Table Of ContentMistakes in
Quality Statistics
and How to Fix Them
Donald W. Benbow
ASQ Quality Press
Milwaukee, Wisconsin
Mistakes in Quality Statistics and How to Fix Them
© 2021 by Donald W. Benbow
All rights reserved. Published 2021
Printed in the United States of Amer i ca
24 23 22 21 SWY 5 4 3 2 1
Publisher’s Cataloging-in-Publication data
Names: Benbow, Donald W., 1936–, author.
Title: Mistakes in quality statistics and how to fix them / by Donald
W. Benbow.
Description: Includes bibliographical references. | Milwaukee, WI: Quality
Press, 2021.
Identifiers: LCCN: 2021935792 | ISBN: 978-1-63694-000-7 (paperback) |
978-1-63694-001-4 (epub)
Subjects: LCSH Statistics. | Statistics—Evaluation. | Statistics—
Methodology. | Auditing. | Quality control. | BISAC BUSINESS &
ECONOMICS / Auditing | MATHEMATICS / Probability & Statistics /
General
Classification: LCC QA276 .B46 2021| DDC 519.5—dc23
No part of this book may be reproduced in any form or by any means,
electronic, mechanical, photocopying, recording, or other wise, without the
prior written permission of the publisher.
ASQ advances individual, orga nizational, and community excellence
worldwide through learning, quality improvement, and knowledge
exchange.
Attention bookstores, wholesalers, schools, and corporations: Quality Press
books are available at quality discounts with bulk purchases for business,
trade, or educational uses. For information, please contact Quality Press at
800-248-1946 or books@asq . org.
To place orders or browse the se lection of ASQ Excellence and Quality
Press titles, visit our website at http:// www . asq . org / quality - press.
Printed on acid- free paper.
About the Author
For nearly 30 years, Don Benbow has conducted industrial sta-
tistics courses for employees of approximately 100 companies,
both face to face and through ICN, Iowa’s closed-c ircuit fiber
optics TV network. Previously, he taught mathe matics, statistics,
and freshman engineering courses at Marshalltown Community
College. He holds BS and MS degrees in mathe matics from Iowa
State University and Michigan State University, respectively. He
is co-author of The Certified Quality Technician Handbook, The Cer
tified Six Sigma Black Belt Handbook, and Introduction to 8D Prob
lem Solving, all available from ASQ Quality Press. He is an ASQ
Certified Quality Auditor (CQA), Certified Quality Technician
(CQT), Certified Quality Engineer (CQE), and Certified Reliabil-
ity Engineer (CRE).
123
List of Figures and Tables
FIGURES
Figure 1.1 Graph of the data from Example 1.1 ......... 2
Figure 1.2 Graph of the data from Example 1.2 ......... 2
Figure 1.3 Graph of the data from increased
sample sizes .............................. 3
Figure 1.4 The distribution of hole sizes showing
standard deviation bound aries and
specification limits ........................ 8
Figure 1.5 χ2 distribution for df = 29 with left
tail 0.05 shaded ........................... 9
Figure 2.1 Spot- welder diagram with strength data ..... 14
Figure 2.2 Graph of spot- weld strength data ........... 14
Figure 2.3 Regression line plotted with data points ...... 14
Figure 2.4 Graph of the data from Example 2.2 ......... 17
Figure 2.5 Examples of vari ous values of the
linear correlation coefficient ................ 17
Figure 2.6 Graph of data from Table 2.3 ................ 18
Figure 2.7 Graph of data from Table 2.4 ................ 21
Figure 2.8 Fermentation tank with beer well ........... 21
Figure 2.9 Graph of data with beer well temperature
set at 140° ................................ 22
Figure 2.10 Sketch for Example 2.5 ..................... 24
Figure 2.11 Plot of data from Table 2.5 .................. 24
vii
viii Mistakes in Quality Statistics
Figure 2.12 Scatter diagram for the data in Table 2.6 ...... 26
Figure 2.13 Comparing predicted with actual DE values .. 29
Figure 4.1 A control chart showing instability .......... 42
Figure 4.2 Control chart without indicators
of instability .............................. 44
Figure 4.3 Histogram showing three distinct humps .... 45
Figure 4.4 Control chart for improved pro cess .......... 46
Figure 4.5 Histogram showing the data after pro cess
improvement ............................. 47
Figure 4.6 Five- by- eight arrays of choco lates on a
waxed- paper- covered con vey or ............. 48
Figure 4.7 Two methods for selecting five choco lates
to weigh ................................. 48
Figure 4.8 Individuals control chart ................... 50
Figure 4.9 X- bar and R chart for a batch pro cess ........ 51
Figure 4.10 X- MR control chart using average values
of percent solids as the individual values ..... 53
Figure 4.11 Scatter plot of the data in Table 4.1 ........... 54
Figure 4.12 Scatter plot of the points in Table 4.2 ......... 57
Figure 4.13 X- bar and R chart using stratified data ....... 60
Figure 4.14 Histogram of the readings in Figure 4.13 ..... 61
Figure 4.15 Illustration of typical pre- control limits ...... 62
Figure 4.16 Pre- control chart for Example 4.7 with four
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Figure 4.17 Illustration for Example 4.10 ................ 66
Figure 4.18 Illustrating the specification limits in
terms of standard deviations from the
pro cess mean ............................. 67
Figure 4.19 Normal distribution, with the portion
of production that violates the lower
specification limit shaded .................. 68
Figure 4.20 NORMSDIST with a z- value of −1.48 ........ 68
Figure 4.21 NORMSDIST value with 5.93 entered ........ 69
Figure 4.22 Illustrating a centered pro cess .............. 70
List of Figures and Tables ix
Figure 4.23 Excel function NORMSDIST with −3.705
input .................................... 71
Figure 6.1 Graphical repre sen ta tion of the average
responses ................................ 86
Figure 7.1 Example of an ac cep tance sampling plan ..... 94
Figure 7.2 Example of a double sampling plan .......... 96
Figure 7.3 Example of sampling plan that uses arrows ... 97
Figure 7.4 Examples of OC curves .................... 98
Figure 7.5 OC curve for 1.5 AQL ...................... 98
Figure 7.6 Details for an OC curve for 1.5 AQL ......... 99
Figure 7.7 Ideal OC curve for an AQL = 1.5% ........... 100
Figure 7.8(a) Example of an OC curve for AQL = 1.5 ....... 100
Figure 7.8(b) Another example of an OC curve
for AQL = 1.5 ............................. 100
Figure 7.9 Table for converting Q- values to percent
defective ................................. 102
TABLES
Table 1.1 Partial χ 2 table ............................ 9
Table 2.1 Humidity and density data ................. 15
Table 2.2 Estimated densities using the regression
equation ................................. 16
Table 2.3 Data relating CO concentration
2
to growth rate ............................ 18
Table 2.4 Data for 18 batches and correlations
with DE ................................. 20
Table 2.5 Fracture data for Example 2.5 ............... 24
Table 2.6 Data relating orifice dia meter
and malleability .......................... 26
Table 2.7 Annealing data ........................... 27
Table 2.8 Annealing data segregated by gauge ......... 28
Table 4.1 Data points for Example 4.5 ................ 55
x Mistakes in Quality Statistics
Table 4.2 Values from Table 4.1 compared
with lag 1 values .......................... 56
Table 4.3 Illustrating batch means for batch size n = 8 ... 58
Table 4.4 Values of d for several sample sizes ......... 63
2
Table 5.1 d for small numbers of subgroups .......... 78
2
Table 6.1 A 23 factorial data collection sheet for
Example 6.1 .............................. 84
Table 6.2 A 23 factorial data collection sheet with
run averages ............................. 84
Table 6.3 A 23 full factorial design using the
+ and − format ............................ 87
Table 6.4 A 23 full factorial design showing
interaction columns ....................... 88
Table 6.5 Half fraction of 23 (also called a 23−1 design) ... 89
Table 6.6 Half fraction of 23 with completed
interaction columns ....................... 90
Preface
This book began as a pres en ta tion titled “Caveats Regarding
the Use of Statistics” that I made during the May 2009 World
Quality Conference. It is not intended to serve as a statistics
textbook. In fact, it assumes the reader has some familiarity with
the subject, perhaps through an introductory course. Texts and
courses tend to emphasize how to perform statistical analy sis
and give little attention to errors that can occur in the pro cess.
The purpose of this book is to show readers how to avoid pit-
falls. The examples and case studies, although based on similar
events, do not include actual data.
xi
Table of Contents
List of Figures and Tables ................................. vii
Preface ................................................ xi
Chapter 1 Hypothesis Tests ........................... 1
1.1 Hypothesis Test for Two Population Means ........ 3
1.2 Hypothesis Test for a Population Proportion ....... 5
1.3 Hypothesis Test for Population
Standard Deviation ............................. 7
Chapter 2 Correlation and Causation .................. 13
2.1 Lurking Variable ................................ 22
2.2 Correlation or Causation? ........................ 23
2.3 Correlation and Prediction ....................... 28
Chapter 3 Margin of Error and Confidence Intervals ..... 31
3.1 Margin of Error and Confidence Intervals for
Population Means ............................... 31
3.2 Hypothesis Test for Two Population Means ........ 34
3.3 Margin of Error and Confidence Intervals
for Proportions ................................. 35
Chapter 4 Control Charts ............................. 41
4.1 Pre- control Charts ............................. 61
4.2 Calculations for a Pro cess with a Normal
Population ..................................... 67
v
vi Mistakes in Quality Statistics
Chapter 5 Mea sure ment Systems ...................... 73
5.1 Mea sure ment System Variation ................... 73
5.2 Evaluating a Mea sur ing System
for Continuous Data ............................. 74
5.3 Sample Procedure for Mea sure ment
System Analy sis ................................ 75
5.4 Diagnosis Using the GRR ........................ 79
5.5 Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.6 Mea sure ment Systems and Capability Analy sis ..... 80
Chapter 6 Designed Experiments ...................... 83
6.1 Main Effects .................................... 85
6.2 Interaction Effects ............................... 87
6.3 Fractional Factorial Designs ...................... 89
6.4 Balanced Designs ............................... 89
Chapter 7 Sampling Plans ............................ 93
7.1 Attribute Sampling Plans ........................ 93
7.2 Double Sampling Plans .......................... 95
7.3 Evaluating Sampling Plans ....................... 97
7.4 Variables Sampling Plans ........................ 101
Chapter 8 Probability ................................ 103
8.1 Summary of Rules .............................. 106
Chapter 9 Some General Caveats ...................... 109
Appendix .............................................. 113
Notes ................................................. 117
Further Study .......................................... 119
Recommended Reading ................................... 121
About the Author ....................................... 123
