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Chapter 16

To accompany Quantitative Analysis for Management, Eleventh Edition, by Render, Stair, and Hanna

Power Point slides created by Brian Peterson and Xiaodong Wu

Statistical Quality Control

16-2

Learning Objectives

After completing this chapter, students will be able to:

Define the quality of a product or service.

Develop four types of control charts: x, R, p, and c.

Understand the basic theoretical underpinnings of statistical quality control, including the central limit theorem.

Know whether a process is in control.

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Chapter Outline

16.1 Introduction

16.2 Defining Quality and TQM

16.3 Statistical Process Control

16.4 Control Charts for Variables

16.5 Control Charts for Attributes

16-4

Introduction

Quality is often the major issue in a purchase decision, as poor quality can be expensive for both the producing firm and the customer.

Quality management, or quality control (QC), is critical throughout the organization,

Quality is important for manufacturing and services.

We will be dealing with the most important statistical methodology, statistical process control (SPC).

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Quality of a product or service is the degree to which the product or service meets specifications.

Increasingly, definitions of quality include an added emphasis on meeting the customer’s needs.

Total quality management (TQM) refers to a quality emphasis that encompasses the entire organization from supplier to customer.

Meeting the customer’s expectations requires an emphasis on TQM if the firm is to complete as a leader in world markets.

Defining Quality and TQM

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Several definitions of quality:

“Quality is the degree to which a specific product conforms to a design or specification.” (Gilmore, 1974)

“Quality is the totality of features and characteristics of a product or service that bears on its ability to satisfy stated or implied needs.” (Johnson and Winchell, 1989)

“Quality is fitness for use.” (Juran, 1974)

“Quality is defined by the customer; customers want products and services that, throughout their lives, meet customers’ needs and expectations at a cost that represents value.” (Ford, 1991)

“Even though quality cannot be defined, you know what it is.” (Pirsig, 1974)

Defining Quality and TQM

Table 16.1

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Statistical Process Control

Statistical process control involves establishing and monitoring standards, making measurements, and taking corrective action as a product or service is being produced.

Samples of process output are examined.

If sample results fall outside certain specific ranges, the process is stopped and the assignable cause is located and removed.

A control chart is a graphical presentation of data over time and shows upper and lower limits of the process we want to control.

This slide introduces the difference between “natural” and “assignable” causes.

The next several slides expand the discussion and introduce some of the statistical issues.

16-8

Patterns to Look for in Control Charts

Figure 16.1

This slide introduces the difference between “natural” and “assignable” causes.

The next several slides expand the discussion and introduce some of the statistical issues.

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Building Control Charts

Control charts are built using averages of small samples.

The purpose of control charts is to distinguish between natural variations and variations due to assignable causes.

This slide introduces the difference between “natural” and “assignable” causes.

The next several slides expand the discussion and introduce some of the statistical issues.

16-10

Building Control Charts

Natural variations

Natural variations affect almost every production process and are to be expected, even when the process is in statistical control.

They are random and uncontrollable.

When the distribution of this variation is normal it will have two parameters.

Mean,  (the measure of central tendency of the average).

Standard deviation,  (the amount by which smaller values differ from the larger ones).

As long as the distribution remains within specified limits it is said to be “in control.”

This slide introduces the difference between “natural” and “assignable” causes.

The next several slides expand the discussion and introduce some of the statistical issues.

16-11

Building Control Charts

Assignable variations

When a process is not in control, we must detect and eliminate special (assignable) causes of variation.

The variations are not random and can be controlled.

Control charts help pinpoint where a problem may lie.

The objective of a process control system is to provide a statistical signal when assignable causes of variation are present.

This slide introduces the difference between “natural” and “assignable” causes.

The next several slides expand the discussion and introduce some of the statistical issues.

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Control Charts for Variables

The x-chart (mean) and R-chart (range) are the control charts used for processes that are measured in continuous units.

The x-chart tells us when changes have occurred in the central tendency of the process.

The R-chart tells us when there has been a change in the uniformity of the process.

Both charts must be used when monitoring variables.

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The Central Limit Theorem

The central limit theorem is the foundation for x-charts.

The central limit theorem says that the distribution of sample means will follow a normal distribution as the sample size grows large.

Even with small sample sizes the distribution is nearly normal.

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The Central Limit Theorem

The central limit theorem says:

The mean of the sampling distribution will equal the population mean.

The standard deviation of the sampling distribution will equal the population standard deviation divided by the square root of the sample size.

We often estimate and with the average of all sample means ( ).

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The Central Limit Theorem

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–3x

–1x

+1x

+2x

–2x

+3x

x = 

(mean)

The Central Limit Theorem

Population and Sampling Distributions

Sampling Distribution of Sample Means (Always Normal)

Normal

Beta

Uniform

 = (mean)

x = S.D.

 = (mean)

x = S.D.

 = (mean)

x = S.D.

99.7% of all x

fall within ±3x

95.5% of all x fall within ±2x

Figure 16.2

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Setting the x-Chart Limits

If we know the standard deviation of the process, we can set the control limits using:

where

= mean of the sample means

z = number of normal standard deviations

= standard deviation of the sampling distribution of the sample means =

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Box Filling Example

A large production lot of boxes of cornflakes is sampled every hour.

To set control limits that include 99.7% of the sample, 36 boxes are randomly selected and weighed.

The standard deviation is estimated to be 2 ounces and the average mean of all the samples taken is 16 ounces.

So and the control limits are:

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Box Filling Example

If the process standard deviation is not available or difficult to compute (a common situation) the previous equations are impractical.

In practice the calculation of the control limits is based on the average range rather than the standard deviation.

where

= average of the samples

A2 = value found in Table 16.2

= mean of the sample means

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Factors for Computing Control Chart Limits

SAMPLE SIZE, n	MEAN FACTOR, A2	UPPER RANGE, D4	LOWER RANGE, D3
2	1.880	3.268	0
3	1.023	2.574	0
4	0.729	2.282	0
5	0.577	2.115	0
6	0.483	2.004	0
7	0.419	1.924	0.076
8	0.373	1.864	0.136
9	0.337	1.816	0.184
10	0.308	1.777	0.223
12	0.266	1.716	0.284
14	0.235	1.671	0.329
16	0.212	1.636	0.364
18	0.194	1.608	0.392
20	0.180	1.586	0.414
25	0.153	1.541	0.459

Table 16.2

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Box Filling Example

Program 16.1

Excel QM Solution for Box-Filling Example

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Super Cola

Super Cola bottles are labeled “net weight 16 ounces.”

The overall process mean is 16.01 ounces and the average range is 0.25 ounces in a sample of size n = 5.

What are the upper and lower control limits for this process?

 16.01 + (0.577)(0.25)

 16.01 + 0.144

 16.154

 16.01 – (0.577)(0.25)

 16.01 – 0.144

 15.866

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Super Cola

Program 16.2

Excel QM Solution for Super Cola Example

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Setting Range Chart Limits

We have determined upper and lower control limits for the process average.

We are also interested in the dispersion or variability of the process.

Averages can remain the same even if variability changes.

A control chart for ranges is commonly used to monitor process variability.

Limits are set at ±3 for the average range

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Setting Range Chart Limits

We can set the upper and lower controls using:

where

UCLR = upper control chart limit for the range

LCLR = lower control chart limit for the range

D4 and D3 = values from Table 16.2

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A process has an average range of 53 pounds.

If the sample size is 5, what are the upper and lower control limits?

From Table 16.2, D4 = 2.114 and D3 = 0.

Range Example

 (2.114)(53 pounds)

 112.042 pounds

 (0)(53 pounds)

 0

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Five Steps to Follow in Using x and R-Charts

Collect 20 to 25 samples of n = 4 or n = 5 from a stable process and compute the mean and range of each.

Compute the overall means ( and ), set appropriate control limits, usually at 99.7% level and calculate the preliminary upper and lower control limits. If process not currently stable, use the desired mean, μ, instead of to calculate limits.

Graph the sample means and ranges on their respective control charts and determine whether they fall outside the acceptable limits.

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Five Steps to Follow in Using x and R-Charts

Investigate points or patterns that indicate the process is out of control. Try to assign causes for the variation and then resume the process.

Collect additional samples and, if necessary, revalidate the control limits using the new data.

R-chart

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Control Charts for Attributes

We need a different type of chart to measure attributes.

These attributes are often classified as defective or nondefective.

There are two kinds of attribute control charts:

Charts that measure the percent defective in a sample are called p-charts.

Charts that count the number of defects in a sample are called c-charts.

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p-Charts

Attributes that are good or bad typically follow the binomial distribution.

If the sample size is large enough a normal distribution can be used to calculate the control limits:

where

= mean proportion or fraction defective in the sample

z = number of standard deviations

= standard deviation of the sampling distribution which is estimated by where n is the size of each sample

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ARCO p-Chart Example

Performance of data-entry clerks at ARCO (n = 100)

SAMPLE NUMBER	NUMBER OF ERRORS	FRACTION DEFECTIVE	SAMPLE NUMBER	NUMBER OF ERRORS		FRACTION DEFECTIVE
1	6	0.06	11		6	0.06
2	5	0.05	12		1	0.01
3	0	0.00	13		8	0.08
4	1	0.01	14		7	0.07
5	4	0.04	15		5	0.05
6	2	0.02	16		4	0.04
7	5	0.05	17		11	0.11
8	3	0.03	18		3	0.03
9	3	0.03	19		0	0.00
10	2	0.02	20		4	0.04
					80

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ARCO p-Chart Example

We want to set the control limits at 99.7% of the random variation present when the process is in control so z = 3.

Percentage can’t be negative.

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ARCO p-Chart Example

p-chart for Data Entry for ARCO

UCLp = 0.10

LCLp = 0.00

0.12 –

0.11 –

0.10 –

0.09 –

0.08 –

0.07 –

0.06 –

0.05 –

0.04 –

0.03 –

0.02 –

0.01 –

0.00 –

Fraction Defective

Sample Number

Figure 16.3

Out of Control

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Excel QM Solution for ARCO p-chart Example

Program 16.3

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c-Charts

In the previous example we counted the number of defective records entered in the database.

But records may contain more than one defect.

We use c-charts to control the number of defects per unit of output.

c-charts are based on the Poisson distribution which has its variance equal to its mean.

The mean is and the standard deviation is equal to

To compute the control limits we use:

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Red Top Cab Company c-Chart Example

The company receives several complaints each day about the behavior of its drivers.

Over a nine-day period the owner received 3, 0, 8, 9, 6, 7, 4, 9, 8 calls from irate passengers, for a total of 54 complaints.

To compute the control limits:

Thus:

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Excel QM Solution for Red Top Cab Company c-Chart Example

Program 16.4

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