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Six Sigma Statistics with Excel and Minitab Issa Bass New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except...

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Mc Graw.Hill.Professional.Lean.Six.Sigma.Using.Sigma XL.and.Minit ab.Jan.2009.e Book-DDU Mohammed Alami Six Sigma Case St udies wit h Minit ab Raja Kumar Chapt er 2 Exercise Solut ions crist ian palacio

Six Sigma Statistics with Excel and Minitab Issa Bass

New York

Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-154268-X The material in this eBook also appears in the print version of this title: 0-07-148969-X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at [email protected] or (212) 904-4069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. DOI: 10.1036/007148969X

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iii

I dedicate this book to My brother, Demba who left me too soon Rest in peace President Leopol Sedar Senghor and Professor Cheikh Anta Diop For showing me the way Monsieur Cisse, my very first elementary school teacher. I will never forget how you used to hold my little five year old fingers to teach me how to write. Thank you! You are my hero.

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ABOUT THE AUTHOR ISSA BASS is a Six Sigma Master Black Belt and Six Sigma project leader for the Kenco Group, Inc. He is the founding editor of SixSigmaFirst.com, a portal where he promotes Six Sigma methodology, providing the public with accessible, easy-to-understand tutorials on quality management, statistics, and Six Sigma.

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Contents Preface ix Acknowledgments

x

Chapter 1. Introduction 1.1

Six Sigma Methodology 1.1.1 Define the organization 1.1.2 Measure the organization 1.1.3 Analyze the organization 1.1.4 Improve the organization 1.2 Statistics, Quality Control, and Six Sigma 1.2.1 Poor quality defined as a deviation from engineered standards 1.2.2 Sampling and quality control 1.3 Statistical Definition of Six Sigma 1.3.1 Variability: the source of defects 1.3.2 Evaluation of the process performance 1.3.3 Normal distribution and process capability

Chapter 2. An Overview of Minitab and Microsoft Excel 2.1

Starting with Minitab 2.1.1 Minitab’s menus 2.2 An Overview of Data Analysis with Excel 2.2.1 Graphical display of data 2.2.2 Data Analysis add-in

Chapter 3. Basic Tools for Data Collection, Organization and Description The Measures of Central Tendency Give a First Perception of Your Data 3.1.1 Arithmetic mean 3.1.2 Geometric mean 3.1.3 Mode 3.1.4 Median 3.2 Measures of Dispersion 3.2.1 Range 3.2.2 Mean deviation 3.2.3 Variance 3.2.4 Standard deviation 3.2.5 Chebycheff’s theorem 3.2.6 Coefficient of variation 3.3 The Measures of Association Quantify the Level of Relatedness between Factors 3.3.1 Covariance 3.3.2 Correlation coefficient 3.3.3 Coefficient of determination 3.4 Graphical Representation of Data 3.4.1 Histograms

1 2 2 6 11 13 14 15 16 16 17 18 19

23 23 25 33 35 37

41

3.1

42 42 47 49 49 49 50 50 52 54 55 55 56 56 58 62 62 62

v

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Contents

3.4.2 Stem-and-leaf graphs 3.4.3 Box plots 3.5 Descriptive Statistics—Minitab and Excel Summaries

Chapter 4. Introduction to Basic Probability 4.1

Discrete Probability Distributions 4.1.1 Binomial distribution 4.1.2 Poisson distribution 4.1.3 Poisson distribution, rolled throughput yield, and DPMO 4.1.4 Geometric distribution 4.1.5 Hypergeometric distribution 4.2 Continuous Distributions 4.2.1 Exponential distribution 4.2.2 Normal distribution 4.2.3 The log-normal distribution

Chapter 5. How to Determine, Analyze, and Interpret Your Samples 5.1

5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

How to Collect a Sample 5.1.1 Stratified sampling 5.1.2 Cluster sampling 5.1.3 Systematic sampling Sampling Distribution of Means Sampling Error Central Limit Theorem Sampling from a Finite Population Sampling Distribution of p Estimating the Population Mean with Large Sample Sizes Estimating the Population Mean with Small Sample Sizes and σ Unknown: t-Distribution Chi Square (χ2 ) Distribution Estimating Sample Sizes 5.10.1 Sample size when estimating the mean 5.10.2 Sample size when estimating the population proportion

Chapter 6. Hypothesis Testing 6.1

6.2.

6.3 6.4 6.5

How to Conduct a Hypothesis Testing 6.1.1 Null hypothesis 6.1.2 Alternate hypothesis 6.1.3 Test statistic 6.1.4 Level of significance or level of risk 6.1.5 Decision rule determination 6.1.6 Decision making Testing for a Population Mean 6.2.1 Large sample with known σ 6.2.2 What is the p-value and how is it interpreted? 6.2.3 Small samples with unknown σ Hypothesis Testing about Proportions Hypothesis Testing about the Variance Statistical Inference about Two Populations 6.5.1 Inference about the difference between two means 6.5.2 Small independent samples with equal variances

64 66 68

73 74 74 79 80 84 85 88 88 90 97

99 100 100 100 100 100 101 102 106 106 108 113 114 117 117 118

121 122 122 122 123 123 123 124 124 124 126 128 130 131 132 133 134

Contents

6.6

6.5.3 Testing the hypothesis about two variances Testing for Normality of Data

Chapter 7. Statistical Process Control 7.1 7.2 7.3

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140 142

145

How to Build a Control Chart The Western Electric (WECO) Rules Types of Control Charts 7.3.1 Attribute control charts 7.3.2 Variable control charts

147 150 151 151 159

Chapter 8. Process Capability Analysis

171

8.1

8.2 8.3 8.4 8.5

Process Capability with Normal Data 8.1.1 Potential capabilities vs. actual capabilities 8.1.2 Actual process capability indices Taguchi’s Capability Indices CPM and PPM Process Capability and PPM Capability Sixpack for Normally Distributed Data Process Capability Analysis with Non-Normal Data 8.5.1 Normality assumption and Box-Cox transformation 8.5.2 Process capability using Box-Cox transformation 8.5.3 Process capability using a non-normal distribution

Chapter 9. Analysis of Variance 9.1 9.2

ANOVA and Hypothesis Testing Completely Randomized Experimental Design (One-Way ANOVA) 9.2.1 Degrees of freedom 9.2.2 Multiple comparison tests 9.3 Randomized Block Design 9.4 Analysis of Means (ANOM)

Chapter 10. Regression Analysis 10.1 Building a Model with Only Two Variables: Simple Linear Regression 10.1.1 Plotting the combination of x and y to visualize the relationship: scatter plot 10.1.2 The regression equation 10.1.3 Least squares method 10.1.4 How far are the results of our analysis from the true values: residual analysis 10.1.5 Standard error of estimate 10.1.6 How strong is the relationship between x and y : correlation coefficient 10.1.7 Coefficient of determination, or what proportion in the variation of y is explained by the changes in x 10.1.8 Testing the validity of the regression line: hypothesis testing for the slope of the regression model 10.1.9 Using the confidence interval to estimate the mean 10.1.10 Fitted line plot 10.2 Building a Model with More than Two Variables: Multiple Regression Analysis 10.2.1 Hypothesis testing for the coefficients 10.2.2 Stepwise regression

174 176 178 183 185 193 194 195 196 200

203 203 204 206 218 222 226

231 232 233 240 241 248 250 250 255 255 257 258 261 263 268

viii

Contents

Chapter 11. Design of Experiment 11.1 The Factorial Design with Two Factors 11.1.1 How does ANOVA determine if the null hypothesis should be rejected or not? 11.1.2 A mathematical approach 11.2 Factorial Design with More than Two Factors (2k )

Chapter 12. The Taguchi Method 12.1 Assessing the Cost of Quality 12.1.1 Cost of conformance 12.1.2 Cost of nonconformance 12.2 Taguchi’s Loss Function 12.3 Variability Reduction 12.3.1 Concept design 12.3.2 Parameter design 12.3.3 Tolerance design

Chapter 13. Measurement Systems Analysis–MSA: Is Your Measurement Process Lying to You? 13.1 Variation Due to Precision: Assessing the Spread of the Measurement 13.1.1 Gage repeatability & reproducibility crossed 13.1.2 Gage R&R nested 13.2 Gage Run Chart 13.3 Variations Due to Accuracy 13.3.1 Gage bias 13.3.2 Gage linearity

Chapter 14. Nonparametric Statistics 14.1 The Mann-Whitney U test 14.1.1 The Mann-Whitney U test for small samples 14.1.2 The Mann-Whitney U test for large samples 14.2 The Chi-Square Tests 14.2.1 The chi-square goodness-of-fit test 14.2.2 Contingency analysis: chi-square test of independence

Chapter 15. Pinpointing the Vital Few Root Causes 15.1 Pareto Analysis 15.2 Cause and Effect Analysis Appendix 1 Appendix 2 Appendix 3 Appendix 4 Appendix 5 Appendix 6 Index

369

Binominal Table P(x) =nCx px qn−x Poisson Table P(x) = ␭x e−␭/x! Normal Z Table Student’s t Table Chi-Square Table F Table ␣ = 0.05

275 276 277 279 285

289 289 290 290 293 295 297 298 300

303 304 305 314 318 320 320 322

329 330 330 333 336 336 342

347 347 350 354 357 364 365 366 367

Preface The role of statistics in quality management in general and Six Sigma in particular has never been so great. Quality control cannot be dissociated from statistics and Six Sigma finds its definition in that science. In June 2005, we decided to create sixsigmafirst.com, a website aimed at contributing to the dissemination of the Six Sigma methodology. The site was primarily focusing on tutorials about Six Sigma. Since statistical analysis is the fulcrum of that methodology, a great deal of the site was slated to enhance the understanding of the science of Statistics. The site has put us in contact with a variety of audiences that range from students who need help with their homework to quality control managers who seek to better understand how to apply some statistics tools to their daily operations. Some of the questions that we receive are theoretical while others are just about how to use some statistics software to conduct an analysis or how to interpret the results of a statistical testing. The many questions that we have been getting have brought about the idea of writing a comprehensive book that covers both statistical theory and helps to better understand how to utilize the most widely used software in statistics. Minitab and Excel are currently the most preponderant software tools for statistical analysis; they are easy to use and provide reliable results. Excel is very accessible because it is found on almost any Windowsbased operating system and Minitab is widely used in corporations and universities. But we believe that without a thorough understanding of the theory behind the analyses that these tools provide, any interpretation made of results obtained from their use would be misleading. That is why we have elected to not only use hundred of examples in this book, with each example, each study case being analyzed from a theoretical standpoint, using algebraic demonstrations, but we also graphically show step by step how to use Minitab and Excel to come to the same conclusions we obtained from our mathematical reasoning. This comprehensive approach does help better understand how the results are obtained and best of all, it does help make a better interpretation of the results. We hope that this book will be a good tool for a better understanding of statistics theory through the use of Minitab and Excel.

ix

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Acknowledgments I would like to thank all those without whom, I would not have written this book. I would especially thank my two sisters Oumou and Amy. Thank you for your support. I am also grateful to John Rogers, my good friend and former Operations’ Director at the Cingular Wireless Memphis Distribution Centre in Memphis, Tennessee. My thanks also go to my good friend Jarrett Atkinson from the Kenco group, at KM Logistics in Memphis. Thank you all for your constant support.

x

Copyright © 2007 by The McGraw-Hill Companies, Inc. Click here for terms of use.

Chapter

1 Introduction

Learning Objectives:


Clearly understand the definition of Six Sigma




Understand the Six Sigma methodology




Understand the Six Sigma project selection methods




Understand balanced scorecards




Understand how metrics are selected and integrated in scorecards




Understand how metrics are managed and aligned with the organization’s strategy




Understand the role of statistics in quality control and Six Sigma




Understand the statistical definition of Six Sigma

A good business performance over a long period of time is never the product of sheer happenstance. It is always the result of a well-crafted and well-implemented strategy. A strategy is a time-bound plan of structured actions aimed at attaining predetermined objectives. Not only should the strategy be clearly geared toward the objectives to be attained, but it should also include the identification of the resources needed and the definition of the processes used to reach the objectives. Over the last decades, several methodologies have been used to improve on quality and productivity and enhance customer satisfaction. Among the methodologies used for these purposes, Six Sigma has so far proved to be one of the most effective.

1

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2

Chapter One

1.1 Six Sigma Methodology Six Sigma is a meticulous, data-driven methodology that aims at generating quasi-perfect production processes that would result in no more than 3.4 defects per 1 million opportunities. By definition, Six Sigma is rooted in statistical analysis because it is data-driven and is a strict approach that drives process improvements through statistical measurements and analyses. The Six Sigma approach to process improvements is project driven. In other words, areas that show opportunities for improvements are identified and projects are selected to proceed with the necessary improvements. The project executions follow a rigorous pattern called the DMAIC (Define, Measure, Analyze, Improve, and Control). At every step in the DMAIC roadmap, specific tools are used, and most of these tools are statistical. Even though Six Sigma is a project-driven strategy, the initiation of a Six Sigma deployment does not start with project selections. It starts with the overall understanding of the organization in terms of how it defines itself, in terms of what its objectives are, how it measures itself, what performance metrics are crucial for it to reach its objectives and how those metrics are analyzed. 1.1.1 Define the organization

Defining an organization means putting it precisely in its context; it means defining it in terms of its objectives, in terms of its internal operations and in terms of its relations with its customers and suppliers. Mission statement. Most companies’ operational strategies are based

on their mission statements. A mission statement (sometimes called strategic intent) is a short inspirational statement that defines the purpose of the organization and its core values and beliefs. It tells why the organization was created and what it intends to achieve in the future. Mission statements are in general very broad in perspective and not very precise in scope. They are mirrors as well as rudders: they are mirrors because they reflect what the organization is about, and they are rudders because they point the direction that the organization should be heading. Even though they do not help navigate through obstacles and certainly do not fix precise quarterly or annual objectives, such as the projected increase of Return On Investment (ROI) by a certain percentage for a coming quarter, mission statements should clearly define the company’s objective so that management can align its strategy with that objective.

Introduction

3

What questions should an organization ask? Every organization’s exis-

tence depends on the profits derived from the sales of the goods or services to its customers. So to fulfill its objectives, an organization must elect to produce goods or services for which it has a competitive advantage and it must produce them at the lowest cost p...