PRACTITIONER'S GUIDEFOR STATISTICS ANDLEAN SIX SIGMA FORPROCESS IMPROVEMENTS

Mikel J. HarrySix Sigma Management InstituteScottsdale, AZ

Prem S. MannDepartment of EconomicsEastern Connecticut State UniversityWillimantic, CT

Ofelia de HodginsInternational Institute for Learning, Inc.New York, NY

Chris LackeMathematics DepartmentRowan UniversityGlassboro, NJ

Richard HulbertBank of New York MellonNew York, NY

WILEYA JOHN WILEY & SONS, INC., PUBLICATION

CONTENTS

Preface xxi

1 Principles of Six Sigma 1

1.1 Overview 11.2 Six Sigma Essentials 1

1.2.1 Driving Need 21.2.2 Customer Focus 21.2.3 Core Beliefs 31.2.4 Deterministic Reasoning 41.2.5 Leverage Principle 5

1.3 Quality Definition 51.4 Value Creation 7

1.4.1 Value 71.5 Business, Operations, Process, and Individual (BOPI) Goals 8

1.5.1 Differences between Product and Process Capabilityfrom a Six Sigma Perspective 9

1.6 Underpinning Economics 91.6.1 Sigma Benchmarking 101.6.2 Breakthrough Goals 111.6.3 Performance Benchmark 11

1.7 Performance Metrics 111.8 Process 12

1.8.1 Process Models 121.9 Design Complexity 121.10 Nature and Purpose of Six Sigma 13

1.10.1 Not Just Defect Reduction 131.11 Needs that Underlie Six Sigma 13

1.11.1 Looking Across the Organization 141.11.2 Processing for Six Sigma 151.11.3 Designing for Six Sigma 151.11.4 Managing for Six Sigma 151.11.5 Risk Orientation 15

1.12 Why Focusing on the Customer is Essential to Six Sigma 161.13 Success Factors 181.14 Software Applications 19

1.14.1 Explore Excel 19

vi CONTENTS

1.14.2 Explore MINITAB 191.14.3 Explore JMP 19

Glossary 20References 21

2 Six Sigma Installation 22

2.1 Overview 222.2 Six Sigma Leadership—the Fuel of Six Sigma 232.3 Deployment Planning 26

2.3.1 Executive Management 262.3.2 Six Sigma Champion 272.3.3 Line Management 282.3.4 Master Black Belts 282.3.5 Black Belts 292.3.6 Green Belts 292.3.7 White Belts 302.3.8 Six Sigma Roadmap 312.3.9 Characteristics of Effective Metrics 322.3.10 The Role of Metrics 322.3.11 Six Sigma Performance Metrics 322.3.12 Profit and Measurement 332.3.13 Twelve Criteria for Performance Metrics 33

2.4 Application Projects 342.5 Deployment Timeline 352.6 Design for Six Sigma [DFSS] Principles 362.7 Processing for Six Sigma [PFSS] Principles 372.8 Managing for Six Sigma [MFSS] Principles 372.9 Project Review 37

2.9.1 Tollgate Criteria 382.9.2 Project Closure 382.9.3 Project Documentation 382.9.4 Personal Recognition 382.9.5 Authenticating Agent 38

2.10 Summary 38Glossary 39References and Notes 40

3 Lean Sigma Projects 41

3.1 Overview 413.2 Introduction 413.3 Project Description 423.4 Project Guidelines 433.5 Project Selection 44

3.5.1 Project Selection Guidelines 443.6 Project Scope 463.7 Project Leadership 483.8 Project Teams 48

CONTENTS vii

3.9 Project Financials 483.10 Project Management 493.11 Project Payback 503.12 Project Milestones 513.13 Project Roadmap 523.14 Project Charters (General) 523.15 Six Sigma Projects 573.16 Project Summary 63Glossary 64References 66

4 Lean Practices 67

4.1 Overview 674.2 Introduction 674.3 The Idea of Lean Thinking 684.4 Theory of Constraints [TOC] 704.5 Lean Concept 714.6 Value-Added Versus Non-Value-Added Activities 714.7 Why Companies Think Lean 714.8 Visual Controls—Visual Factory 724.9 The Idea of Pull (Kanban) • 724.10 5S-6S Approach 734.11 The Idea of Perfection (Kaizen) 764.12 Replication—Translate 764.13 Poka-Yoke System—Mistakeproofing 774.14 SMED System 784.15 7W + 1 Approach—Seven Plus One Deadly Waste(s) 794.16 6M Approach 824.17 Summary 83Glossary 83References and Notes 85

5 Value Stream Mapping 87

5.1 Overview 875.2 Introduction 875.3 Value Stream Mapping . 88

5.3.1 Waste Review 895.3.2 Value-Added and Non-Value-Added Activities 905.3.3 Elements of a Value Stream Map 92

5.4 Focused Brainstorming 945.5 Graphical Representation of a Process in a Value

Stream Map 965.6 Effective Working Time 995.7 Customer Demand 995.8 Takt Time 99

viii CONTENTS

5.9 Pitch Time 1005.10 Queuing Time 1015.11 Cycle Time 1015.12 Total Cycle Time 1015.13 Calculation of Total Lead Time(s) 1015.14 Value-Added Percentage and Six Sigma Level 1055.15 Drawing the Current-Value-Stream Map 105

5.15.1 Drawing Tips 1055.15.2 Common Failure Modes 1065.15.3 Common Definitions 106

5.16 Drawing the Value Stream Map 1065.17 What Makes a Value Stream Lean 1135.18 The Future Value Stream Map 1135.19 Summary 114Glossary 114References and Notes 116

6 Introductory Statistics and Data 118

118118119119121122126128128129129130132132133133135135136136138

Quality Tools 139

7.1 Overview 1397.2 Introduction 1397.3 Nature of Six Sigma Variables 140: 7.3.1 CT Concept 142

7.3.2 CTQ and CTP Characteristics 143

6.16.26.36.46.56.6

6.76.8

6.9

6.106.116.12

OverviewIntroductionGenetic Code of StatisticsPopulations and SamplesThe Idea of DataNature of Data6.6.1 Quantitative Variables and Data6.6.2 Qualitative/Categorical Variables and DataData CollectionThe Importance of Data Collection6.8.1 Control Cards6.8.2 Data Collection SheetSampling in Six Sigma6.9.1 Random Sampling6.9.2 Sequential Sampling6.9.3 Stratified SamplingSources of DataDatabaseSummary

GlossaryReferences

CONTENTS

7.3.3 CTX Tree (Process Tree)7.3.4 CTY Tree (Product Tree)7.3.5 The Focus of Six Sigma7.3.6 The Leverage Principle

7.4 Quality Function Deployment (QFD)7.5 Scales of Measurement

7.5.1 Likert Scale7.5.2 Logarithm Scale

7.6 Diagnostic Tools7.6.1 Elements for Problem Solving—Diagnostic Tools and Methods7.6.2 Problem Definition—Defining Project Objective

7.7 Analytical Methods7.7.1 Cause-Effect (CE) Analysis

7.7.1.1 How to Construct the CE Diagram7.7.1.2 How to Construct the CE Diagram Using MINITAB

7.7.2 Failure Mode-Effects Analysis (FMEA)7.7.2.1 FMEA Diagram

7.7.3 XY Matrix7.8 Graphical Tools

7.8.1 Graphical Summary7.8.2 Boxplot or Box-and-Whisker Plot7.8.3 Normal Probability Plot7.8.4 Main-Effects Plot7.8.5 Pareto Chart7.8.6 Run Chart7.8.7 Time-Series Plot7.8.8 Multi-Vari Charts7.8.9 Scatterplot

7.9 Graphical Representation of a Process7.9.1 Process Flowcharts7.9.2 Process Mapping7.9.3 Cross-Functional Mapping7.9.4 Process Mapping—Deployment Diagram

7.10 SIPOC Diagram7.11 IPO Diagram—General Model of a Process System7.12 Force-Field Analysis7.13 Matrix Analysis—The Importance of Statistical Thinking7.14 Checksheets7.15 Scorecards7.16 Affinity Diagram7.17 Concept IntegrationGlossaryReferences

IX

144145146146148149149150152152153154154154156157158161161163164166169171173174175176179180182184184186187188190190191191192192194

8 Making Sense of Data in Six Sigma and Lean 195

8.1 Overview 1958.2 Summarizing Quantitative Data: Graphical Methods 195

CONTENTS

8.2.1 Analytical Charts 1958.2.2 Dotplots 1968.2.3 Stem-and-Leaf Plots 1988.2.4 Frequency Tables 1998.2.5 Histograms and Performance Histograms 2018.2.6 Run Charts 2028.2.7 Time-Series Plots 203

8.3 Summarizing Quantitative Data: Numerical Methods 2048.3.1 Measures of Center 2048.3.2 Measures of Variation 207

8.3.2.1 Range 2078.3.2.2 Variance and Standard Deviation 2088.3.2.3 Coefficient of Variation (CV) 2108.3.2.4 The Interquartile Range 2108.3.2.5 Boxplots 211

8.3.3 Identifying Potential Outliers 2128.3.4 Measures of Position and the Idea of z Scores in Six Sigma 214

8.3.4.1 Percentiles 2148.3.4.2 The Use of z Scores 215

8.3.5 Measures of Spread and Lean Sigma 2158.4 Organizing and Graphing Qualitative Data 217

8.4.1 Organizing Qualitative Data 2178.4.2 Graphing Qualitative Data 219

• 8.4.2.1 Pie Chart 2198.4.2.2 Bar Graph 219

8.4.3 Pareto Analysis with Lorenz Curve 2208.5 Summarizing Bivariate Data 222

8.5.1 Scatterplot 2228.5.2 Correlation Coefficient 224

8.5.2.1 Pearson Correlation Coefficient 2258.5.2.2 Spearman's Rho (p) 2268.5.2.3 Kendall's Tau (x) Rank Correlation 228

8.6 Multi-Vari Charts 229Glossary 230Exercises 232

9 Fundamentals of Capability and Rolled Throughput Yield 237

9.1 Overview 2379.2 Introduction 2379.3 Why Capability 238

9.3.1 Performance Specifications 2399.3.2 Fundamental Concepts of Defect-Based Measurement 240

9.4 Six Sigma Capability Metric 2419.4.1 Criteria for Performance Metrics 241

CONTENTS xi

9.4.2 Computing the Sigma Level from Discrete Data 2429.4.3 Defective Proportions 2449.4.4 Six-Sigma-Level Calculations (DPU, DPO, DPMO,

PPM)—Examples 2449.5 Discrete Capability 2469.6 Continuous Capability—Example 247

9.6.1 Data Collection for Capability Studies 2499.7 Fundamentals of Capability 2499.8 Short- Versus Long-Term Capability 251

9.8.1 Short-Term Capability 2519.8.2 Long-Term Capability 2529.8.3 Introduction to Calibrating the Shift 253

9.9 Capability and Performance 2539.10 Indices of Capability 256

9.10.1 Cp Index 2569.10.2 Cpk Index 2589.10.3 Pp Index 2609.10.4 Ppk Index 261

9.11 Calibrating the Shift 2629.12 Applying the 1.5a shift Concept 2649.13 Yield 265

9.13.1 Final Test Yield (FTY) 2679.13.2 Yield Related to Defects 2679.13.3 Rolled Throughput Yield (RTY) 2699.13.4 In-Process Yield (IPY) 2699.13.5 In-Process Yield (IPY) and Rolled Throughput Yield (RTY) 270

9.14 Hidden Factory 273

9.14.1 Hidden Factory Composition 275Glossary 276References 278

279

279279280281281283283284285285286287288

10 Probability

10.110.210.3

10.410.510.6

10.7

OverviewExperiments, Outcomes, and Sample SpaceCalculating Probability10.3.1 Equally Likely Events10.3.2 Probability as Relative Frequency10.3.3 Subjective ProbabilityCombinatorial ProbabilityMarginal and Conditional ProbabilitiesUnion of Events10.6.1 Addition Rule10.6.2 Mutually Exclusive Events10.6.3 Complementary EventsIntersection of Events

xii CONTENTS

10.7.1 Independent Versus Dependent Events 28810.7.2 Multiplication Rule 289

Glossary 292Exercises 292

11 Discrete Random Variables and their Probability Distributions 296

11.1 Overview 29611.2 Six Sigma Performance Variables 29611.3 Six Sigma Leverage Variables 29811.4 Random Variables 299

11.4.1 Discrete Random Variables 30011.4.2 Continuous Random Variables 300

11.5 Probability Distributions of a Discrete Random Variable 30111.6 Mean of a Discrete Random Variable 30211.7 Standard Deviation of a Discrete Random Variable 30411.8 The Binomial Distribution 306

11.8.1 Factorials and Combinations 30611.8.2 The Binomial Experiment 30811.8.3 The Binomial Probability Distribution And Binomial

Formula 30911.8.4 Probability of Success and Shape of the Binomial

Distribution 31311.8.5 Mean and Standard Deviation of the Binomial Distribution 314

11.9 The Poisson Probability Distribution 31411.9.1 Mean and Standard Deviation of the Poisson Probability

Distribution 31711.10 The Geometric Distribution 31711.11 The Hypergeometric Probability Distribution 318Glossary 320Exercises 320

12 Continuous Random Variables and Their Probability Distributions 324

12.1 Overview 32412.2 Continuous Probability Distriutions 32412.3 The Normal Distribution 326

12.3.1 The Empirical Rule 32812.3.2 The Standard Normal Distribution . 329

12.3.2.1 Using a Normal Probability Table to CalculateNormal Distribution Probabilities 330

12.3.2.2 Determining Percentiles of the Standard NormalDistribution 333

12.3.3 Applications of the Normal Distribution 33612.4 The Exponential Distribution 340Glossary 345Exercises 346

CONTENTS xiii

13 Sampling Distributions 349

13.1 Overview 34913.2 Sampling Distribution of a Sample Mean 349

13.2.1 Sampling and Nonsampling Errors 35113.3 Sampling Distribution of a Sample Proportion 35413.4 The Central-Limit Theorem (CLT) 356

13.4.1 The CLT and Sampling Distribution of the Sample Mean 35613.4.2 The CLT and Sampling Distribution

of the Sample Proportion 359Glossary 362Exercises 362

14 Single-Population Estimation 364

14.1 Overview 36414.2 Meaning of a Confidence Level 36514.3 Estimating a Population Mean 366

14.3.1 Confidence Interval for a Population Mean Usingthe Normal Distribution 36614.3.1.1 Underlying Conditions for Using the z Interval

for a Population Mean 36814.3.2 Confidence Interval for a Population Mean Using

the t Distribution 36914.3.2.1 Underlying Conditions for Using the / Procedure 37014.3.2.2 Using the t Procedure When the Normality

Assumption is Questionable 37214.4 Estimating a Population Proportion 374

14.4.1 Traditional Large-Sample Method 37414.4.2 Wilson Estimator 375

14.5 Estimating a Population Variance 377Glossary 380Exercises 380

15 Control Methods 384

15.1 Overview 38415.2 Introduction 38415.3 Control Logic 38515.4 Statistical Control Systems 386

15.4.1 Mistakeproofing 38615.5 Statistical Control 39415.6 Prevention Versus Detection 39515.7 A Process Control System Definition 39515.8 Variation 396

15.8.1 Common Causes 39615.8.2 Special Causes 396

CONTENTS

15.9 Process Out of Control 39715.10 Fundamentals of Process Control 40015.11 Continuous Statistical Process Control (SPC) Tools 40115.12 Interpreting Process Control 40215.13 Statistical Process Control and Statistical Process Monitoring 40315.14 The Foundation of SPC 40415.15 Tools for Process Controls - Control Charts 40515.16 Control Limits 40615.17 Process Out-of-Control Condition 40715.18 Western Electric Rules 40715.19 Control Charts and How They are Used 41115.20 Precontrol Method 413

15.20.1 The Foundations of Precontrol 41415.20.2 Precontrol Charts 416

15.21 Control Charts for Variables 41815.21.1 X Chart 41815.21.2 R Chart (Range Chart) 42015.21.3 X-R Chart 42115.21.4 Moving Range (MR) Chart 42215.21.5 Standard Deviation Chart 423

15.22 Control Chart for Attributes 42415.22.1 p Chart 42415.22.2 Control Chart—np Chart 42415.22.3 c Chart • 42615.22.4 u Chart 427

Glossary 429References and Notes 432

16 Single-Population Hypothesis Tests 433

16.1 Overview 43316.2 Introduction to Hypothesis Testing 43316.3 Testing a Claim About a Population Mean 442

16.3.1 Hypothesis Test Using the Normal Distribution 44216.3.2 Hypothesis Test Using the t Distribution 44416.3.3 Hypothesis Test About the Median 448

16.4 Hypothesis Test About a Population Proportion 450Glossary 453Exercises . 454

17 Estimation and Hypothesis Tests: Two Populations 457

17.1 Overview 45717.2 Inferences About Differences Between Two Population

Means for Independent Samples 45717.2.1 Two-Sample t Test 45817.2.2 Mann-Whitney Test 466

CONTENTS xv

17.3 Inferences About Differences Between Two PopulationMeans for Paired Samples 47017.3.1 Paired? Test 47017.3.2 Wilcoxon Signed-Rank Test 477

17.4 Inferences About Differences Between Two Population Proportions 47917.4.1 Large-Sample Procedure 480

Glossary 485Exercises 486

18 Chi-Square Tests 489

18.1 Overview 48918.2 A Goodness-of-Fit Test 49018.3 Contingency Tables 49518.4 Tests of Independence and Homogeneity 495

18.4.1 Test of Independence 49518.4.2 Test of Homogeneity 501

Glossary 503Exercises 504

19 Analysis of Variance 507

19.1 Overview 50719.2 The F Distribution 50719.3 One-Way Analysis of Variance 510

19.3.1 Variance between Groups 51119.3.2 Variance within Groups 51219.3.3 Total Sum of Squares (SST) 51219.3.4 Relationships within Sums of Squares and Degrees

of Freedom 51319.3.5 Equal Sample Sizes 51319.3.6 Calculating the Value of the Test Statistic 51319.3.7 The One-Way ANOVA Table 514

19.4 Pairwise Comparisons 51919.5 Multifactor Analysis of Variance 520

19.5.1 Two-Way ANOVA 52019.5.2 N-Way ANOVA 520

19.6 What to Do When the Assumptions Are Unreasonable 521Glossary 522Exercises 522

20 Linear and Multiple Regression 525

20.1 Overview 52520.2 Simple Regression Model 52520.3 Linear Regression • 526

xvi CONTENTS

20.3.1 Simple Linear Regression Analysis 52820.3.2 Scatterplots 529

20.3.2.1 Least Squares Line 53020.3.2.2 Interpretations of a and b 533

20.3.3 Assumptions of the Regression Model 53420.3.4 Standard Deviation of Random Errors 536

20.4 Coefficient of Determination and Correlation 53820.5 Multiple Regression 544

20.5.1 Assumptions of the Multiple Regression Model 54620.5.2 Standard Deviation of Errors 54620.5.3 Coefficient of Multiple Determination 547

20.6 Regression Analysis 54820.6.1 Testing for Overall Significance of Multiple Regression

Model 54820.6.2 Inferences about a Single Regression Coefficient, Bt 549

20.6.2.1 Sampling Distribution of b 54920.6.2.2 Estimation of B for a Simple Linear Regression 55020.6.2.3 Estimation of B, for a Multiple Linear Regression 55020.6.2.4 Hypothesis Testing about B for a Simple Linear

Regression 55020.6.2.5 Hypothesis Testing about Individual Coefficients

for a Multiple Linear Regression 55120.7 Using the Regression Model 55120.8 Residual Analysis 55920.9 Cautions in Using Regression 561

20.9.1 Determining whether a Model is Good or Bad 56120.9.2 Outliers and Influential Observations 56220.9.3 Multicollinearity 56320.9.4 Extrapolation . 56320.9.5 Causality ' 564

Glossary 564Exercises 565

21 Measurement Analysis 570

21.1 Overview 57021.2 Introduction 57021.3 Measurement 57121.4 Measurement Error 57221.5 Accuracy and Precision 57321.6 Measurement System as a Process 57621.7 Categories of Measurement Error that Affect Location 57821.8 Categories of Measurement that Affect Spread 57921.9 Gage Accuracy and Precision 58021.10 Exploring Linearity Error 58121.11 Gage Repeatability and Reproducibility (R&R) 582

21.11.1 Variable Gage R&R 584

CONTENTS xvii

21.11.2 Crossed Gage R&R 58621.11.3 Attribute Gage R&R 590

21.12 ANOVA Method Versus X-R Method 59721.13 ANOVA/Variance Component Analysis 59921.14 Rules of Thumb 60121.15 Acceptability Criteria 60321.16 Chapter Review 604Glossary 605References 609

22 Design of Experiments 610

22.1 Overview 61022.2 Introduction 61022.3 Design of Experiments (DOE) Definition 61122.4 Role of Experimental Design in Process Improvement 61422.5 Experiment Design Tools 61622.6 Principles of an Experimental Design 61722.7 Different Types of Experiments 619

22.7.1.1 Main Effects 62022.8 Introduction to Factorial Designs 62322.9 Features of Factorial Designs—Orthogonality 62522.10 Full Factorial Designs 62722.11 Residual Analysis (22) 62822.12 Modeling (22) 63022.13 Multifactor Experiment 63022.14 Fractional Factorial Designs 63122.15 The ANOVA Table 63622.16 Normal Probability Plot of the Effects 63622.17 Main-Effects Plot 63722.18 Blocking Variable 63722.19 Statistical Significance 63822.20 Practical Significance 63822.21 Fundamentals of Residual Analysis 63922.22 Centerpoints 64022.23 Noise Factors 64122.24 Strategy of Good Experimentation 64122.25 Selecting the Variable Levels 64122.26 Selecting the Experimental Design 64222.27 Replication 64222.28 Analyzing the Data (ANOVA) 64322.29 Recommendations 64322.30 Achieving the Objective 64322.31 Chapter Summary 64422.32 Chapter Examples 645Glossary 652References 658

xviii CONTENTS

23 Design for Six Sigma (DFSS), Simulation, and Optimization 659

23.1 Overview 65923.2 Introduction 65923.3 Six Sigma as Stretch Target 66023.4 Producibility 66223.5 Statistical Tolerances 66323.6 Design Application 66523.7 Design Margin 66723.8 Design Qualification 66923.9 Design for Six Sigma (DFSS) Principles 671

23.9.1 DFSS Leverage in Product Design 67123.9.2 Importance of DFSS for Product Design 672

23.10 Decision Power 67423.11 Experimentation 67623.12 Experiment Design 67723.13 Response Surface Designs 67923.14 Factorial Producibility 68723.15 Toolbox Overview 68823.16 Monte Carlo Simulations 689

23.16.1 Monte Carlo Simulation Defined 68923.16.2 When Simulation is an Appropriate Tool 69023.16.3 Defining Distributions and Outputs in Crystal Ball 691

23.17 Design for Six Sigma Project Selection Example 70323.18 Defining Simulation Inputs 70423.19 Defining Outputs and Running a Simulation 705

23.19.1 Analyzing a Simulation 70623.20 Stochastic Optimization: Discovering the Best Portfolio

with the Least Risk 70823.21 Conclusions 709Glossary 709References 713

24 Survey Methods and Sampling Techniques 715

24.1 Overview 71524.2 Introduction 71524.3 The Sample Survey 71524.4 The Survey System 71624.5 Clear Goals 71824.6 Target Population and Sample Size 71824.7 Interviewing Method 72124.8 Response Rate, Respondents and Nonrespondents 72224.9 Survey Methods 72324.10 Sources of Information and Data 72424.11 Order of the Questions 72424.12 Pilot Testing the Questionnaire 72424.13 Biased Sample or Response Error 72424.14 Sampling—Random and Nonrandom Samples 725

CONTENTS xix

24.15 Population Distribution 72724.16 Sampling Distribution 72824.17 Sampling and Nonsampling Errors 730Glossary 733References 736

Appendix A Statistical Tables 737

Table I Table of Binomial Probabilities 738Table II Standard Normal Distribution Table 750Table III The t Distribution Table 752Table IV Chi-Square Distribution Table 753Table V The F Distribution Table 754Table VI Critical Values for the Mann-Whitney Test 762Table VII Critical Values for the Wilcoxon Signed-Rank Test 763Table VIII Sigma Conversion Table 763

Appendix B Answers to Selected Odd-Numbered Exercises 765

Index 777