# Nonparametric Statistical Process Control

# Nonparametric Statistical Process Control

ISBN: 978-1-118-89056-1

Oct 2018

424 pages

## Description

**A unique approach to understanding the foundations of statistical quality control with a focus on the latest developments in nonparametric control charting methodologies**

Statistical Process Control (SPC) methods have a long and successful history and have revolutionized many facets of industrial production around the world. This book addresses recent developments in statistical process control bringing the modern use of computers and simulations along with theory within the reach of both the researchers and practitioners. The emphasis is on the burgeoning field of nonparametric SPC (NSPC) and the many new methodologies developed by researchers worldwide that are revolutionizing SPC.

Over the last several years research in SPC, particularly on control charts, has seen phenomenal growth. Control charts are no longer confined to manufacturing and are now applied for process control and monitoring in a wide array of applications, from education, to environmental monitoring, to disease mapping, to crime prevention. This book addresses quality control methodology, especially control charts, from a statistician’s viewpoint, striking a careful balance between theory and practice. Although the focus is on the newer nonparametric control charts, the reader is first introduced to the main classes of the parametric control charts and the associated theory, so that the proper foundational background can be laid.

- Reviews basic SPC theory and terminology, the different types of control charts, control chart design, sample size, sampling frequency, control limits, and more
- Focuses on the distribution-free (nonparametric) charts for the cases in which the underlying process distribution is unknown
- Provides guidance on control chart selection, choosing control limits and other quality related matters, along with all relevant formulas and tables
- Uses computer simulations and graphics to illustrate concepts and explore the latest research in SPC

Offering a uniquely balanced presentation of both theory and practice, *Nonparametric Methods for Statistical Quality Control *is a vital resource for students, interested practitioners, researchers, and anyone with an appropriate background in statistics interested in learning about the foundations of SPC and latest developments in NSPC.

Nonparametric Statistical Process Control 1

Contents 3

**Chapter 1: Background/Review of Statistical Concepts 11**

1.0 Chapter overview 11

1.1 Basic probability 11

1.2 Random variables and their distributions 12

1.3 Random sample 20

1.4 Statistical inference 24

1.5 Role of the computer 29

**Chapter 2: Basics of Statistical Process Control 30**

2.0 Chapter Overview 30

2.1 Basic concepts 30

2.1.1 Types of variability 30

2.1.2 The control chart 31

2.1.3 Construction of control charts 36

2.1.4 Variables and attributes control charts 37

2.1.5 Sample size or subgroup size 40

2.1.6 Rational subgrouping 40

2.1.7 Nonparametric or distribution-free 41

2.1.8 Monitoring process location and/or process scale 43

2.1.9 Case K and Case U 44

2.1.10 Control charts and hypothesis testing 44

2.1.11 General steps in designing a control chart 47

2.1.12 Measures of control chart performance 47

2.1.12.1 False alarm probability (FAP) 49

2.1.12.2 False alarm rate (*FAR*) 50

2.1.12.3 The average run-length (*ARL*) 50

2.1.12.4 Standard deviation of run-length (SDRL) 52

2.1.12.5 Percentiles of run-length 52

2.1.12.6 Average number of samples to signal (*ANSS*) 55

2.1.12.7 Average number of observations to signal (*ANOS*) 56

2.1.12.8 Average time to signal (*ATS*) 56

2.1.12.9 Number of individual items inspected (*I*) 57

2.1.13 Operating characteristic curves (*OC*-curves) 57

2.1.14 Design of control charts 59

2.1.15 Sample size, sampling frequency and variable sample sizes 59

2.1.16 Size of a shift 66

2.1.17 Choice of control limits 67

2.1.17.1 *k*-sigma limits 67

2.1.17.2 Probability limits 68

**Chapter 3: Parametric univariate variables control charts 72**

3.0 Chapter overview 72

3.1 Introduction 72

3.2 Parametric variables control charts in Case K 73

3.2.1 Shewhart control charts 73

3.2.2 CUSUM control charts 76

3.2.3 EWMA control charts 81

3.3 Types of parametric variables charts in Case K: Illustrative examples 86

3.3.1 Shewhart control charts 86

3.3.2 CUSUM control charts 93

3.3.3 EWMA control charts 96

3.4 Shewhart, EWMA and CUSUM charts: Which to use when 98

3.5 Control chart enhancements 100

3.5.1 Sensitivity rules 100

3.5.2 Runs-type signalling rules 104

3.5.2.1 Signalling indicators 105

3.6 Run-length distribution in the specified parameter case (Case K) 114

3.6.1 Methods of calculating the run-length distribution 114

3.6.1.1 The exact approach (for Shewhart and some Shewhart-type charts) 114

3.6.1.2 The Markov chain approach 115

3.6.1.3 The integral equation approach 132

3.6.1.4 The computer simulations (the Monte Carlo) approach 132

3.6.1.5 A note on the number of subintervals to be used for the EWMA chart 133

3.7 Parameter estimation problem and its effects on the control chart performance 135

3.8 Parametric variables control charts in Case U 137

3.8.1 Shewhart control charts in Case U 137

3.8.1.1 Shewhart control charts for the mean in Case U 137

3.8.1.2 Shewhart control charts for the standard deviation in Case U 138

3.8.2 CUSUM chart for the mean in Case U 141

3.8.3 EWMA chart for the mean in Case U 141

3.9 Types of parametric control charts in Case U: Illustrative examples 141

3.9.1 Charts for the mean 142

3.8.2 Chart for the standard deviation 145

3.10 Run-length distribution in the unknown parameter case (Case U) 155

3.10.1 Methods of calculating the run-length distribution and its properties – the conditioning – unconditioning method 155

3.10.1.1 The Shewhart chart for the mean in Case U 155

3.10.1.2 The Shewhart chart for the variance in Case U 171

3.10.1.3 The CUSUM chart for the mean in Case U 172

3.10.1.3 The EWMA chart for the mean in Case U 172

3.11 Control chart enhancements 173

3.11.1 Run-length calculation for runs-type signaling rules in Case U 173

3.12 Phase I Control Charts 175

3.13 Size of Phase I data 177

3.14 Robustness of parametric control charts 178

APPENDIX 3.1: Some derivations for the EWMA control chart 179

Result 3.1.1: The variance of the EWMA charting statistic 179

Result 3.1.2: The EWMA is an unbiased estimator of the process mean 181

APPENDIX 3.2 181

Markov chains 181

Definition 3.1 181

Definition 3.2 181

Definition 3.3 182

Definition 3.4 182

Definition 3.5 183

Definition 3.6 183

APPENDIX 3.3 185

Some derivations and tables for the dispersion charts 185

**Chapter 4: Nonparametric (Distribution-free) univariate variables control charts 188**

4.0 Chapter overview 188

4.1 Introduction 188

4.2 Distribution-free variables control charts in Case K 190

4.2.1 Shewhart control charts 190

4.2.1.1 Shewhart control charts based on signs 190

4.2.1.2 Shewhart control charts based on signed-ranks 197

4.2.2 CUSUM control charts 203

4.2.2.1 CUSUM control charts based on signs 204

4.2.2.2 A CUSUM sign control chart with runs-type signalling rules 204

4.2.2.2 CUSUM control charts based on signed-ranks 207

4.2.3 EWMA control charts 210

4.2.3.1 EWMA control charts based on signs 210

4.2.3.2 EWMA control charts based on signs with runs-type signalling rules 212

4.2.3.3 Methods of calculating the run-length distribution 212

4.2.4 EWMA control charts based on signed-ranks 216

4.3 Distribution-free control charts in Case K: Illustrative examples 220

4.4 Distribution-free variables control charts in Case U 252

4.4.1 Shewhart control charts 252

One-sided Shewhart-Prec control charts 272

4.4.2 CUSUM control charts 278

4.4.3 EWMA control charts 285

4.4.3.2 EWMA control charts based on the Wilcoxon rank-sum statistic 288

4.5 Distribution-free control charts in Case U: Illustrative examples 291

4.6 Effects of parameter estimation 303

4.7 Size of Phase I data 304

4.8 Control chart enhancements 304

4.8.1 Sensitivity and runs-type signalling rules 304

4.9 Appendix 307

**Chapter 5: Miscellaneous univariate distribution-free (nonparametric) variables control charts 320**

5.0. Chapter overview 320

5.1. Introduction 320

5.2. Other univariate distribution-free (nonparametric) variables control charts 320

**Appendix A: Tables 361**

Table A: Binomial distribution – probabilities for the in-control case 361

Table B: Probabilities for the Wilcoxon signed-rank statistic 362

Table C: Unbiasing charting constants for the construction of variables control charts 365

Table D1: Cumulative probabilities for the standard normal distribution 366

Table D2: Cumulative probabilities for the standard normal distribution continued 367

Table E: Upper tail probabilities for the t distribution 368

Table F: Upper tail probabilities for the Chi-square distribution 370

Table G: Charting constants for the Phase II Shewhart control chart in Case UU for *n* = 5, varying *m* and *ARL*_{IC} = 370 and 500 371

Table H: Charting constants for Phase II Shewhart *R* and *S* control charts in Case UU with three Phase I estimators of standard deviation for nominal *ARL*_{IC} values of 370 and 500 with varying and = 5, 10. 372

Appendix B: Programmes 373

References 397