Applied Statistics for the Six Sigma Green Belt Bhisham C. Gupta H. Fred Walker ASQ Quality Press Milwaukee, Wisconsin American Society for Quality, Quality Press, Milwaukee 53203 ©2005 by American Society for Quality All rights reserved. Published 2005 Printed in the United States of America 12 11 10 09 08 07 06 05 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Gupta, Bhisham C., 1942– Applied statistics for the Six Sigma Green Belt / Bhisham C. Gupta, H. Fred Walker.— 1st ed. p. cm. Includes bibliographical references and index. ISBN 0-87389-642-4 (hardcover : alk. paper) 1. Six sigma (Quality control standard) 2. Production management. 3. Quality control. I. Walker, H. Fred, 1963– II. Title. TS156.G8673 2005 658.4'013—dc22 2004029760 ISBN 0-87389-642-4 No part of this book may be reproduced in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Publisher: William A. 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Printed on acid-free paper Introduction W henever a process is not producing products or services at a desired level of quality, an investigation is launched to better understand and improve the process. In many instances such investigations are launched to rapidly identify and correct underlying problems as part of a problem solving methodology—one such methodology is commonly known as “root cause analysis.” Many problem-solving methodologies, such as root cause analysis, rely on the study of numerical (quantitative) or non-numeri- cal (qualitative) data as a means of discovering the true cause to one or more problems negatively impacting product or service quality. The problem- solving methodologies, however, are all too commonly used to investigate problems that need a quick solution and thus are not afforded the time or resources needed for a particularly detailed or in-depth analysis. Further, problem-solving methodologies are also all too commonly used to investi- gate problems without sufficient analysis of a series of costs associated with a given problem as they relate to lost profit or opportunity, human resources needed to investigate the problem, and so forth. Let us not have the wrong impression of problem-solving methodologies such as root cause analysis! Each of these methodologies has a proper place in quality and process improvement; however, the scope or size of the problem needs also to be considered. In this context, when problems are smaller and are easier to understand, we can effectively use less rigorous, complicated, and thorough problem-solving methodologies. When problems become large, complex, and expensive, a more detailed and robust problem-solving methodology is needed, and that problem-solving methodology is Six Sigma. While it is beyond the intended scope of this book to discuss, in detail, the Six Sigma methodology as an approach to problem solving, it is the explicit intent of this book to describe the concepts and application of tools and techniques used to support the Six Sigma methodology. Next we give a brief description of the topics discussed in the book, followed by where in the Six Sigma methodology you can expect to use these tools and techniques. In Chapter 1 we introduce the concept of Six Sigma from both statistical and quality perspectives. We briefly describe what we need for converting data into information. In statistical applications we come across various xxiv Introduction xxv types of data that require specific analyses that depend upon the types of data we are working with. It is therefore important to distinguish between differ- ent types of data. In Chapter 2 we discuss and provide examples for different types of data. In addition, terminology such as population and sample are introduced. InChapter3weintroduceseveralgraphicalmethodsfoundindescriptive statistics. These graphical methods are some of the basic tools of statistical qualitycontrol(SQC).Thesemethodsarealsoveryhelpfulinunderstanding thepertinentinformationcontainedinverylargeandcomplexdatasets. InChapter4welearnaboutthenumericalmethodsofdescriptivestatis- tics.Numericalmethodsthatareapplicabletobothsampleaswellaspopula- tiondataprovideuswithquantitativeornumericalmeasures.Suchmeasures furtherenlightenusabouttheinformationcontainedinthedata. In Chapter 5 we proceed to study the basic concepts of probability theo- ry and see how probability theory relates to applied statistics. We also intro- duce the random experiment and define sample space and events. In addition, we study certain rules of probability and conditional probability. In Chapter 6 we introduce the concept of a random variable, which is a vehicle used to assign some numerical values to all the possible outcomes of a random experiment. We also study probability distributions and define mean and standard deviation of random variables. Specifically, we study some special probability distributions of discrete random variables such as Bernoulli, binomial, hypergeometric, and Poisson distributions, which are encountered frequently in many statistical applications. Finally, we discuss under what conditions (e.g., the Poisson process) these probability models are applicable. In Chapter 7 we continue studying probability distributions of random variables. We introduce the continuous random variable and study its proba- bility distribution. We specifically examine uniform, normal, exponential, and Weibull continuous probability distributions. The normal distribution is the backbone of statistics and is extensively used in achieving Six Sigma quality characteristics. The exponential and Weibull distributions form an important part of reliability theory. The hazard or failure rate function is also introduced. Having discussed probability distributions of data as they apply to dis- crete and continuous random variables in Chapters 6 and 7, in Chapter 8 we expand our study to the probability distributions of sample statistics. In par- ticular, we study the probability distribution of the sample mean and sample proportion. We then study Student’s t, chi-square, and F distributions. These distributions are an essential part of inferential statistics and, therefore, of applied statistics. Estimation is an important component of inferential statistics. In Chapter 9 we discuss point estimation and interval estimation of population mean and of difference between two population means, both when sample size is large and when it is small. Then we study point estimation and interval estimation of population proportion and of difference between two population propor- tions when the sample size is large. Finally, we study the estimation of a pop- ulation variance, standard deviation, ratio of two population variances, and ratio of two population standard deviations. xxvi Introduction Table 1 Applied statistics and the Six Sigma methodology. Six Sigma Phase Tool or Technique Where in this book? Define Descriptive Statistics Chapter 2 Graphical Methods Chapter 3 Numerical Descriptions Chapter 4 Measure Sampling Chapter 8 Point & Interval Estimation Chapter 9 Analyze Probability Chapter 5 Discrete & Continuous Distributions Chapters 6 & 7 Hypothesis Testing Chapters 10 Improve Control In Chapter 10 we study another component of inferential statistics, which is the testing of statistical hypotheses. The primary aim of statistical hypotheses is to either refute or support the existing theory, which is, in other words, what is believed to be true based upon the information contained in sample data. This further enhances good procedures. In this chapter we dis- cuss the techniques of testing statistical hypotheses for one population mean and for differences between two population means, both when sample sizes are large and when they are small. We also discuss techniques of testing hypotheses for one population proportion and for differences between two population proportions when sample sizes are large. Finally, we discuss test- ing of statistical hypotheses for one population variance and for ratio of two population variances under the assumption that the populations are normal. The results of Chapter 9 and this chapter are frequently used in statistical quality control (SQC) and design of experiments (DOE). In Chapter 11 we consider computer-based tools for applied statistical support. Computing resources were purposefully included at the end of the book so as to encourage readers not to rely on computers until afterthey have gained a mastery of the statistical content presented in the preceding chapters. But where in the Six Sigma methodology do we use these tools and tech- niques? The answer is throughout the methodology! Let’s take a closer look. The information contained in Table 1 will help us better relate specific tools and techniques to phases of the Six Sigma methodology as they relate to the intended scope and purpose of this book—a basic level of applied statistics. Additional topics will be discussed in later books in this series. As topics are discussed in later books, these topics will be added to content of Table 1 and readers can use the table to help associate specific tools to the Six Sigma methodology. The array of topics as they relate to the Six Sigma methodology is help- ful in understanding where you may use these tools and techniques. It is important to note however, that any of these tools and techniques may come into play in more than one phase of the Six Sigma methodology, and in fact, should be expected to do so. What is presented in Table 1 is a first point in the methodology you may expect to use these tools and techniques. From here it’s time to get started! Enjoy! Preface A pplied Statistics for the Six Sigma Green Belt was written as a desk reference and instructional aid for individuals involved with Six Sigma project teams. As Six Sigma team members, green belts will help select appropriate statistical tools, collect data for those tools, and assist with data interpretation within the context of the Six Sigma methodology. Composed of steps or phases titled Define, Measure, Analyze, Improve, and Control (DMAIC), the Six Sigma methodology calls for the use of many more statistical tools than is reasonable to address in one large book. Accordingly, the intent of this book is to provide Green Belts with the bene- fit of a thorough discussion relating to the underlying concepts of “basic sta- tistics.” More advanced topics of a statistical nature will be discussed in three other books that, together with this book, will comprise a four-book series. The other books in the series will discuss statistical quality control, intro- ductory design of experiments and regression analysis, and advanced design of experiments. While it is beyond the scope of this book and series to cover the DMAIC methodology specifically, we do focus this book and series on concepts, applications, and interpretations of the statistical tools used during, and as part of, the DMAIC methodology. Of particular interest in this book, and indeed the other books in this series, is an applied approach to the topics covered while providing a detailed discussion of the underlying concepts. This level of detail in providing the underlying concepts is particularly important for individuals lacking a recent study of applied statistics as well as for individu- als who may never have had any formal education or training in statistics. In fact, one very controversial aspect of Six Sigma training is that, in many cases, this training is targeted at the Six Sigma Black Belt and is all too commonly delivered to large groups of people with the assumption that all trainees have a fluent command of the underlying statistical concepts and theory. In practice this assumption commonly leads to a good deal of con- cern and discomfort for trainees who quickly find it difficult to keep up with and successfully complete black belt–level training. This concern and dis- comfort becomes even more serious when individuals involved with Six xx Preface xxi Sigma training are expected to pass a written and/or computer-based exami- nation that so commonly accompanies this type of training. So if you are beginning to learn about Six Sigma and are either prepar- ing for training or are supporting a Six Sigma team, the question is: How do I get up to speed with applied statistics as quickly as possible so I can get the most from training or add the most value to my Six Sigma team? The answer to this question is simple and straightforward—get access to a book that pro- vides a thorough and systematic discussion of applied statistics, a book that uses the plain language of application rather than abstract theory, and a book that emphasizes learning by examples. Applied Statistics for the Six Sigma Green Belt has been designed to be just that book. This book was organized so as to expose readers to applied statistics in a thorough and systematic manner. We begin by discussing concepts that are the easiest to understand and that will provide you with a solid foundation upon which to build further knowledge. As we proceed with our discussion, and as the complexity of the statistical tools increases, we fully intend that our readers will be able to follow the discussion by understanding that the use of any given statistical tool, in many cases, enables us to use additional and more powerful statistical tools. The order of presentation of these tools in our discussion then will help you understand how these tools relate to, mutually support, and interact with one another. We will continue this logic of the order in which we present topics in the remaining books in this series. Getting the most benefit from this book, and in fact from the complete series of books, is consistent with how many of us learn most effectively—start at the beginning with less complex topics, proceed with our discussion of new and more powerful statistical tools once we learn the “basics,” be sure to cover all the statistical tools needed to support Six Sigma, and emphasize examples and applications throughout the discussion. So let us take a look together at Applied Statistics for the Six Sigma Green Belt. What you will learn is that statistics aren’t mysterious, they aren’t scary, and they aren’t overly difficult to understand. As in learning any topic, once you learn the “basics” it is easy to build on that knowledge—try- ing to start without a knowledge of the basics, however, is generally the beginning of a difficult situation! Contents List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Chapter 1 Setting the Context for Six Sigma . . . . . . . . . . . . . . . . 1 1.1 Six Sigma Defined as a Statistical Concept . . . . . . . . . . . . . . . 1 1.2 Now, Six Sigma Explained as a Statistical Concept . . . . . . . . . 2 1.3 Six Sigma as a Comprehensive Approach and Methodology for Problem Solving and Process Improvement . . . . . . . . . . . . 3 1.4 Understanding the Role of the Six Sigma Green Belt as Part of the Bigger Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Converting Data into Useful Information . . . . . . . . . . . . . . . . . 6 Chapter 2 Getting Started with Statistics. . . . . . . . . . . . . . . . . . . 9 2.1 What Is Statistics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Populations and Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Classification of Various Types of Data . . . . . . . . . . . . . . . . . . 11 2.3.1 Nominal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Ordinal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.3 Interval Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.4 Ratio Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 3 Describing Data Graphically. . . . . . . . . . . . . . . . . . . . 15 3.1 Frequency Distribution Table . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.1 Qualitative Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.2 Quantitative Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Graphical Representation of a Data Set . . . . . . . . . . . . . . . . . . 20 3.2.1 Dot Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.2 Pie Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.3 Bar Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.4 Histograms and Related Graphs . . . . . . . . . . . . . . . . . . . 27 vii viii Contents 3.2.5 Line Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.6 Stem and Leaf Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.7 Measure of Association . . . . . . . . . . . . . . . . . . . . . . . . . 39 Chapter 4 Describing Data Numerically. . . . . . . . . . . . . . . . . . . . 45 4.1 Numerical Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Measures of Centrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2.1 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2.2 Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.3 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3.1 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3.2 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3.3 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.4 Coefficient of Variation . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4 Measures of Central Tendency and Dispersion for Grouped Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4.1 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4.2 Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4.3 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4.4 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 Empirical Rule (Normal Distribution) . . . . . . . . . . . . . . . . . . . 60 4.6 Certain Other Measures of Location and Dispersion . . . . . . . . . 63 4.6.1 Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.6.2 Quartiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.6.3 Interquartile Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.7 Box Whisker Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.7.1 Construction of a Box Plot . . . . . . . . . . . . . . . . . . . . . . 66 4.7.2 How to Use the Box Plot . . . . . . . . . . . . . . . . . . . . . . . . 67 Chapter 5 Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1 Probability and Applied Statistics . . . . . . . . . . . . . . . . . . . . . . 71 5.2 The Random Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.3 Sample Space, Simple Events, and Events of Random Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.4 Representation of Sample Space and Events Using Diagrams . . 75 5.4.1 Tree Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.4.2 Permutation and Combination . . . . . . . . . . . . . . . . . . . . 77 5.5 Defining Probability Using Relative Frequency . . . . . . . . . . . . 83 5.6 Axioms of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.7 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Chapter 6 Discrete Random Variables and Their Probability Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.1 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Mean and Standard Deviation of a Discrete Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2.1 Interpretation of the Mean and the Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 Contents ix 6.3 The Bernoulli Trials and the Binomial Distribution . . . . . . . . .101 6.3.1 Mean and Standard Deviation of a Bernoulli Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102 6.3.2 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . .102 6.3.3 Binomial Probability Tables . . . . . . . . . . . . . . . . . . . . . .105 6.4 The Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . .107 6.4.1 Mean and Standard Deviation of a Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110 6.5 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110 Chapter 7 Continuous Random Variables and Their Probability Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.1 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . .115 7.2 The Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .118 7.2.1 Mean and Standard Deviation of the Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120 7.3 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121 7.3.1 Standard Normal Distribution Table . . . . . . . . . . . . . . . .123 7.4 The Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . .129 7.4.1 Mean and Standard Deviation of an Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130 7.4.2 Distribution Function F(x) of the Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130 7.5 The Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132 7.5.1 Mean and Variance of the Weibull Distribution . . . . . . . .133 7.5.2 Distribution Function F(t) of Weibull . . . . . . . . . . . . . . .133 Chapter 8 Sampling Distributions . . . . . . . . . . . . . . . . . . . . . . . . 137 8.1 Sampling Distribution of Sample Mean . . . . . . . . . . . . . . . . . .138 8.2 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .141 8.2.1 Sampling Distribution of Sample Proportion . . . . . . . . . .147 8.3 Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148 8.4 The Student’s t-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . .153 8.5 Snedecor’s F-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 8.6 The Poisson Approximation to the Binomial Distribution . . . . .158 8.7 The Normal Approximation to the Binomial Distribution . . . . .159 Chapter 9 Point and Interval Estimation. . . . . . . . . . . . . . . . . . . 165 9.1 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166 9.1.1 Properties of Point Estimators . . . . . . . . . . . . . . . . . . . .167 9.2 Interval Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171 9.2.1 Interpretation of a Confidence Interval . . . . . . . . . . . . . .172 9.3 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .172 9.3.1 Confidence Interval for Population Mean (cid:1)When the Sample Size Is Large . . . . . . . . . . . . . . . . . . . . . . . .173 9.3.2 Confidence Interval for Population Mean (cid:1)When the Sample Size Is Small . . . . . . . . . . . . . . . . . . . . . . . .177 9.4 Confidence Interval for the Difference between Two Population Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .180