (i.e., there actually

is a relationship).

universe The total population of cases.

by means of a study.

researcher hopes to be able to answer

research question The question the

relationship in the population).

sis is true (i.e., there is actually no such

E X E R C I S E S 141

S y m b o l s a n d F o r m u l a s

H0 Null hypothesis

H1 Research hypothesis

� Significance level of a test

� Population mean

p Observed significance level of a test

�2 Population variance

E x e r c i s e s

6.1 For each of the following random samples, describe the population to which the results could be generalized:

a. Subscribers to a magazine on hunting are asked about gun legislation.

b. Youth aged 15 to 19 years in the United States are asked about per- sonal drug use.

c. Registered voters in Los Angeles are asked whom they will vote for in the next election for mayor.

d. Visitors to a domestic violence shelter are assessed for psychologi- cal distress.

e. Members of the National Organization for Women (NOW) are asked about sexual harassment in the workplace.

f. Judges in New Jersey are asked about their sentencing philosophy.

6.2 A foundation sponsored a review of all studies carried out over the past 15 years into the link between smoking and juvenile delinquency. Eric, a criminologist commissioned by the foundation, unearthed five studies that sought to determine at what age delinquents who smoke began their habit. The five studies, conducted at the same time, use identical sampling techniques and sample sizes and draw their sam- ples from a fixed database of delinquents. The mean age, however, is different for each of the samples:

Study sample no. 1: mean age � 12.2

Study sample no. 2: mean age � 11.6

Study sample no. 3: mean age � 14.0

Study sample no. 4: mean age � 11.3

Study sample no. 5: mean age � 12.8

Overall computed mean � 12.38

142 C H A P T E R S I X : T H E L O G I C O F S T A T I S T I C A L I N F E R E N C E

a. During Eric’s presentation, one member of the foundation asks him to explain how it can be that each of the study samples produced a different result: Does this mean that there was something wrong with the sampling techniques used? How should Eric respond?

b. Another foundation member also seems confused. She asks if the overall computed mean is the mean age of the population. How should Eric respond?

6.3 A researcher collects data on the families of 20 delinquent and 20 nondelinquent children from the records of a school and checks how many children from each group come from broken homes.

a. State the null hypothesis.

b. State a nondirectional research hypothesis.

c. State a directional research hypothesis.

6.4 A researcher is interested in whether electronic monitoring of offend- ers on probation is an effective means of reducing crime. To test for an effect of electronic monitoring, probationers are randomly assigned either to a monitored group or to a control group that has no elec- tronic monitoring.

a. State the null hypothesis.

b. State a nondirectional research hypothesis.

c. State a directional research hypothesis.

6.5 A study published in a distinguished journal reported the results of a series of tests carried out on 50 convicted burglars. One of the claims of the investigators is that the average IQ of convicted burglars is 120.

a. From the following list of options, choose an appropriate null hy- pothesis and research hypothesis for testing this claim.

IQ � 120 IQ � 120 IQ � 120 IQ � 120

b. Explain your choice of null and research hypotheses.

6.6 A gang of criminals is planning to rob a supermarket. Eddy, the gang leader, reports that he “staked the store out” the day before and saw the store detective going for a 15-minute coffee break at 9.15 A.M. He sug- gests that this would be the best time to strike. Clive, a gang member, thinks that this plan is too risky—how do they know that the detective takes his break at the same time each day? Eddy, who is desperate for the money, thinks that the plan is safe enough and wants to carry out the robbery the next day. After an argument, they agree to compromise and watch the supermarket for three more days. On each of the three days, the store detective indeed takes his 15-minute break at 9:15 A.M. The gang decides to go ahead with the robbery on the fourth day.

The robbers can be seen as having set themselves a research ques- tion and having made a statistical decision based on a simple study.

E X E R C I S E S 143

a. How would you frame the robbers’ null hypothesis and research hypothesis?

b. Based on these hypotheses, what is their decision?

c. How would the robbers make a Type I error? How would the rob- bers make a Type II error? What type of statistical error ought the robbers to be most concerned with making? Explain why.

d. How does the argument between Eddy and Clive relate to the con- cept of statistical significance?

6.7 The government wishes to launch a pre-Christmas advertising cam- paign warning about the dangers of drunk driving. It suspects that dri- vers aged 18 to 21 are most likely to drive while under the influence of alcohol and is considering targeting the campaign specifically at this age group. A preliminary study gathers data on the ages of drunk dri- vers apprehended in a particular district over a six-month period.

b. How might the government make a Type I error?

c. How might the government make a Type II error?

d. The government accepts that targeting this specific age group in the advertising campaign will not cost any extra money. There is a feel- ing that the new campaign will be “worth a try,” even if the study doesn’t find enormous differences between the offending rate of 18- to 21-year-olds and that of other ages. How should these con- siderations affect the researchers’ decision on what level of signifi- cance to set?

6.8 The head of the police force in the city of Cheadle suspects that in- creasing the pay of his officers might increase their efficiency. A police researcher is assigned to test whether there is a difference between the crime-solving rates of a group of detectives who have been ran- domly awarded pay raises and a control group of detectives who have not been awarded pay raises. In writing up his report, the researcher concludes as follows:

The results show that the observed significance level is 0.14, mean- ing that rejecting the null hypothesis would run a 14% risk of a Type I error. Although a 5% significance level is considered stan- dard, in light of the potential benefits of salary increases for crime control rates, a higher 15% threshold is justified here, and the H0

may therefore be rejected.

a. What is the null hypothesis to which the researcher refers?

b. Explain why the researcher’s statistical reasoning is problematic.

research hypothesis is directional or nondirectional. a. What are the null and research hypotheses? Explain why the

144 C H A P T E R S I X : T H E L O G I C O F S T A T I S T I C A L I N F E R E N C E

6.9 A study explored whether there is a link between male aggression and climate. The researcher recorded her results as follows: 5% signifi- cance level set, H0 could not be rejected.

a. Explain these results in plain English.

b. Why is it important for the researcher to set the significance level at the beginning of the research and not at the end?

6.10 A private research foundation claims that increased regulation of handguns would reduce homicides in the United States. To examine this relationship, the foundation funds a study to assess the impact of handgun legislation on homicides in four states that recently passed laws restricting handgun ownership. The foundation expects that the reduced availability of handguns following the change in law will re- duce the number of opportunities for lethal violence. The researcher collects data on homicides in the four states for one year before and one year after the change in handgun laws.

b. Explain how the researcher could make a Type I error.

c. Explain how the researcher could make a Type II error.

dation wants the researcher to increase the significance level of the study to 10% and reject the null hypothesis. Should the researcher increase the significance level of the study? Why?

6.11 A group of researchers at a private think tank claims that the increased use of incarceration in the United States has not been harmful to the social fabric of the country. To support this claim, the researchers con- duct a study looking at rates of incarceration and rates of divorce in all fifty states for a 20-year period. The group of researchers tests for a relationship between rates of incarceration and rates of divorce, but does not expect to find a relationship between these two variables.

b. Explain how the research group could make a Type I error.

c. Explain how the research group could make a Type II error.

d. The observed significance level developed in their study is 0.03. The research group initially set a 5% risk of Type I error. One member of the research group suggests that they simply decrease the significance level of the study to 1% and fail to reject the null hypothesis. Should the research group decrease the significance level of the study? Why?

a. What are the null and research hypotheses? Explain why the research hypothesis is directional or nondirectional.

d. The results show an observed significance level of 0.06. The researcher wants to conclude that the null hypothesis cannot be rejected based on a 5% risk of Type I error. An official from the foun-

research hypothesis is directional or nondirectional. a. What are the null and research hypotheses? Explain why the

Defining the Observed Significance Level

of a Test: A Simple Example Using

the Binomial Distribution

What Role Do They Play in Inferential Statistics?

How Does One Calculate the Probability of a Given Outcome?

C h a p t e r s e v e n

S a m p l i n g d i s t r i b u t i o n s

P r o b a b i l i t i e s a n d p r o b a b i l i t y d i s t r i b u t i o n s

T h e b i n o m i a l d i s t r i b u t i o n

What are They?

What are Probability Distributions?

How are They Used?

What is It?

How is It Calculated?

What are Its Characteristics?

D. Weisburd and C. Britt, Statistics in Criminal Justice, DOI 10.1007/978-1-4614-9170-5_7,

© Springer Science+Business Media New York 2014

WHEN WE MAKE INFERENCES to a population, we rely on a statistic in

our sample to make a decision about a population parameter. At the

heart of our decision is a concern with Type I error. Before we reject

our null hypothesis, we want to be fairly confident that it is in fact false

for the population we are studying. For this reason, we want the ob-

served risk of a Type I error in a test of statistical significance to be as

small as possible. But how do statisticians calculate that risk? How do

they define the observed significance level associated with the outcome

of a test?

The methods that statisticians use for calculating the observed sig-

nificance level of a test of statistical significance vary depending on

the statistics examined. Sometimes these methods are very complex.

But the overall logic that underlies these calculations is similar, irre-

spective of the statistic used. Thus, we can take a relatively simple

example and use it as a model for understanding how the observed

significance level of a test is defined more generally in statistics. This

is fortunate for us as researchers, because it means that we do not

have to spend all of our time developing complex calculations to de-

fine risks of error. Once we understand how risks of error are defined

for one problem, we can let statisticians calculate the risks for other

more complex problems. Our concern is not with the calculations

themselves, but with understanding the general logic that underlies

them.

We begin this chapter by discussing a very simple decision. When

should we begin to suspect that a coin used in a coin toss is unfair or bi-

ased? Ordinarily, we might come to a conclusion based on common

sense or intuition. In statistics, we take a more systematic approach, rely-

ing on the logic of hypothesis testing and a type of distribution called a

sampling distribution. Using this example of the coin toss and a sam-

pling distribution called the binomial distribution, we illustrate how sta-

tisticians use probability theory to define the observed significance level,

or risk of Type I error, for a test of statistical significance.

146

T H E F A I R C O I N T O S S 147

T h e F a i r C o i n T o s s

Imagine that you and your friends play a volleyball game each week

against a group of criminal justice students from another school. You al-

ways begin the game with a coin toss to decide who will serve the ball

first. Your opponents bring an old silver dollar, which you have agreed

to use for the toss. They choose heads and continue to choose heads

each time you play. At first, this does not seem like a problem. However,

each week you play, the coin comes up heads and they serve the ball.

Suppose that this happened for four straight weeks. Would you begin

to become suspicious? What if it went on for six weeks? How many

times in a row would they have to win the coin toss before you and

your team accused them of cheating? Would they have to win for ten or

twenty weeks? You might worry about accusing them too quickly, be-

cause you know that even if the coin is fair it sometimes happens that

someone is lucky and just keeps on winning. You would want to be

fairly certain that the coin was biased before concluding that something

was wrong and taking some action.

In everyday life, you are likely to make this decision based on intu-

ition or prior experience. If you ask your classmates, each one is likely

to come up with a slightly different number of coin tosses before he or

she would become suspicious. Some students may be willing to tolerate

only four or five heads in a row before concluding that they have

enough evidence to accuse their opponents of cheating. Others may be

unwilling to reach this conclusion even after ten or fifteen tosses that

come up heads. In part, the disagreement comes from personality differ-

ences. But more important, guesswork or common sense does not give

you a common yardstick for deciding how much risk you take in coming

to one conclusion or another.

Sampling Distributions and Probability Distributions

Statistical inference provides a more systematic method for making deci-

sions about risk. The coin toss can be thought of as a simple test of statisti-

cal significance. The research hypothesis is that the coin is biased in favor

of your opponents. The null hypothesis is that the coin is fair. Each toss of

the coin is an event that is part of a sample. If you toss the coin ten times,

you have a sample of ten tosses. Recall from Chapter 6 that Type I error

is the error of falsely rejecting the null hypothesis that the coin is fair. If

you follow the common norm in criminal justice, then you are willing to

reject the null hypothesis if the risk of a Type I error is less than 5%.

But how can we calculate the risk of a Type I error associated with a

specific outcome in a test of statistical significance, or what we generally

term the observed significance level of a test? One simple way to gain an

148 C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K

estimate of the risk of unfairly accusing your friends is to check how

often a fair coin would give the same result as is observed in your series

of volleyball games. For example, let’s say that you played ten games in a

season and in all ten games the old silver dollar came up heads (meaning

that the opposing team won the toss). To check out how often this might

happen just by chance when the coin is in fact a fair one, you might go

to a laboratory with a fair coin and test this out in practice. One problem

you face is deciding how many samples or trials you should conduct. For

example, should you conduct one trial or sample by flipping your fair

coin just ten times and stopping? Or should you conduct multiple trials or

samples, each with ten tosses of the fair coin? Clearly, one trial or sample

of ten tosses will not tell you very much. Indeed, one of the reasons you

have gone to the laboratory is that you know it sometimes happens that a

fair coin will come out heads ten times in a row. What you want to know

is how rare an event this is. How often would you gain ten heads in a

row in a very large number of samples or trials of a fair coin?

The distribution that is gained from taking a very large number of

samples or trials is called a sampling distribution. In principle, one

could create a sampling distribution by drawing thousands and thou-

sands of samples from a population. For example, in the case of our

coin toss, we might conduct thousands of trials of ten flips of a fair coin.

If we recorded the outcome for each trial and placed our results in a fre-

quency distribution, we would have a sampling distribution for a sample

of ten tosses of a fair coin.

This sampling distribution would allow us to define the risk of a Type

I error we would face in rejecting the null hypothesis that the old silver

dollar is fair. For example, suppose that in the sampling distribution we

gained a result of ten heads in only 1 in 1,000 samples. If we reject the

null hypothesis in this case, our risk of making a Type I error, according

to the sampling distribution, is only 0.001. This is the observed signifi-

cance level of our test of statistical significance. In only 1 in 1,000 sam-

ples of ten tosses of a fair coin would we expect to gain a result of ten

heads. If the old silver dollar was indeed a fair coin, it would seem very

unlikely that on our one trial of ten tosses of the coin each toss would

come out heads. Of course, in making our decision we cannot be certain

that the silver dollar used in the volleyball toss is not a fair coin. While

the occurrence of ten heads in ten tosses of a fair coin is rare, it can hap-

pen about once in every 1,000 samples.

Building a sampling distribution provides a method for defining our

risk of a Type I error. However, it is very burdensome to create a sam-

pling distribution by hand or even in the laboratory. If you try out our

example of ten tosses of a fair coin, you will see that developing even

100 samples is not easy. If we had to actually construct a sampling distri-

bution every time we wanted to make a decision about a hypothesis, it

would be virtually impossible to make statistical inferences in practice.

T H E F A I R C O I N T O S S 149

Fortunately, there is another method we can use for creating sampling

distributions. This method relies on probability theory, rather than a bur-

densome effort to collect samples in the real world. Because we use

probabilities, the distributions that are created using this method are

called probability distributions. Importantly, though we rely on prob-

ability theory because it is very difficult to develop sampling distributions

in practice, we do not suffer for our approach. This is because probabil-

ity theory allows us to calculate the outcomes one would expect in a

perfect world. In the real world, we might flip the coin slightly differ-

The Multiplication Rule

In order to estimate the risk of a Type I error in the case of a series of

tosses of a fair coin, we can use the multiplication rule, a simple rule

about probabilities drawn from probability theory. The multiplication

rule tells you how likely we are to gain a series of events one after

another—in this case, a series of outcomes in a toss of a coin. It allows

us to estimate theoretically how often we would gain a specific series of

events if we drew an infinite number of samples. The multiplication rule

generally used to establish probabilities in statistics is based on the as-

sumption that each event in a sample is independent of every other

event. In the case of the coin toss, this means that the outcome of one

toss of a coin is unaffected by what happened on the prior tosses. Each

time you toss the coin, it is as if you started with a clean slate. That

would seem a fairly reasonable assumption for our problem. What wor-

ries us is that the coin is unfair overall, not that it is becoming less or

more unfair as time goes on.

An example of a series of events that are not independent is draws

from a deck of cards. Each time you draw a card, you reduce the num-

ber of cards left in the deck, thus changing the likelihood of drawing any

card in the future. For example, let’s say that on your first draw from a

deck of 52 cards you drew an ace of spades. On your second draw, you

cannot draw an ace of spades because you have already removed it from

the deck. The likelihood of drawing an ace of spades on the second

draw has thus gone from 1 in 52 to 0 in 51. You have also influenced the

likelihood of drawing any other card because there are now 51, not 52,

cards left in the deck. If you want a series of draws from a deck to be

independent of one another, you have to return each card to the deck

after you draw it. For example, if you returned the ace of spades to the

deck, the chance of choosing it (assuming the deck was mixed again)

would be the same as it was on the first draw. The chances of choosing

any other card would also be the same because you once again have all

52 cards from which to draw.

another, thus affecting the outcomes we gain. In probability theory, we

remove the imperfections of the real world from our estimates.

ently as we got tired or the coin might become worn on one side or

150 C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K

The multiplication rule for four independent events is stated in Equa-

tion 7.1. It says that the likelihood of any series of events, represented as

A, B, C, and D, happening one after another is equal to the probability

of event A times the probability of event B times the probability of event

C times the probability of event D. The rule can be extended to as many

events as you like. We have chosen four here, because this was the

number of volleyball games we began with. If you wanted to extend the

rule, you would simply include the additional events on the left side of

the equation (for example, E and F ) and include the probability of each

on the right side of the equation [e.g., P (E ) • P (F )].

P (A & B & C & D) � P (A) • P (B) • P (C ) • P (D) Equation 7.1

Extending this logic to our example of the coin toss is straightforward.

The probability of A and B and C and D can represent the probability

that four tosses in a row come up heads. Our main problem is to estab-

lish what the probability is of a head coming up on any particular toss of

the coin. In this, we are helped by our null hypothesis, which states that

the coin is fair. If the coin is fair, then there should be an even chance of

a head or a tail coming up on any particular toss of the coin. Put differ-

ently, under the assumption of the null hypothesis that the coin is fair,

the likelihood of a head coming up is 0.50.

What, then, does the multiplication rule tell us about the chances of

getting four heads in a row on four tosses of a fair coin? In part a of

Table 7.1, we calculate that probability by multiplying 0.50 (the likeli-

hood of gaining a head on any toss of a fair coin) by itself four times, to

represent four tosses of an unbiased coin. The result is 0.0625. If you

had decided at the outset to make a decision about the null hypothesis—

that the coin is fair—after four tosses of the coin, then you have con-

ducted a type of test of statistical significance. For this test, the observed

significance level (or risk of a Type I error) of rejecting the null hypothe-

sis on the basis of four heads is 0.0625.

If you use the norms of criminal justice research, this is not enough,

however, for you to reject the null hypothesis that the coin is fair. Crimi-

nal justice researchers generally want the risk of falsely rejecting the null

hypothesis to be less than 5%. A bit over 6% is still more than the 5% sig-

nificance criterion that is used by convention. So if you had decided to

Probabilities Associated with Tosses of a Fair Coin

a. P (A & B & C & D) � P (A) • P (B) • P (C) • P (D) � (0.50)(0.50)(0.50)(0.50) � 0.0625

b. P (A & B & C & D & E) � P (A) • P (B) • P (C) • P (D) • P (E) � (0.50)(0.50)(0.50)(0.50)(0.50) � 0.0313

c. P (A & B & C & D & E & F & G & H & I & J) � P (A) • P (B) • P (C) • P (D) • P (E) • P (F ) • P (G) • P (H) • P (I ) • P (J) � (0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50) � 0.0010

Table 7.1

D I F F E R E N T W A Y S O F G E T T I N G S I M I L A R R E S U L T S 151

make a decision about the fairness of the coin after four coin tosses, you

would probably not want to reject the null hypothesis that the coin is fair

and confront your opponents. Under this criterion, the likelihood of

falsely rejecting the null hypothesis, or the observed significance level of

your test, would have to be below 0.05.

What if you had decided at the outset to make a decision about the

null hypothesis after five tosses of a coin? Would a result of five heads in

a row lead you to reject the null hypothesis? As illustrated in part b of

Table 7.1, the multiplication rule tells you that the likelihood of getting

five heads in a row if the coin is fair is 0.0313. This is less than our

threshold of 0.05, and thus would lead you to reject the null hypothesis.

Is this consistent with your earlier commonsense conclusions? Students

are usually surprised at how quickly they reach the 0.05 significance

threshold in this example.

If you had decided at the outset that you would need ten or fifteen

heads in a row, you may want to reconsider, given what we have

learned from the multiplication rule. The likelihood of getting ten heads

in a row in ten tosses of a fair coin is only 1 in 1,000 (see part c of Table

7.1). The likelihood of getting fifteen heads in a row in fifteen tosses of a

fair coin is even lower: about 3 in 100,000. In both of these cases, you

would take a very small risk of a Type I error if you rejected the null hy-

pothesis. Nonetheless, the multiplication rule tells us that, even if the

coin is fair, it is possible to get ten or even fifteen heads in a row. It just

does not happen very often.

The multiplication rule allows us to estimate how often we would ex-

pect to get a series of specific outcomes in a very large number of trials

or samples, without actually going out and doing the hard work of con-

structing a sampling distribution in the real world. However, the problem

as examined so far assumes that the coin will come up heads every time.

What if the coin comes up heads generally, but not all the time? For ex-

ample, what if you play ten games and the coin comes up heads nine

times? The situation here is not as one-sided. Nonetheless, it still seems

unlikely that your opponents would win most of the time if the coin

were fair. The multiplication rule alone, however, does not allow us to

define how likely we are to get such a result.

D i f f e r e n t W a y s o f G e t t i n g S i m i l a r R e s u l t s

The multiplication rule allows us to calculate the probability of getting a

specific ordering of events. This is fine so far in our coin toss because in

each example we have chosen there is only one way to get our out-

come. For example, there is only one way to get five heads in five coin

152 C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K

tosses or ten heads in ten coin tosses. In each case, your opponents

must toss a head before each game. This would be the situation as well

if your opponents tossed tails ten times in ten coin tosses. However, for

any outcome in between, there is going to be more than one potential

way to achieve the same result.

For example, if your opponents tossed nine heads in ten coin tosses,

they could win the coin toss nine times (with a head) and then lose the

toss in the tenth game (with a tail). Or they could lose the first toss (with

a tail) and then win the remaining nine. Similarly, they could lose the

second, third, fourth, fifth, sixth, seventh, eighth, or ninth coin toss and

win all the others. Each of these possible ordering of events is called an

arrangement. As is illustrated in Table 7.2, there are ten possible

arrangements, or different ways that you could get nine heads in ten

coin tosses. In the case of ten heads in ten coin tosses, there is only one

possible arrangement.

It is relatively simple to list all of the arrangements for our example of

nine heads in ten coin tosses, but listing becomes very cumbersome in

practice as the split of events becomes more even. For example, if we

were interested in how many ways there are of getting eight heads in ten

coin tosses, we would have to take into account a much larger number

of arrangements. As Table 7.3 illustrates, it takes a good deal of effort to

list every possible arrangement even for eight heads. In the case of a

more even split of events—for example, five heads in ten tosses—it be-

comes extremely cumbersome to list each arrangement one by one. Be-

cause of this, we generally use the formula in Equation 7.2 to define the

number of arrangements in any series of events.

Equation 7.2

On the left side of this equation we have N “choose” r, where N is the

number of events in the sample and r is the number of successes in the

�N

r� � N !

r !(N � r)!

Arrangements for Nine Successes in Ten Tosses of a Coin

Arrangement 1 � � � � � � � � � � Arrangement 2 � � � � � � � � � � Arrangement 3 � � � � � � � � � � Arrangement 4 � � � � � � � � � � Arrangement 5 � � � � � � � � � � Arrangement 6 � � � � � � � � � � Arrangement 7 � � � � � � � � � � Arrangement 8 � � � � � � � � � � Arrangement 9 � � � � � � � � � � Arrangement 10 � � � � � � � � � �

� � Head; � � Tail

Table 7.2

D I F F E R E N T W A Y S O F G E T T I N G S I M I L A R R E S U L T S 153

total number of events. In our case, N is the number of coin tosses and r

is the number of times that the coin comes up heads. Put together, this

statement establishes our question: How many ways are there of gaining

r heads in N tosses of a coin? To answer our question, we need to solve

the right side of the equation. Each of the terms in the equation is de-

fined as a factorial, indicated by the symbol !. When we take a factorial

of a number, we merely multiply it by all of the whole numbers smaller

than it. For example, 3! is equal to (3)(2)(1), or 6. Because factorials get

very large very quickly, a table of factorials is presented in Appendix 1.

Note that 0! � 1. Applied to our problem of nine heads in ten coin

tosses, Equation 7.2 is worked out below:

Arrangements for Eight Successes in Ten Tosses of a Coin

1: ���������� 16: ���������� 31: ���������� 2: ���������� 17: ���������� 32: ���������� 3: ���������� 18: ���������� 33: ���������� 4: ���������� 19: ���������� 34: ���������� 5: ���������� 20: ���������� 35: ���������� 6: ���������� 21: ���������� 36: ���������� 7: ���������� 22: ���������� 37: ���������� 8: ���������� 23: ���������� 38: ���������� 9: ���������� 24: ���������� 39: ����������

10: ���������� 25: ���������� 40: ���������� 11: ���������� 26: ���������� 41: ���������� 12: ���������� 27: ���������� 42: ���������� 13: ���������� 28: ���������� 43: ���������� 14: ���������� 29: ���������� 44: ���������� 15: ���������� 30: ���������� 45: ����������

� � Head; � � Tail

Table 7.3

W orking It Out

� 10

� 3,628,800

362,880(1)

� 10!

9! 1!

�10 9 � �

10! 9!(10 � 9)!

�N

r� � N !

r !(N � r)!

154 C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K

Using this method, we get the same result as before. There are ten possi-

ble arrangements to get nine heads in ten tosses of a coin. When we

apply Equation 7.2 to the problem of five heads in ten coin tosses, its

usefulness becomes even more apparent. There are 252 different ways of

getting five heads in ten tosses. Listing each would have taken us consid-

erably longer than the calculation below.

W orking It Out

� 252

� 3,628,800

120(120)

� 10!

5! 5!

�10 5 � �

10! 5!(10 � 5)!

�N

r� � N !

r !(N � r)!

S o l v i n g M o r e C o m p l e x P r o b l e m s

Now that we have a method for calculating arrangements, we can return

to our original problem, which was to define the probability of your op-

ponents tossing the coin in ten games and getting heads nine times. Be-

cause there are ten different ways of getting nine heads in ten coin

tosses, you need to add up the probabilities associated with these ten se-

quences. This is what is done in Table 7.4. The multiplication rule is

used to calculate the probability for each sequence, or arrangement,

under the assumption of the null hypothesis that the coin is fair. Because

our null hypothesis states that the coin is fair, we can assume that the

chances of gaining a head and a tail are even. The probability of any

event, whether a head or a tail, is 0.50, and the probability of a sequence

of ten events is always the same. This makes our task easier. But it is im-

portant to note that if the null hypothesis specified an uneven split (for

example, 0.75 for a head and 0.25 for a tail), then each of the sequences

would have a different probability associated with it. In any case, the

likelihood of getting any one of these sequences is about 0.001, rounded

to the nearest thousandth. When we add together the ten sequences, we

get a probability of 0.010.

T H E B I N O M I A L D I S T R I B U T I O N 155

This means that we would expect to get nine heads in ten coin tosses

of a fair coin in only about 1 in 100 samples in a very large number of

trials of a fair coin. But is this the observed significance level of a test of

statistical significance in which we gain nine heads in ten tosses of a

coin? Or put in terms of Type I error, is this the total amount of risk we

face of falsely rejecting the null hypothesis when we gain nine heads?

The answer to this question is no, although it may be difficult at first to

understand why. If we are willing to reject the null hypothesis based on

an outcome of nine heads in ten trials, then we are, by implication, also

willing to reject the null hypothesis if our outcome is ten heads in ten tri-

als. In calculating our total risk of a Type I error, we must add together

the risk of all potential outcomes that would lead us to reject the null hy-

pothesis. This is why, when testing hypotheses, we generally do not

begin with an estimate of the specific probability associated with a single

outcome, but rather with the sampling distribution of probabilities of all

possible outcomes.

T h e B i n o m i a l D i s t r i b u t i o n

To construct a probability or sampling distribution for all of the possible

outcomes of ten coin tosses, we could continue to compute the number

of permutations and the likelihood of any particular arrangement. How-

ever, Equation 7.3 provides us with a more direct method for calculating

the probability associated with each of the potential outcomes in our

The Sum of Probabilities for All Arrangements of Nine Heads in Ten Tosses of a Fair Coin

PROBABILITY

Arrangement 1 � � � � � � � � � � 0.001 Arrangement 2 � � � � � � � � � � 0.001 Arrangement 3 � � � � � � � � � � 0.001 Arrangement 4 � � � � � � � � � � 0.001 Arrangement 5 � � � � � � � � � � 0.001 Arrangement 6 � � � � � � � � � � 0.001 Arrangement 7 � � � � � � � � � � 0.001 Arrangement 8 � � � � � � � � � � 0.001 Arrangement 9 � � � � � � � � � � 0.001 Arrangement 10 � � � � � � � � � � 0.001

Total Probability: 0.01

Probability of throwing each arrangement of 10 throws � P (A) • P (B) • P (C) • P (D) • P (E) • P (F ) • P (G) • P (H) • P (I ) • P (J ) � (0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50) � 0.001

Table 7.4

156 C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K

sample. Equation 7.3 is generally defined as the binomial formula, and

the distribution created from it is called the binomial distribution. As

the name suggests, the binomial distribution is concerned with events in

which there are only two possible outcomes—in our example, heads

and tails.

Equation 7.3

The binomial formula may look confusing, but most of it is familiar from

material already covered in this chapter. The left-hand side of the equa-

tion represents the quantity in which we are interested—the probability

of getting r successes (in our case, r heads) in a sample of N events (for

us, ten tosses of a coin). The first part of the equation provides us with

the number of arrangements for that number of heads. This quantity is

then multiplied by pr(1 � p)N�r, where p is the probability of a successful

outcome (a head) under the null hypothesis and r is the number of suc-

cesses. This formula gives us the probability associated with each

arrangement. Although this part of the equation looks somewhat differ-

ent from the multiplication rule we used earlier, it provides a shortcut for

getting the same result, as the example below illustrates.

We have already calculated the likelihood of getting nine or ten heads

in ten coin tosses if the coin is fair. To complete our sampling distribu-

tion, we need to compute probabilities associated with zero through

eight heads as well. Let’s begin with eight heads in ten coin tosses of an

unbiased coin:

Step 1: Calculating the number of arrangements

P �10 8 � �

10! 8!(10 � 8)!

(0.50)8(1 � 0.50)10�8

P �N

r� � N !

r !(N � r)! pr(1 � p)N�r

W orking It Out

� 45

� 3,628,800

40,320(2)

� 10!

8! 2!

�10 8 � �

10! 8!(10 � 8)!

T H E B I N O M I A L D I S T R I B U T I O N 157

In step 1 we simply follow the same method as we did earlier in estab-

lishing the number of ways of getting eight heads in ten tosses of a coin.

Our conclusion is that there are 45 different arrangements.

Step 2: Calculating the probability of any specific arrangement

W orking It Out

� 0.00098

� (0.50)10

� (0.50)8(0.50)2

pr(1 � p)N�r � (0.50)8(1 � 0.50)10�8

Step 2 provides us with the likelihood of getting any particular arrange-

ment under the assumption of the null hypothesis that the coin is fair. By

the null hypothesis, p is defined as 0.50, and r is the number of suc-

cesses (heads) in our example, or 8. So pr is (0.50)8, and (1 � p)N�r is

(1 � 0.50)10�8, or (0.50)2. The outcome of this part of the equation can

be simplified to (0.50)10. This in turn is the same outcome that we would

obtain using the multiplication rule, because the expression (0.50)10

means that we multiply the quantity 0.50 by itself 10 times. Using the

multiplication rule, we would have done just that.

Step 3: Combining the two outcomes

W orking It Out

� 0.0441

P �10 8 � � 45(0.00098)

P �N

r� � N !

r !(N � r)! pr(1 � p)N�r

of tossing eight heads in ten tosses of a fair coin is about 0.044. In

Combining the two parts of the equation, we find that the likelihood

Table 7.5, we calculate the probabilities associated with all the potential

158 C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K

The probability or sampling distribution for ten tosses of a fair coin il-

lustrates how likely you are to get any particular outcome. All of the out-

comes together add up to a probability of 1.1 Put differently, there is a

100% chance that in ten tosses of a coin you will get one of these 11 po-

tential outcomes. This is obvious, but the sampling distribution allows

you to illustrate this fact. Following what our common sense tells us, it

also shows that an outcome somewhere in the middle of the distribution

Computation of Probability Distribution for Ten Tosses of a Fair Coin

0 heads

1 head

2 heads

3 heads

4 heads

5 heads

6 heads

7 heads

8 heads

9 heads

10 heads

� � 1.0*

*The total in the last column is in fact slightly greater than 100%. This is due to rounding the numbers to the nearest decimal place in order to make the calculation more manageable.

3,628,800

3,628,800(10 � 10)! �

3,628,800

3,628,800 � 1 1(0.00098) � 0.0010

3,628,800

362,880(10 � 9)! �

3,628,800

362,880 � 10 10(0.00098) � 0.0098

3,628,800

40,320(10 � 8)! �

3,628,800

80,640 � 45 45(0.00098) � 0.0441

3,628,800

5,040(10 � 7)! �

3,628,800

30,240 � 120 120(0.00098) � 0.1176

3,628,800

720(10 � 6)! �

3,628,800

17,280 � 210 210(0.00098) � 0.2058

3,628,800

120(10 � 5)! �

3,628,800

14,400 � 252 252(0.00098) � 0.2470

3,628,800

24(10 � 4)! �

3,628,800

17,280 � 210 210(0.00098) � 0.2058

3,628,800

6(10 � 3)! �

3,628,800

30,240 � 120 120(0.00098) � 0.1176

3,628,800

2(10 � 2)! �

3,628,800

80,640 � 45 45(0.00098) � 0.0441

3,628,800

1(10 � 1)! �

3,628,800

362,880 � 10 10(0.00098) � 0.0098

3,628,800

1(10 � 0)! �

3,628,800

3,628,800 � 1 1(0.00098) � 0.0010

�N r �pr (1 � p)N�r�N

r � � N !

r ! (N � r )!

Table 7.5

1Because of rounding error, the total for our example is actually slightly larger than 1

(see Table 7.5).

outcomes in this binomial distribution. The resulting sampling distribution

is displayed in Table 7.6.

U S I N G T H E B I N O M I A L D I S T R I B U T I O N 159

is most likely. If the coin is fair, then we should more often than not get

about an even split of heads and tails.

The largest proportion (0.247) in the sampling distribution is found at

five heads in ten tosses of a coin. As you move farther away from the

center of the distribution, the likelihood of any particular result declines.

The smallest probabilities are associated with gaining either all heads or

no heads. Like many of the distributions that we use in statistics, this dis-

tribution is symmetrical. This means that the same probabilities are asso-

ciated with outcomes on both sides.

In Chapter 6, we talked about the fact that samples vary from one an-

other. This is what makes it so difficult to make inferences from a sam-

ple to a population. Based on a sample statistic, we can never be sure

about the actual value of the population parameter. However, as illus-

trated in this sampling distribution, samples drawn from the same popu-

lation vary in a systematic way in the long run. It is very unlikely to draw

a sample with ten heads in ten tosses of a fair coin. On the other hand, it

is very likely to draw a sample with four, five, or six heads in ten tosses.

U s i n g t h e B i n o m i a l D i s t r i b u t i o n t o E s t i m a t e t h e O b s e r v e d S i g n i f i c a n c e L e v e l o f a T e s t

Using the sampling distribution, we can now return to the problem of

identifying the risks of error associated with rejecting the null hypothesis

that the coin brought by the other volleyball team is fair. Earlier we sug-

gested that you might want to use a 5% significance level for this test, in

part because it is the standard or conventional significance level used by

Probability or Sampling Distribution for Ten Tosses of a Fair Coin

0 heads 0.001 1 head 0.010 2 heads 0.044 3 heads 0.118 4 heads 0.206 5 heads 0.247 6 heads 0.206 7 heads 0.118 8 heads 0.044 9 heads 0.010 10 heads 0.001

Table 7.6

most criminal justice researchers. This means that in order to reject the null

hypothesis you would require that the observed significance level (p) of your

test (or the risk of making a Type1 error by incorrectly rejecting the null hypo-

to reject the null hypothesis that the coin is fair and confront your opponents?

thesis) be less than 5% (or p < .05). Using this level, when would you be willing

Applying the Binomial Distribution to Situations Where p � 0.5

The examples in the text focus on applying the binomial distribution to sit-

uations where the probability of a success is equal to 0.5. There are other

situations where we are interested in the probability of multiple suc-

cesses (or failures), but success and failure are not equally likely. For ex-

ample, many of the games of chance that a person might play at a casino

are constructed in such a way that winning and losing are not equally

likely—the chances of losing are greater than the chances of winning—

but use of the binomial distribution would allow for calculation of the

chances of winning over several plays of the game.

Consider the following more detailed example. Suppose that we have a

quiz with five questions and we are interested in the probability of a student

sponse on any single question is p � 1/2 � 0.5. We can then apply the bino-

mial in the same way as we have in the previous examples to determine the

probability of some number of correct answers. The following table pre-

sents the numbers of correct answers and the corresponding probabilities.

NUMBER OF CORRECT ANSWERS

0 correct

1 correct

2 correct

3 correct

4 correct

5 correct

Now suppose that the questions are worded as multiple-choice items

and the student has to choose one answer from four possibilities. For any

single question, the probability of guessing the correct answer is p � 1/4

� 0.25. Given that we have multiple questions, we can again calculate the

5!

5!(5 � 5)! 0.55 (1 � 0.5)5�5 � 0.03125

5!

4!(5 � 4)! 0.54 (1 � 0.5)5�4 � 0.15625

5!

3!(5 � 3)! 0.53 (1 � 0.5)5�3 � 0.3125

5!

2!(5 � 2)! 0.52 (1 � 0.5)5�2 � 0.3125

5!

1!(5 � 1)! 0.51 (1 � 0.5)5�1 � 0.15625

5!

0!(5 � 0)! 0.50 (1 � 0.5)5�0 � 0.03125

�N r �pr (1 � p)N�r

Computation

of Binomial

Probabilities for

Five True-False

Questions

answers are true or false, then the probability of guessing the correct re-

correctly guessing all of the answers on the quiz. If the only possible

probability for the number of correct responses using the binomial distri-

bution, but we need to replace p � 0.5 with p � 0.25 in the equations to

reflect the different probability of a correct answer. The following table

presents the numbers of correct responses and the corresponding prob-

abilities for the multiple-choice response set.

NUMBER OF CORRECT ANSWERS

0 correct

1 correct

2 correct

3 correct

4 correct

5 correct

It is important to note that the distribution presented in the second table

is no longer symmetrical, reflecting the fact that the probability of a cor-

rect response is no longer equal to the probability of an incorrect re-

sponse. For the true-false questions, where the probabilities of correct

and incorrect answers are the same, we see that the probabilities of zero

and five correct responses are equal, the probabilities of one and four cor-

rect responses are equal, and the probabilities of two and three correct

responses are equal. In contrast, when we look at the probabilities for

multiple-choice questions with four possible answers, there is no such

symmetry. The most likely outcome is one correct response, with a proba-

bility of 0.3955. The probability of guessing four or five correct multiple-

choice answers is much lower than the probability of guessing four or five

correct true-false answers. In general, the probabilities in the table show

that increasing the number of possible answers makes it much more diffi-

cult for the student to correctly guess all the answers and increases the

chances of getting no correct responses or only one correct response.

5!

5!(5 � 5)! 0.255 (1 � 0.25)5�5 � 0.0010

5!

4!(5 � 4)! 0.254 (1 � 0.25)5�4 � 0.0146

5!

3!(5 � 3)! 0.253 (1 � 0.25)5�3 � 0.0879

5!

2!(5 � 2)! 0.252 (1 � 0.25)5�2 � 0.2637

5!

1!(5 � 1)! 0.251 (1 � 0.25)5�1 � 0.3955

5!

0!(5 � 0)! 0.250 (1 � 0.25)5�0 � 0.2373

�N r �pr (1 � p)N�r

Computation

of Binomial

Probabilities

for Five

Multiple-

Choice

Questions

162 C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K

At first glance, you might decide to reject the null hypothesis for out-

comes of zero, one, two, eight, nine, and ten heads. Each of these is

below the threshold of 0.05 that we have suggested. However, at the

outset we stated in our research hypothesis that we were concerned not

that the coin was biased per se, but that it was biased against your team.

This means that we set up our research hypotheses in such a way that

we would reject the null hypothesis only if the outcomes were mostly

heads. Although tossing zero, one, or two heads is just as unlikely as

tossing eight, nine, or ten heads, our research hypothesis states our in-

tention not to consider the former outcomes.

What about the risk of falsely rejecting the null hypothesis in the case

of eight, nine, or ten heads? As we noted earlier, in calculating the risk of

a Type I error, we must add up the probabilities associated with all the

outcomes for which we would reject the null hypothesis. So, for exam-

ple, if we want to know the risk of falsely rejecting the null hypothesis

on the basis of eight heads in ten coin tosses, we have to add together

the risks associated with eight, nine, and ten heads in ten tosses. The

question we ask is, What is the risk of falsely rejecting the null hypothe-

sis if we gain eight or more heads in ten coin tosses? The total risk, or

observed significance level, would be about 0.055 (that is, 0.044 � 0.010

� 0.001), which is greater than our threshold of 0.05 for rejecting the

null hypothesis. It is too large an outcome for you to confront your op-

ponents and accuse them of cheating.

In the case of nine heads, the outcome is well below the threshold of

a Type I error we have chosen. By adding together the probabilities as-

sociated with gaining nine or ten heads in ten coin tosses, we arrive at a

risk of 0.011 of falsely rejecting the null hypothesis. If we decided to re-

ject the null hypothesis that the coin is fair on the basis of an outcome of

nine heads, then the observed significance value for our test would be

0.011. For ten heads, as we noted earlier, the risk of a Type I error is

even lower (p � 0.001). Because there are no outcomes more extreme

than ten heads in our distribution, we do not have to add any probabili-

ties to it to arrive at an estimate of the risk of a Type I error.

You would take a very large risk of a Type I error if you decided in

advance to reject the null hypothesis that the coin is fair based on six

heads in ten tosses of a coin. Here, you would have to add the probabil-

ities associated with six (0.206), seven (0.118), eight (0.044), nine

(0.010), and ten heads (0.001).

As the coin toss example illustrates, sampling distributions play a very

important role in inferential statistics. They allow us to define the ob-

served significance level, or risk of a Type I error, we take in rejecting

the null hypothesis based on a specific outcome of a test of statistical

significance. Although most sampling distributions we use in statistics are

considerably more difficult to develop and involve much more complex

K E Y T E R M S 163

mathematical reasoning than the binomial distribution, they follow a

logic similar to what we have used here. For each distribution, statisti-

cians use probabilities to define the likelihood of gaining particular out-

comes. These sampling distributions provide us with a precise method

for defining risks of error in tests of statistical significance.

What you have learned here provides a basic understanding of how

sampling distributions are developed from probability theory. In later

chapters, we will rely on already calculated distributions. However, you

should keep in mind that steps similar to those we have taken here have

been used to construct these distributions.

C h a p t e r S u m m a r y

Whereas a sample distribution is the distribution of the results of one

sample, a sampling distribution is the distribution of outcomes of a

very large number of samples, each of the same size. A sampling distrib-

ution that is derived from the laws of probability (without the need to

take countless samples) may also be called a probability distribution.

A sampling distribution allows us to define the observed significance

level of a test of statistical significance, or the estimated risk of a Type I

error we take in rejecting the null hypothesis based on sample statistics.

To guide our decision as to whether to reject or fail to reject the null hy-

pothesis, we compare the observed significance level with the criterion

significance level set at the outset of the test of statistical significance.

By using the multiplication rule, we can calculate the probability of

obtaining a series of results in a specific order. The number of arrange-

ments is the number of different ways of obtaining the same result. The

total probability of obtaining any result is the individual probability mul-

tiplied by the number of different possible arrangements.

The binomial distribution is the sampling distribution for events

with only two possible outcomes—success or failure, heads or tails, etc.

It is calculated using the binomial formula. When deciding whether

the result achieved, or observed significance level, passes the desired

threshold for rejecting the null hypothesis, it is important to remember to

take a cumulative total of risk.

K e y T e r m s

arrangements The different ways events

can be ordered and yet result in a single

outcome. For example, there is only one

arrangement for gaining the outcome of ten

heads in ten tosses of a coin. There are,

however, ten different arrangements for

gaining the outcome of nine heads in ten

tosses of a coin.

164 C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K

S y m b o l s a n d F o r m u l a s

! Factorial

r Number of successes

N Number of trials

p

To determine the probability of events A, B, C, and D occurring jointly

under the assumption of independence (the multiplication rule):

P (A & B & C & D) � P (A) • P (B) • P (C ) • P (D)

To determine the number of arrangements of any combination of

events:

To determine the probability of any binomial outcome occurring in all

its possible arrangements (the binomial formula):

P�N

r� � N !

r !(N � r)! pr(1 � p)N�r

�N

r� � N !

r !(N � r)!

binomial distribution The probability or

sampling distribution for an event that has

only two possible outcomes.

binomial formula The means of deter-

mining the probability that a given set of

binomial events will occur in all its possi-

ble arrangements.

factorial The product of a number and all

the positive whole numbers lower than it.

independent Describing two events when

the occurrence of one does not affect the

occurrence of the other.

multiplication rule The means for deter-

mining the probability that a series of

events will jointly occur.

probability distribution A theoretical

distribution consisting of the probabilities

expected in the long run for all possible

outcomes of an event.

sampling distribution A distribution

of all the results of a very large number

of samples, each one of the same size

and drawn from the same population

under the same conditions. Ordinarily,

sampling distributions are derived using

probability theory and are based on prob-

ability distributions.

The probability of a success in the binomial formula. It is also used as a

significance. symbol of the observed significance level of a test of statistical

E X E R C I S E S 165

E x e r c i s e s

7.1 Calculate the probability for each of the following:

a. Two tails in two tosses of a fair coin.

b. Three heads in three tosses of a fair coin.

c. Four heads in four tosses of an unfair coin where the probability of a head is 0.75.

d. Three sixes in three rolls of a fair die.

e. Five fours in five rolls of an unfair die where the probability of a four is 0.25.

7.2 All of Kate’s children are boys.

a. Intuitively, how many boys do you think Kate would have to have in succession before you would be willing to say with some cer- tainty that, for some biological reason, she is more likely to give birth to boys than girls?

b. Now calculate the number of successive births required before you could make such a decision statistically with a 5% risk of error.

c. How many successive boys would have to be born before you would be prepared to come to this conclusion with only a 1% risk of error?

7.3 The Federal Bureau of Investigation trains sniffer dogs to find explo- sive material. At the end of the training, Lucy, the FBI’s prize dog, is let loose in a field with four unmarked parcels, one of which contains Semtex explosives. The exercise is repeated three times, and on each occasion, Lucy successfully identifies the suspicious parcel.

a. What is the chance of an untrained dog performing such a feat? (Assume that the untrained dog would always approach one of the parcels at random.)

b. If there had been five parcels instead of four and the exercise had been carried out only twice instead of three times, would the chances of the untrained dog finding the single suspicious parcel have been greater or less?

7.4 Alex, an attorney, wishes to call eight witnesses to court for an impor- tant case. In his mind, he has categorized them into three “strong” wit- nesses and five “weaker” witnesses. He now wishes to make a tactical decision on the order in which to call the strong and the weaker witnesses.

166 C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K

For example, one of his options is

Strong Weak Weak Strong Weak Weak Weak Strong

a. In how many different sequences can he call his strong and weaker witnesses?

b. If Alex decides that one of his three strong witnesses is in fact more suited to the weaker category, how many options does he now have?

7.5 In a soccer match held at a low-security prison, the inmates beat the guards 4 to 2.

a. How many different arrangements are there for the order in which the goals were scored?

b. What would your answer be if the final score were 5 to 1?

7.6 At the end of each year, the police force chooses its “Police Officer of the Year.” In spite of the fact that there are equal numbers of men and women on the force, in the last 15 years, 11 of the winners have been men and 4 have been women. Paul has been investigating whether women and men are treated differently in the police force.

a. Do these figures provide Paul with a reasonable basis to suspect that the sex of the officer is an active factor? Explain your answer.

b. Looking back further into the records, Paul discovers that for the three years before the 15-year span initially examined, a woman was chosen each time. Does this affect his conclusion? Explain your answer.

7.7 Use the binomial distribution to calculate each of the following probabilities:

a. Three heads in eight tosses of a fair coin.

b. Six tails in thirteen tosses of a fair coin.

c. Four fives in five rolls of a fair die.

d. Two ones in nine rolls of a fair die.

e. Five sixes in seven rolls of a fair die.

7.8 Tracy, a teacher, gives her class a ten-question test based on the homework she assigned the night before. She strongly suspects that Mandy, a lazy student, did not do the homework. Tracy is surprised to see that of the ten questions, Mandy answers seven correctly. What is the probability that Mandy successfully guessed seven of the ten an- swers to the questions if

a. The questions all required an answer of true or false?

b. The questions were all in the multiple-choice format, with students having to circle one correct answer from a list of five choices?

E X E R C I S E S 167

7.9 After a supermarket robbery, four eyewitnesses each report seeing a man with glasses fleeing from the scene. The police suspect Eddy and make up an identity parade of five men with glasses. Eddy takes his place in the parade alongside four randomly chosen stooges. Of the four eyewitnesses who are brought in, three identify Eddy and the fourth points to one of the stooges. The detective in charge decides that there is enough evidence to bring Eddy to trial.

a. The detective’s superior wishes to know the probability that Eddy would have been chosen by three out of the four eyewitnesses if each witness had chosen a member of the identity parade entirely at random. What is the probability?

b. What is the probability of Eddy being chosen at random by only two of the four witnesses?

7.10 A gang of five child thieves draws straws each time before they go shoplifting. Whoever draws the short straw is the one who does the stealing. By tradition, Anton, the leader, always draws first. On the four occasions that the gang has performed this ritual, Anton has drawn the short straw three times.

a. Construct a table to illustrate the binomial distribution of Anton’s possible successes and failures for each of the four draws.

b. Should he accuse his fellow gang members of rigging the draw if

i. He is willing to take a 5% risk of falsely accusing his friends?

ii. He is willing to take only a 1% risk of falsely accusing his friends?

7.11 Baron, a gambler, plays 11 rounds at a casino roulette wheel, each time placing a $100 note on either black or red.

a. Construct a table to illustrate the binomial distribution of Baron’s possible successes and failures for each of the 11 rounds.

b. The casino croupiers have been told to inform the management if a client’s winning streak arouses suspicion that he might be cheating. The threshold of suspicion is set at 0.005. How many successes does Baron need on 11 trials to arouse the management’s suspicion?

7.12 Nicola is playing roulette on an adjacent table. On 12 successive spins of the wheel, she places a $100 note on either the first third (numbers 1–12), the second third (numbers 13–24), or the final third (numbers 25–36).

a. Construct a table to illustrate the binomial distribution of Nicola’s possible successes and failures for each of the 12 spins.

b. How many times out of the 12 would Nicola need to win in order to arouse the suspicion of the casino manager that she was cheat- ing, if the management policy is to limit the risk of falsely accusing a customer to 0.001?

7.13 A security consultant hired by store management thinks that the prob- ability of store security detecting an incident of shoplifting is 0.1. Sup - pose the consultant decides to test the effectiveness of security by try - ing to steal an item ten different times.

a. Construct a table to illustrate the binomial distribution of possible detections for each of the ten attempted thefts.

b. Store management claims that the chances of detection are greater than 0.1. If the consultant set the threshold for detection at 0.05, how many times would she have to be detected to increase the probability of detection?

7.14 In a crime spree, Joe commits six robberies.

a. If the probability of arrest for a single robbery is 0.7, what is the probability that Joe will be arrested for three of the robberies?

b. If the probability of detection for a single robbery is 0.4, what is the probability that Joe will not be arrested for any of his crimes?

7.15 The arrest histories for a sample of convicted felons revealed that, with ten previous arrests, the probability of a drug arrest was 0.25. If an offender has been arrested ten times, what is the probability of

a. two drug arrests?

b. five drug arrests?

c. seven drug arrests?

C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K168

C o m p u t e r E x e r c i s e s

The computation of binomial probabilities by hand can be quite tedious and time

consuming. Spreadsheet packages typically include a function that can be inserted

into a cell that will allow for the computation of a binomial probability. To com-

pute the probability correctly, you will need to enter three pieces of information:

the number of “successes” (i.e., events of interest), the total number of trials,

and the probability of success on a single trial. Another item that you will need

to pay attention to is whether the binomial function you are using computes a

cumulative probability—the default in many spreadsheet packages. Throughout

this chapter, we have not computed cumulative probabilities, but rather, what are

labeled, “probability densities” in many spreadsheet packages.

Although not quite as flexible as a spreadsheet package, the computation of

binomial probabilities in both SPSS and Stata is not complicated. We illustrate

how the various commands work in each program.

The computation of binomial probabilities in SPSS requires the use of the

COMPUTE command that was noted in the Chapter 5 Computer Exercises. As

a means of illustrating the computation of binomial probabilities in SPSS, we will

SPSS

169C O M P U T E R E X E R C I S E S

a new data set in SPSS that contains one variable: the number of success. This

new variable will have values ranging from 0 to 10—enter them in order for ease

of interpretation later. (After you enter these data, you should have 11 values for

your new variable. For ease of illustration below, rename this variable to “suc-

cesses”—without the quotation marks.)

To compute the binomial probabilities, the general form of the COMPUTE

command will be:

The PDF.BINOM function will compute binomial probabilities for any given

combination of successes (q), number of trials (n), and probability of success on

a single trial (p). Prior to executing this command, we need to insert values for

each item in the PDF.BINOM function.

of successes—this is the variable that you created with values ranging from 0

to 10. Enter the variable name here.

The value for “n” (the second value referenced in the parentheses) is the total

number of trials. For our example, enter the number 10.

The value for “p” (the third value referenced in the parentheses) is the

probability of success for a single trial. For our example, enter the value 0.5.

Assuming that you named the new variable successes, the COMPUTE command

would look like:

We have named the new variable “binom_prob” and once this command has

been run, you should see a new variable in the second column of the SPSS data

window that contains binomial probabilities. With the exception of rounding

differences, these values are identical to those presented in Table 7.5.

Stata

The computation of binomial probabilities in Stata is nearly identical to that in

SPSS and requires the use of the gen command that was noted in the Chapter

5 Computer Exercises. We walk through the same illustration of the computa-

tion of binomial probabilities as we did in SPSS, and will reproduce the binomial

probabilities listed in Table 7.5 with the commands in Stata.

To create a new data set in Stata, click on the “Data Editor” button at the

top center of the Stata window. You should see a spreadsheet layout, much like

in SPSS. Again, we will begin by creating a new variable representing the number

of success. Enter values for this new variable that range from 0 to 10. (After you

reproduce the binomial probabilities listed in Table 7.5. To begin, we will create

COMPUTE new_var_name = PDF.BINOM(q,n,p).

EXECUTE.

COMPUTE binom_prob = PDF.BINOM(successes,10,0.5).

EXECUTE.

C H A P T E R S E V E N : D E F I N I N G T H E O B S E R V E D R I S K170

enter these data, you should have 11 values for your new variable. For ease of

illustration below, rename this variable to “successes”—without the quotation

marks.)

To compute the binomial probabilities, the general form of the gen com-

mand will be:

Where the binomialp function will compute binomial probabilities for any

given combination of number of trials (n), of number of successes (k), and

probability of success on a single trial (p). Prior to executing this command, we

need to insert values for each item in the binomialp function.

The value for n -

ber of trials. For our example, enter the number 10.

The value for k (the second value referenced in the parentheses) is the number

of successes—this is the variable that you created with values ranging from 0

to 10. Enter the variable name here.

The value for p (the third value referenced in the parentheses) is the

probability of success for a single trial. For our example, enter the value 0.5.

Assuming that you named the new variable successes, the gen command would

look like:

The new variable is named “binom_prob” and once this command has been run,

you should see a new variable in the second column of the data window that

contains binomial probabilities. Similar to the analysis with SPSS, these values

are identical to those presented in Table 7.5, with the exception of rounding

differences.

Problems

2. Verify the probabilities you calculated for any of the Exercises you worked

through at the end of Chapter 7.

3. Construct a table of binomial probabilities for each of the following

combinations:

a. Number of trials = 10, probability of success = 0.2.

b. Number of trials = 10, probability of success = 0.7.

c. Number of trials = 15, probability of success = 0.3.

d. Number of trials = 15, probability of success = 0.5.

e. Number of trials = 15, probability of success = 0.8.

gen new_var_name = binomialp(n,k,p)

gen binom_prob = binomialp(10,successes,0.5)

1. Reproduce the tables of binomial probabilities on pages 160 and 161 in

the box applying binomial probabilities when p ≠ 0.5.

Steps in a Statistical Test:

Using the Binomial Distribution

to Make Decisions About Hypotheses

Are Assumptions Made About the Population Distribution?

C h a p t e r e i g h t

S t a t i s t i c a l a s s u m p t i o n s

S a m p l i n g d i s t r i b u t i o n

S i g n i f i c a n c e l e v e l

T e s t s t a t i s t i c a n d d e c i s i o n

What Type of Measurement is Being Used?

What Sampling Method is Being Used?

What are the Hypotheses?

Which Sampling Distribution is Appropriate?

What is the Rejection Region?

Should a One-Tailed or a Two-Tailed Test be Used?

What is the Test Statistic?

How is a Final Decision Made?

Where is It Placed?

D. Weisburd and C. Britt, Statistics in Criminal Justice, DOI 10.1007/978-1-4614-9170-5_8,

© Springer Science+Business Media New York 2014

IN THE PREVIOUS CHAPTER, you saw how probability theory is used to

identify the observed significance level in a test of statistical significance.

But you cannot simply rely on mathematical calculations to determine

whether to reject the null hypothesis. You must make sure at the outset

that the methods used are appropriate to the problem examined. You

must clearly state the assumptions made. You must define the specific

hypotheses to be tested and the specific significance criteria to be used.

It is best to take a careful step-by-step approach to tests of statistical sig-

nificance. Using this approach, you will be much less likely to make seri-

ous mistakes in developing such tests.

In this chapter, we introduce the basic elements of this step-by-step

approach. To place this approach in context, we illustrate each step with

a specific research problem that can be addressed using the binomial

distribution. Although we use the binomial distribution as an example,

you should not lose sight of the fact that our purpose here is to establish

a general model for presenting tests of statistical significance, which can

be used whichever sampling distribution is chosen.

T h e P r o b l e m : T h e I m p a c t o f P r o b l e m - O r i e n t e d P o l i c i n g o n D i s o r d e r l y A c t i v i t y a t V i o l e n t - C r i m e H o t S p o t s

In Jersey City, New Jersey, researchers developed a problem-oriented

policing program directed at violent-crime hot spots.1 Computer map-

ping techniques were used to identify places in the city with a very high

level of violent-crime arrests or emergency calls to the police. Jersey City

police officers, in cooperation with staff of the Rutgers University Center

for Crime Prevention Studies, developed strategies to solve violent-crime

1See Anthony Braga, “Solving Violent Crime Problems: An Evaluation of the Jersey

City Police Department’s Pilot Program to Control Violent Crime Places,” unpublished

dissertation, Rutgers University, Newark, NJ, 1996.

172

T H E P R O B L E M

problems at a sample of 11 places. The strategies followed a problem-

oriented policing (POP) approach, in which police collect a wide variety

of information about each hot spot, analyze that information to identify

the source of the problem, develop tailor-made responses to do some-

thing about the problem, and finally assess whether their approach actu-

ally had an impact.2

The evaluation involved a number of different components. One part

of the research sought to identify whether “disorderly” activity at the hot

spots had declined during the period of the study. For example, the re-

searchers wanted to see whether the number of loiterers or homeless

people had been reduced as a result of the efforts of the police. The

treatment areas were compared to a matched group, or control group, of

similar but untreated violent-crime places. Table 8.1 presents the overall

2Problem-oriented policing is an important new approach to police work formulated

by Herman Goldstein of the University of Wisconsin Law School. See H. Goldstein,

Problem-Oriented Policing (New York: McGraw-Hill, 1990).

Results at Treatment and Control Locations Derived from Observations

of Disorderly Behavior Before and After Intervention

TRIAL PLACE OUTCOME

1 Journal Square East �

Newport Mall

2 Stegman & Ocean �

Clerk & Carteret

3 Glenwood & JFK �

Journal Square West

4 Bergen & Academy �

Westside & Duncan

5 Westside & Clendenny �

Franklin & Palisade

6 Belmont & Monticello �

MLK & Wade

7 MLK & Atlantic �

Neptune & Ocean

8 MLK & Armstrong �

Ocean & Eastern

9 Westside & Virginia �

JFK & Communipaw

10 Park & Prescott �

Dwight & Bergen

11 Old Bergen & Danforth �

Bramhall & Arlington

Note: Experimental or treatment hot spots are listed in boldface type. � � Relative improvement in experimental locations � � Relative improvement in control locations

Table 8 .1

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C H A P T E R E I G H T : S T E P S I N A S T A T I S T I C A L T E S T

results of pre- and posttest comparisons of outcomes for the 11 matched

pairs of locations. In 10 of the 11 pairs, the experimental hot spots (those

receiving POP intervention) improved relative to the control locations.

The research question asked by the evaluator was whether the POP

approach has an impact on disorderly activity at violent-crime hot spots.

The statistical problem faced is that the 11 comparisons are only a sample

of such comparisons. What conclusions can the researcher make regard-

A s s u m p t i o n s : L a y i n g t h e F o u n d a t i o n s f o r S t a t i s t i c a l I n f e r e n c e

The first step in a test of statistical significance is to establish the as-

sumptions on which the test is based. These assumptions form the

foundation of a test. No matter how elegant the statistics used and the

approach taken, if the assumptions on which they are built are not solid,

then the whole structure of the test is brought into question.

Level of Measurement

Our first assumption is related to the type of measurement used. Differ-

ent types of tests of statistical significance demand different levels of

measurement.

Accordingly, it is important to state at the outset the type of measure-

ment required by a test. For the binomial test, which is based on the bi-

nomial distribution, a nominal-level binary measure is required. A binary

measure has only two possible outcomes, as was the case with the coin

toss example in Chapter 7. The type of outcome measure used to evalu-

ate the impact of problem-oriented policing on disorderly activity—

whether the treatment hot spot improved (or got worse) relative to the

control location—fits this assumption. In stating our assumptions (as is

done at the end of this section), we include a specific definition of the

level of measurement required:

Level of Measurement: Nominal binary scale.

Shape of the Population Distribution

The second assumption refers to the shape of the population distribu-

tion. In statistical inference, we are generally concerned with two types

of tests. In the first type—termed parametric tests—we make an as-

sumption about the shape of the population distribution. For example, in

a number of tests we will examine in later chapters, there is a require-

appropriate for our problem is based on the binomial sampling distribution.

question, we use a test of statistical significance. The specific test that is

ing the larger population of violent-crime hot spots? To answer this

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A S S U M P T I O N S : L A Y I N G T H E F O U N D A T I O N S

ment that for the population to which you infer, the scores on the vari-

able be normally distributed.

The second type of test of statistical significance does not make a spe-

cific assumption regarding the population distribution. These tests are

called nonparametric tests or distribution-free tests. The advantage

of nonparametric tests is that we make fewer assumptions. The disad-

vantage is that nonparametric tests do not allow us to analyze data at

higher levels of measurement. They are generally appropriate only for

nominal and ordinal scales. The binomial test is a nonparametric test. Ac-

Population Distribution: No assumption made.

Sampling Method

The third assumption concerns the sampling method used. When we

conduct a test of statistical significance, we want our sample to be a

good representation of the population from which it is drawn. Put in sta-

tistical terms, we want our study to have high external validity.

Let’s suppose you are interested in attitudes toward the death penalty.

Would a sample of your friends provide an externally valid sample of all

Americans? Clearly not, because a sample of only your friends is not

likely to include age or ethnic or class differences that typify the U.S.

population. Even if we used your friends as a sample of U.S. college stu-

dents, we could still identify threats to the external validity of the study.

Colleges have differing criteria for admission, so it is not likely that one

college will be representative of all colleges. Even as a sample of stu-

dents at your college, your friends may not provide a valid sample. They

may be drawn primarily from a specific year of college or have other

characteristics that make them attractive as friends but also mean that

they are a poor representation of others in the college.

How can we draw a representative sample? The most straightfor-

ward approach is to choose cases at random from the population. This

type of sampling is called random sampling. Random samples are as-

sumed to have high external validity compared with what may be termed

convenience samples. A convenience sample consists of whatever sub-

jects are readily available to the researcher. Your friends form a conve-

nience sample of students at your college or of all college students.

It is important to note that convenience samples are not always bad

samples. For example, if you choose to examine prisoners in one prison on

the assumption that prisoners there provide a cross section of the different

types of prisoners in the United States, you might argue that it is a repre-

sentative sample. However, if you use a convenience sample, such as pris-

oners drawn from a single prison, you must always be wary of potential

threats to external validity. Convenience samples are prone to systematic

biases precisely because they are convenient. The characteristics that make

175

cordingly, in stating our assumptions we write:

C H A P T E R E I G H T : S T E P S I N A S T A T I S T I C A L T E S T

them easy for the researcher to define are likely as well to differentiate

them in one way or another from the population the researcher seeks to

study.

Statistical tests of significance generally assume that the researcher has

used a type of random sampling called independent random sampling.

Independent random sampling requires not only that cases be identified at

random, but also that the selection of cases be independent. As discussed

in the previous chapter, two events are statistically independent when the

occurrence of one does not affect the occurrence of the other. In sam-

pling, this means that the choice of one case or group of cases will not

have any impact on the choice of another case or group of cases. This is a

useful assumption in assuring the external validity of a study because it

prevents biases that might be brought into the process of sampling.

For example, suppose you want to select 1,000 prisoners from the

population of all prisoners in the United States. Each time you select a

prisoner for your sample, you use a random method of selection. How-

ever, prison officials have told you that if you select one prisoner from a

cell then you cannot select any other prisoner from that cell. Accord-

In order to ensure independent random sampling, the same popula-

the deck. If we didn’t replace the card, we would influence the likeli-

hood of a specific card being chosen on the next draw from the deck.

For example, if we started with a full deck of 52 cards, the likelihood of

getting the queen of spades would be 1 in 52. However, if we drew, say,

a jack of hearts and didn’t return it to the deck, what would be the likeli-

hood of getting a queen of spades on our next draw? This time we

would have only 51 cards to draw from, so our likelihood would change

to 1 in 51. In order to gain a fully independent random sample, we must

use a method of sampling called sampling with replacement. This

means that we must use the same population each time we select a case.

For every selection, the sampling frame must remain exactly the same. In

this way, we can ensure that the choice of one case cannot have any im-

pact on the choice of another.

Though this method ensures independence, it also means that a par-

ticular case may be selected more than once. For example, suppose you

choose a particular prisoner as case number five in your sample. Be-

cause you must use the same sampling frame each time you select a

case, that prisoner is returned to the sampling frame after selection. Later

ingly, after each selection of a prisoner, you must remove all of his

cellmates from your sampling frame, or universe of eligible cases. The

result is that there are now systematic reasons why you might suspect

that your sample is not representative of the population.

tion of cases must be used in drawing each case for a sample. As we

discussed in Chapter 7, if we want each draw from a deck of cards to be

independent, we have to return the card chosen on any specific draw to

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A S S U M P T I O N S : L A Y I N G T H E F O U N D A T I O N S

in your study, you might choose that prisoner again. Accordingly, while

sampling with replacement, or returning sampled cases to the sampling

frame after each selection, makes statistical sense, it often does not make

practical sense when you are carrying out research in the real world. If

you are conducting an interview study, for example, independent ran-

dom sampling would allow individuals to be interviewed more than

once. It is likely that subjects would find it strange to be reinterviewed

using the same interview schedule. Moreover, their responses would

likely be influenced by their knowledge of the survey. Similarly, if a sub-

ject or place is chosen twice in a study that involves a specific treatment

or intervention, then that subject or place should be given the treatment

after each selection. Here there is the difficulty that it may be harmful to

provide the treatment more than once.

Even when there are no specific practical barriers to sampling with re-

placement, it is difficult to explain to practitioners or even many re-

searchers why an individual may appear twice in the same sample. As a

result, many, if not most, criminal justice studies do not replace individu-

als in the sampling frame once they have been selected. Although this

represents a formal violation of the assumptions of your test, in most

cases its impact on your test result is negligible. This is because samples

are generally very small relative to populations, and thus in practice

there is little chance of selecting a case more than once even when sam-

pling with replacement. If, however, your sample reaches one-fifth or

more of the size of your population, you may want to include a correc-

tion factor in your test.3

For this test of statistical significance, we assume that researchers in

the Jersey City POP study sampled cases randomly from a large popula-

tion of hot spots during the sample selection month. Because it would

not have been practical to implement treatments more than once at any

site, the researchers did not sample with replacement.

3The correction factor adjusts your test to account for the fact that you have not al-

lowed individuals to be selected from the population more than once. Not including a

correction factor makes it more difficult to reject the null hypothesis. That is, the in-

clusion of a correction factor will make it easier for you to reject the null hypothesis.

One problem criminal justice scholars face in using a correction factor is that they

often want to infer to populations that are beyond their sampling frame. For example,

a study of police patrol at hot spots in a particular city may sample 50 of 200 hot

spots in the city during a certain month. However, researchers may be interested in

making inferences to hot spots generally in the city (not just those that exist in a par-

ticular month) or even to hot spots in other places. For those inferences, it would be

misleading to adjust the test statistic based on the small size of the sampling frame.

For a discussion of how to correct for sampling without replacement, see Paul S. Levy

and Stanley Lemeshow, Sampling of Populations: Methods and Applications (New

York: Wiley, 1991).

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C H A P T E R E I G H T : S T E P S I N A S T A T I S T I C A L T E S T

The binomial test, however, like most tests of statistical significance

examined in this book, assumes independent random sampling. Accord-

ingly, in stating our assumptions, it is important to note both the require-

ment for this test and our failure to meet that requirement. Therefore we

state our assumption:

Sampling Method: Independent random sampling (no replacement; sam-

ple is small relative to population).

Throughout this text, we state the assumptions of a test and then place

any violations of assumptions in parentheses. This is good practice, as it

will alert you to the fact that in many studies there are violations of one

type or another of assumptions. Some of these violations are not impor-

tant. For example, not sampling with replacement in this study does not

affect the test outcome because the population of hot spots is assumed

to be very large relative to the sample. However, you will sometimes

find more serious violations of assumptions. In those cases, you will

have to take a more critical view of the results of the test. It is good practice to define not only the sampling method used but

also the sampling frame of your study. In our example, we can make in-

ferences based on our random sample to the population of hot spots in

Jersey City during the month of sample selection. Accordingly, we state

in our assumptions:

Sampling Frame: Hot spots of violent crime in one month in Jersey

City.

Our sampling frame reminds us of the specific population to which our

sample infers. However, researchers usually want to infer beyond the

specific population identified by their sampling frame. For example, the

population of interest for the POP study is likely to be hot spots through-

out the year, not just those in a specific month. Researchers may even

want to infer to violent-crime hot spots generally, not just those in Jersey

City.

We cannot assume that our sample is a representative sample for

these inferences based on our sampling method, since these populations

did not constitute our sampling frame. However, we can ask whether

our sample is likely to provide valid inferences to those populations. In

the case of hot spots in Jersey City, we would need to question whether

there is any reason to suspect that hot spots chosen in the month of

study were different from those that would be found in other months of

the year. For inferences to the population of hot spots in other locations,

we would have to assume that Jersey City hot spots are similar to those

in other places and would respond similarly to POP interventions. In

making any inference beyond your sampling frame, you must try to

identify all possible threats to external validity.

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The Hypotheses

The final assumptions we make in a test of statistical inference refer to

the hypotheses of our study. As discussed in Chapter 6, hypotheses are

developed from the research questions raised in a project. Hypotheses

must be stated before the researcher collects outcome data for a study. If

hypotheses are stated only after data have been collected and analyzed,

the researcher might be tempted to make changes in the hypotheses that

unfairly affect the tests of statistical significance that are conducted.

As discussed in Chapter 6, the researcher ordinarily begins by defin-

ing the research hypothesis. In the problem-oriented policing study, we

might state our research hypothesis in three different ways:

Hypothesis 1. Incivilities in treatment hot spots decline relative to incivil-

ities in control hot spots after POP intervention.

Hypothesis 2. Incivilities in treatment hot spots increase relative to inci-

vilities in control hot spots after POP intervention.

Hypothesis 3. The level of incivilities in treatment hot spots relative to

incivilities in control hot spots changes after POP intervention.

hypotheses. The first two research hypotheses are directional hypotheses

because they specify the direction, or type of relationship, that is ex-

pected. For example, hypothesis 1 is concerned only with whether the

POP program is successful in reducing incivilities. If the researcher

adopts this hypothesis, then he or she is stating that the statistical test

employed will not be concerned with the second hypothesis—that the

intervention makes matters worse and increases incivilities. The third hy-

pothesis is a nondirectional hypothesis. In this case, the researcher is in-

terested in testing the possibility that the intervention improves hot spots

or makes them worse.

In the POP study, researchers wanted to assess both positive and neg-

ative outcomes. Although they believed that problem-oriented policing

should reduce incivilities at violent-crime hot spots, they did not want to

preclude at the outset a finding that the program actually made matters

worse. Accordingly, they used a nondirectional research hypothesis:

“The level of incivilities in treatment hot spots relative to incivilities in

control hot spots changes after POP intervention.” The null hypothesis is

“The level of incivilities in treatment hot spots does not change relative

to incivilities in control hot spots after POP intervention.”

In practice, the null hypothesis may be stated in terms of probabili-

ties, just as we could state the coin toss hypothesis in the last chapter

in terms of probabilities. In this study, the researchers examined (for

each matched pair of hot spots) whether the hot spot that received the

problem-oriented policing intervention improved or worsened relative to

Recall from Chapter 6 that we distinguish directional from nondirectional

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the control location. The null hypothesis suggests that the treatment and

control hot spots are equally likely to improve. Put in terms of probabili-

ties, there is a 0.50 chance of success (P � 0.50) for the intervention

under the null hypothesis. The research hypothesis represents all other

possible outcomes (P � 0.50). Remember that our hypotheses are state-

ments about the populations examined. Accordingly, in stating the hy-

potheses, we use symbols appropriate for population parameters—in

this case P rather than p. Stating our assumptions, we write

Hypotheses:

H0: The level of incivilities in treatment hot spots does not change rela-

tive to incivilities in control hot spots after POP intervention, P � 0.50.

H1: The level of incivilities in treatment hot spots relative to incivilities in

control hot spots changes after POP intervention, P � 0.50.

Stating All of the Assumptions

Our assumptions may be stated as follows:

Assumptions:

Level of Measurement: Nominal binary scale.

Population Distribution: No assumption made.

Sampling Method: Independent random sampling (no replacement; sam-

ple is small relative to population).

Sampling Frame: Hot spots of violent crime in one month in Jersey City.

Hypotheses:

H0: The level of incivilities in treatment hot spots does not change rela-

tive to incivilities in control hot spots after POP intervention, P � 0.50.

H1: The level of incivilities in treatment hot spots relative to incivilities in

control hot spots changes after POP intervention, P � 0.50.

S e l e c t i n g a S a m p l i n g D i s t r i b u t i o n

In stating our hypotheses, we already noted the specific requirements of

the binomial sampling distribution. Now we must state why we have

chosen the binomial distribution and identify the specific characteristics

of the sampling distribution that will be used to assess the risk of falsely

rejecting the null hypothesis in our problem-oriented policing example.

Choosing a sampling distribution is one of the most important decisions

that researchers make in statistical inference. As we will show in later

chapters, there are a number of different types of sampling distributions.

Moreover, as with the binomial distribution, a single type of sampling

distribution may have different forms depending on the problem

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S E L E C T I N G A S A M P L I N G D I S T R I B U T I O N

Because our measure is nominal and binary (see assumptions), we se-

lected the binomial distribution for our test. The specific distribution that we

use is based on our null hypothesis and the size of our sample. As illus-

trated in Chapter 7, the binomial distribution provides the likelihood of gain-

ing a particular number of successes (heads in the example of the coin toss)

in a fixed number of trials. In order to assess that likelihood, we also need

to know what the probability of a success or failure is on any particular trial.

In our example, there are 11 trials, or 11 matched comparisons. Our

null hypothesis states that the likelihood of a success for any comparison

is 0.50. To build our sampling distribution, we apply the binomial for-

mula to each of the 12 possible outcomes that could be gained in our

study, under the assumption that P � 0.50. This is done in Table 8.2.

The resulting distribution is presented in Table 8.3.

Computation of Sampling Distribution of Success or Failure in 11 Trials

0 successes 1(0.00049) � 0.00049

1 success 11(0.00049) � 0.00537

2 successes 55(0.00049) � 0.02686

3 successes 165(0.00049) � 0.08057

4 successes 330(0.00049) � 0.16113

5 successes 432(0.00049) � 0.22638*

6 successes 432(0.00049) � 0.22638*

7 successes 330(0.00049) � 0.16113

8 successes 165(0.00049) � 0.08057

9 successes 55(0.00049) � 0.02686

10 successes 11(0.00049) � 0.00537

11 successes 1(0.00049) � 0.00049

*Probabilities contain rounding error.

39,916,800

39,916,800(11 � 11)! �

39,916,800

39,916,800 � 1

39,916,800

3,628,800(11 � 10)! �

39,916,800

3,628,800 � 11

39,916,800

362,880(11 � 9)! �

39,916,800

725,760 � 55

39,916,800

40,320(11 � 8)! �

39,916,800

241,920 � 165

39,916,800

5,040(11 � 7)! �

39,916,800

120,960 � 330

39,916,800

720(11 � 6)! �

39,916,800

86,400 � 462

39,916,800

120(11 � 5)! �

39,916,800

86,400 � 462

39,916,800

24(11 � 4)! �

39,916,800

120,960 � 330

39,916,800

6(11 � 3)! �

39,916,800

241,920 � 165

39,916,800

2(11 � 2)! �

39,916,800

725,760 � 55

39,916,800

1(11 � 1)! �

39,916,800

3,628,800 � 11

39,916,800

1(11 � 0)! �

39,916,800

39,916,800 � 1

�N r �pr (1 � p)N�r�N

r � � N !

r !(N � r )!

Table 8.2

research problem examined, then the conclusion reached will be suspect.

examined. If the sampling distribution used is inappropriate for the

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S i g n i f i c a n c e L e v e l a n d R e j e c t i o n R e g i o n

Having selected the distribution that will be used to assess Type I error,

we are ready to define the outcomes that will lead us to reject the null

hypothesis. Our first step is to choose the significance level of our test.

As described in Chapter 6, the significance level of a test is the amount

of Type I error we are willing to risk in rejecting the null hypothesis. By

convention, criminal justice researchers use a 5% significance threshold.

But, as discussed in Chapter 6, we should consider at the outset whether

a more lenient or more stringent significance level is appropriate for our

study.

As researchers in the problem-oriented policing study do not present

any special reason for altering conventionally accepted levels of signifi-

cance, we will set a 5% significance threshold for our test of statistical

significance. As noted in Chapter 6, in articles and books the significance

level is often expressed by the Greek letter �. For our test, � � 0.05.

The significance level defines the Type I error we are willing to risk in

our test. But it does not tell us directly what outcomes in our sample

would lead us to reject the null hypothesis. For this, we need to turn to

our sampling distribution and define an area within it called a rejection

region. The rejection region of a test is the area in the sampling distribu-

tion that includes those outcomes that would lead to rejection of the null

hypothesis. If the observed significance level of a test, or the p value of

the test, falls within the rejection region, then the researcher rejects the

null hypothesis and concludes that the outcome is statistically significant.

The area covered by the rejection region is equivalent to the significance

level of a test. The point at which the rejection region begins is called

the critical value because it is the point at which the test becomes criti-

cal and leads the researcher to reject the null hypothesis.

Sampling Distribution of Success or Failure in 11 Trials

OUTCOME OF TRIALS OVERALL PROBABILITY

0 successes 0.00049 1 success 0.00537 2 successes 0.02686 3 successes 0.08057 4 successes 0.16113 5 successes 0.22559 6 successes 0.22559 7 successes 0.16113 8 successes 0.08057 9 successes 0.02686

10 successes 0.00537 11 successes 0.00049

Table 8.3

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S I G N I F I C A N C E L E V E L A N D R E J E C T I O N R E G I O N

In the problem-oriented policing example, the rejection region in-

cludes 5% of the sampling distribution. Our initial problem is to define

which 5%. Should we define the rejection region to be in the middle of

the distribution represented in Table 8.3—for example, at 5 or 6 suc-

cesses in 11 comparisons? Or should we look only at the extreme values

on the positive side of the distribution, where there are mostly successes?

Or should we include the area on the negative side of the distribution,

where there are no successes?

Choosing a One-Tailed or a Two-Tailed Rejection Region

The answer to our questions comes in part from common sense and in

part from our assumptions. It just would not make sense to place the re-

jection region in the middle of the sampling distribution. We are trying

to decide whether the outcomes observed in our sample are very differ-

ent from the outcomes that would be expected if problem-oriented

policing had no impact. Putting the rejection region in the middle of the

distribution would place it among those outcomes that are most likely

under the null hypothesis. Clearly, we want the rejection region to be on

the edges of the distribution, or in what statisticians call the tails of the

distribution. These are the unlikely events—those that we would not

expect if the null hypothesis were true. As indicated in our sampling dis-

tribution in Table 8.3, we would expect to get 11 successes in a row in

about 5 of 10,000 samples if the program had no impact on the popula-

tion. This is a very unlikely event and one that would lead us to reject

the null hypothesis.

But zero successes is also an unlikely event, with the same probability

of occurrence as 11 successes. Should we include only one tail of the dis-

tribution in our rejection region—the tail that assesses whether the pro-

gram was a success? Or should we also include the opposite side of the

distribution, which suggests that the program led to more disorder? Our

answer is drawn from the research hypothesis that we stated in our as-

sumptions. We chose a nondirectional research hypothesis, meaning that

we are interested in evaluating both the possibility that the experimental

sites improved relative to the control hot spots and the potential outcome

that they got worse relative to the control hot spots. In terms of the sam-

pling distribution, our research hypothesis suggests that the rejection re-

gion for our test should be split between both tails of the distribution.

This type of test is called a two-tailed test of significance. If we had

stated a directional research hypothesis, we would be concerned with out-

comes on only one side of the sampling distribution. Such a test is called a

one-tailed test of significance. For example, if our research hypothesis

were that incivilities in treatment hot spots decrease relative to incivilities

in control hot spots after POP intervention, we would be concerned only

with outcomes on the side of the distribution that shows program success.

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The choice of a one-tailed or two-tailed test of statistical significance

has important implications for the types of study outcomes that will lead

to rejection of the null hypothesis. Because our test is a two-tailed test,

the rejection region must be divided between both sides of the sampling

distribution. This means in practice that the total significance level of

0.05 must be divided in half. Half of the rejection region, or 0.025, is

found in the tail associated with success of the program, and half, or

0.025, in the tail associated with failure.

What outcomes would lead to rejection of the null hypothesis in our

example? When we add 0 and 1 successes or 10 and 11 successes, we

gain a probability value of 0.00586 (in each tail of the distribution,

0.00049 � 0.00537). This is less than the 0.025 value that we have de-

fined as the rejection region for each tail of our test. Accordingly, an out-

come of 0, 1, 10 or 11 would lead to an observed significance level less

than the significance level of 0.05 that we have set, and thus we would

reject the null hypothesis (p � 0.05). However, including 2 or 9 suc-

cesses, each of which has a probability value of 0.027, increases the area

of the distribution to 0.066. This area is larger than our rejection region.

An outcome of 9 or 2 would result in an observed significance level

greater than 0.05, and thus we would fail to reject the null hypothesis.

Figure 8.1 presents the binomial probabilities for our example and high-

lights the two tails of the distribution that are used to test our nondirec-

tional hypothesis.

P ro

b ab

il it

y

0.25

0.2

0.15

0.1

0.05

0

Number of Successes

0 1 2 3 4 5 6 7 8 9 10 11

0.0005 0.005

0.03 0.03

0.08 0.08

0.16 0.16

0.23 0.23

0.005 0.0005

Outcomes in the rejection region

Outcomes in the rejection region

Outcomes That Would Lead to Rejecting the Null Hypothesis

for a Two-Tailed Test of Significance (� � 0.05) Figure 8.1

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S I G N I F I C A N C E L E V E L A N D R E J E C T I O N R E G I O N

But what if we state a directional research hypothesis? How does

this affect our rejection region? In this case, we calculate the area of the

rejection region on only one side of the sampling distribution. Parts a

and b of Figure 8.2 present the binomial probabilities for our two dis-

tinct directional hypotheses and highlight the tail of the distribution

that is potentially of interest. For example, if our research hypothesis is

that incivilities in treatment hot spots decline relative to incivilities in

control hot spots after POP intervention, we look at outcomes only on

the tail of the distribution that shows program success (Figure 8.2b).

Because we are concerned only about these outcomes, all 5% of the re-

jection region is placed in this one tail of the distribution. We do not

have to split the area of the rejection region. In this example, outcomes

of 9, 10, and 11 successes are all within the rejection region, because

adding their probabilities results in a value of 0.033 (0.00049 � 0.00537

� 0.02686). An outcome of 9, 10, or 11 results in an observed signifi-

cance level that is less than the 5% significance threshold of our test

(see Figure 8.2b). Adding the probability of 8 successes (or 0.08057)

puts us above that threshold. If our research hypothesis is that incivili-

ties increase in treatment hot spots relative to control hot spots, then

we look at outcomes only on the opposite tail of the distribution (Fig-

ure 8.2a). In this case, outcomes of 0, 1, and 2 successes lead us to re-

ject the null hypothesis.

This example reinforces a rule described earlier: It is important to

state the research hypothesis before you gain study outcomes. What if

the problem-oriented policing hot spots improved relative to control

locations in nine comparisons? With a one-tailed test, the result would

fall within our rejection region and lead to rejection of the null hypoth-

esis. With a two-tailed test, the result would be outside our rejection re-

gion. The choice of a directional or nondirectional research hypothesis

can have an important impact on our conclusions. Merely by stating

the research hypothesis a bit differently, we can change the outcome of

the test.

A one-tailed test makes it easier to reject the null hypothesis based on

outcomes on one side of a sampling distribution because it precludes re-

jection of the null hypothesis based on outcomes on the opposite side.

The price of a larger rejection region in one-tail of the sampling distribu-

tion is no rejection region in the other tail. Similarly, the price of being

able to examine outcomes on both sides of the distribution, as is the

case with a two-tailed test, is that the rejection region will be smaller on

each side. The benefit is that you can assess results in both directions. If

you already know the outcomes of a test, you might be tempted to ad-

just the direction of the test according to the observed outcomes of a

study. Taking such an approach unfairly adjusts the rejection region to

your advantage.

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P ro

b ab

il it

y

0.25

0.2

0.15

0.1

0.05

0

Number of Successes

0 1 2 3 4 5 6 7 8 9 10 11

0.0005 0.005

0.03 0.03

0.08 0.08

0.16 0.16

0.23 0.23

0.005 0.0005

Outcomes in the rejection region

P ro

b ab

il it

y

0.25

0.2

0.15

0.1

0.05

0

Number of Successes

0 1 2 3 4 5 6 7 8 9 10 11

0.0005 0.005

0.03 0.03

0.08 0.08

0.16 0.16

0.23 0.23

0.005 0.0005

Outcomes in the rejection region

Outcomes That Would Lead to Rejecting the Null Hypothesis

for a One-Tailed Test of Significance (� � 0.05) Figure 8.2

(b) Focus on Program Successes

(a) Focus on Program Failures

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M A K I N G A D E C I S I O N

T h e T e s t S t a t i s t i c

In most tests of statistical significance, it is necessary to convert the spe-

cific outcome of a study to a test statistic. A test statistic expresses the

value of your outcome in units of the sampling distribution employed in

your test. For the binomial distribution, the units are simply the number

of successes in the total number of trials. The test statistic for our POP

intervention example is 10.

M a k i n g a D e c i s i o n

The final step in a test of statistical significance is making a decision. If

you have laid out all of the steps discussed above, then your choice

should be easy. If your test statistic falls within the rejection region, then

you reject the null hypothesis. This means in practice that the observed

significance level of your test is less than the criterion significance level

that you set when you defined the significance level and rejection region

for your test. If the test statistic does not fall in the rejection region, you

cannot reject the null hypothesis. In our example, the test statistic (10)

does fall in the rejection region, which includes 0, 1, 10, and 11 suc-

cesses. In this case, our observed significance level is less than the 0.05

threshold we set earlier. Our decision, then, is to reject the null hypothe-

sis that incivilities in treatment hot spots do not change relative to incivil-

ities in control hot spots after POP intervention. We conclude that the

differences observed are statistically significant.

But what does this mean? When we say that a result is statistically

significant, we are not claiming that it is substantively important. The

importance of a result depends on such issues as whether the research

affects real-life criminal justice decision making or whether it con-

tributes new knowledge to a specific area of criminology or criminal

justice. We also are not stating that we are certain that the null hypothe-

sis is untrue for the population. Without knowledge of the population

parameter, we cannot answer this question with certainty. Statistical sig-

nificance has a very specific interpretation. The fact that an outcome is

statistically significant means that it falls within the rejection region of

your test. This happens when the observed significance level for a test

is smaller than the significance criterion, or significance level, set at the

outset of the test. A statistically significant result is one that is unlikely if

the null hypothesis is true for the population. Whenever we make a

statement that a result is statistically significant, we do it with the recog-

nition that we are risking a certain level of Type I error. In this test, as

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C H A P T E R E I G H T : S T E P S I N A S T A T I S T I C A L T E S T

in most tests of statistical significance in criminal justice, we were will-

ing to take a 5% risk of falsely rejecting the null hypothesis.

C h a p t e r S u m m a r y

The first stage in a test of statistical significance is to state one’s as-

sumptions. The first assumption is about the type of measurement

used. The second assumption concerns the shape of the population

distribution. A parametric test is one that makes assumptions about

the shape of the population distribution. A nonparametric test

makes no such assumptions. Although nonparametric tests have the

advantage of making fewer assumptions, they are generally used only

for nominal and ordinal scales. The third assumption relates to the

sampling method. A random sample is generally considered to be

pling, it is in theory necessary to return the subject to the sampling

frame after selection. Sampling with replacement creates practical

problems, however, and is generally not required if the sample is

small relative to the population. The fourth assumption states the null

and research hypotheses. Care should be taken in framing them and in

deciding whether the research hypothesis should be directional.

The second stage is to select an appropriate sampling distribution.

The third stage is to select a significance level. The significance level

determines the size of the rejection region and the location of the

critical values of the test. If a test result falls within the rejection re-

gion, the researcher is prepared to reject the null hypothesis. This

means that the observed significance level of the test is less than the

significance level the researcher set at the outset of the test. If the hy-

potheses are directional, then the researcher will be concerned only

with one tail of the distribution, and the entire rejection region will

be placed on one side of the distribution (a one-tailed test of signifi-

cance). If the hypotheses are nondirectional, then the researcher is

concerned with results in both tails, and the rejection region will be

divided equally between both sides of the distribution (a two-tailed

test of significance).

The fourth stage involves calculating a test statistic. The study re-

sult is now converted into the units of the sampling distribution. Fi-

nally, a decision is made: The null hypothesis will be rejected if the

test statistic falls within the rejection region. When such a decision can

be made, the results are said to be statistically significant.

accepted form of sampling. To ensure the independence of the sam-

more representative, or to have greater external validity, than a

convenience sample. Independent random sampling is the most

188

K E Y T E R M S

assumptions Statements that identify the

requirements and characteristics of a test of

statistical significance. These are the foun-

dations on which the rest of the test is built.

convenience sample A sample chosen not

at random, but according to criteria of expe-

dience or accessibility to the researcher.

critical value The point at which the re-

jection region begins.

distribution-free tests Another name for

nonparametric tests.

external validity The extent to which a

study sample is reflective of the population

from which it is drawn. A study is said to

have high external validity when the sam-

ple used is representative of the population

to which inferences are made.

independent random sampling A form

of random sampling in which the fact that

one subject is drawn from a population in

no way affects the probability of drawing

any other subject from that population.

nonparametric tests Tests of statistical

significance that make no assumptions as

to the shape of the population distribution.

one-tailed test of significance A test of

statistical significance in which the region

for rejecting the null hypothesis falls on

only one side of the sampling distribution.

One-tailed tests are based on directional re-

search hypotheses.

parametric tests Tests of statistical signifi-

cance that make assumptions as to the

shape of the population distribution.

random sampling Drawing samples from

the population in a manner that ensures

every individual in that population an

equal chance of being selected.

rejection region The area of a sampling

distribution containing the test statistic val-

ues that will cause the researcher to reject

the null hypothesis.

representative sample A sample that

reflects the population from which it is

drawn.

sampling frame The universe of eligible

cases from which a sample is drawn.

sampling with replacement A sampling

method in which individuals in a sample

are returned to the sampling frame after

they have been selected. This raises the

possibility that certain individuals in a pop-

ulation may appear in a sample more than

once.

statistically significant Describing a test

statistic that falls within the rejection region

defined by the researcher. When this oc-

curs, the researcher is prepared to reject the

null hypothesis and state that the outcome

or relationship is statistically significant.

tails of the distribution The extremes on

the sides of a sampling distribution. The

events represented by the tails of a sam-

pling distribution are those deemed least

likely to occur if the null hypothesis is true

for the population.

test statistic The outcome of the study,

expressed in units of the sampling distribu-

tion. A test statistic that falls within the re-

jection region will lead the researcher to re-

ject the null hypothesis.

two-tailed test of significance A test of

statistical significance in which the region

for rejecting the null hypothesis falls on

both sides of the sampling distribution.

Two-tailed tests are based on nondirec-

tional research hypotheses.

K e y T e r m s

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E x e r c i s e s

8.1 Answer the following conceptual questions:

a. Is it better to have more or fewer assumptions at the beginning of a test of statistical significance? Explain your answer.

b. Why is it important to state all of the assumptions at the outset of the test?

c. In what sense can stating the null and research hypotheses be seen as making assumptions?

8.2 Gatley University is an elite university of 1,000 students. Nadia, a stu- dent studying Chinese at the university, wishes to determine the aver- age IQ of students at Gatley. She has decided that her sample size will be 50, and she is considering several different sampling methods. For each method, state the sampling frame and discuss whether the sam- pling method is random and whether it is independent.

a. Nadia chooses 50 names at random from the list of language stu- dents at the university.

b. Nadia asks 50 of her acquaintances at the university if they would mind taking an IQ test.

c. Nadia chooses the first two students from the alphabetical list of each of the 25 university departments.

d. Nadia takes all 1,000 names and puts them into a hat. She draws out a name, writes it down, and then puts it back in the hat and draws again. This procedure is repeated 50 times.

8.3 Hale Prison is renowned for its poor internal discipline. The new prison governor wants to tackle this problem and decides to investi- gate whether removing prisoners’ visiting privileges will act as a deter- rent against future misbehaving. From 100 prisoners who recently took part in a violent prison riot, he selects the 25 inmates with the worst disciplinary records, removes their visiting privileges, and begins to monitor their progress relative to the others.

a. Does this method meet the criteria of independent random sampling?

b. Is independent sampling possible in this case?

c. Describe a more appropriate sampling method.

8.4 For each of the following hypotheses, state whether a one-tailed or a two-tailed test of statistical significance would be appropriate. In each case, explain your choice.

a. H1: Citizens over the age of 50 are more likely to be the victims of assault than citizens under the age of 50.

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b. H1: Children raised by adopted parents have rates of delinquency different from those of children raised by their biological parents.

c. H1: The experience of imprisonment has an impact on the chances of an ex-convict reoffending.

d. H1: Women are more likely than men to support increased sentences for rapists.

e. H1: Persons who are not victims of assault have lower levels of anger than persons who have been victims of assault.

f. H1: White offenders are less likely to be sentenced to prison than Hispanic offenders.

g. H1: Teenagers have rates of crime that are different from adult rates of crime.

h. H1: Defendants charged with property crimes have different rates of pretrial misconduct than defendants charged with violent crimes.

i. H1: Male defendants are more likely to be held on bail than female defendants.

j. H1: Women are more supportive of capital punishment than men.

k. H1: States with higher unemployment rates have higher rates of property crime.

l. H1: The level of poverty in a neighborhood affects the neighbor- hood’s crime rate.

m. H1: Democrats are less supportive of cutting taxes than Republicans.

n. H1: Graduates from private law schools are more likely to become federal judges than graduates from state law schools.

8.5 In Chapter 7, we constructed a binomial distribution showing the chances of success and failure for ten tosses of a fair coin. The distri- bution was as follows:

0 heads 0.001

1 head 0.010

2 heads 0.044

3 heads 0.118

4 heads 0.206

5 heads 0.247

6 heads 0.206

7 heads 0.118

8 heads 0.044

9 heads 0.010

10 heads 0.001

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Consider the following alternative hypotheses:

Alternative 1: H0: The coin is fair. H1: The coin is biased.

Alternative 2: H0: The coin is fair. H1: The coin is biased in favor of heads.

a. Would a one-tailed or a two-tailed test be more appropriate for a researcher who chose alternative 1? Explain why.

b. For a sequence of ten throws, what results would cause a re- searcher operating under the hypotheses listed under alternative 1 to reject the null hypothesis at a significance level of 5%?

c. Would a one-tailed or a two-tailed test be more appropriate for a researcher who chose alternative 2? Explain why.

d. For a sequence of ten throws, what results would cause a re- searcher operating under the hypotheses listed under alternative 2 to reject the null hypothesis at a significance level of 5%?

8.6 Use the following binomial distribution showing the chances of suc- cess and failure for 12 trials.

Number of Successes Probability

0 successes 0.00118

1 success 0.01065

2 successes 0.04418

3 successes 0.11110

4 successes 0.18857

5 successes 0.22761

6 successes 0.20032

7 successes 0.12953

8 successes 0.06107

9 successes 0.02048

10 successes 0.00463

11 successes 0.00064

12 successes 0.00004

Using a significance level of 0.05, what outcomes would lead you to reject the null hypothesis for each of the following pairs of hypotheses?

a. H0: P � 0.50 H1: P � 0.50

b. H0: P � 0.50 H1: P � 0.50

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c. H0: P � 0.50 H1: P � 0.50

d. If you changed the significance level to 0.01, how would your an- swers to parts a, b, and c change?

8.7 Use the following binomial distribution showing the chances of suc- cess and failure for 15 trials.

Number of Successes Probability

0 successes 0.00000

1 success 0.00000

2 successes 0.00001

3 successes 0.00006

4 successes 0.00042

5 successes 0.00228

6 successes 0.00930

7 successes 0.02928

8 successes 0.07168

9 successes 0.13650

10 successes 0.20051

11 successes 0.22313

12 successes 0.18210

13 successes 0.10288

14 successes 0.03598

15 successes 0.00587

Using a significance level of 0.05, what outcomes would lead you to reject the null hypothesis for each of the following pairs of hypotheses?

a. H0: P � 0.50 H1: P � 0.50