# Statistics with JMP: Graphs, Descriptive Statistics and Probability

## Books Peter Goos, Department of Statistics, University of Leuven, Faculty of Bio-Science Engineering and University of Antwerp, Faculty of Applied Economics, Belgium

David Meintrup, Department of Mathematics and Statistics, University of Applied Sciences Ingolstadt, Faculty of Mechanical Engineering, Germany

Thorough presentation of introductory statistics and probability theory, with numerous examples and applications using JMP

JMP: Graphs, Descriptive Statistics and Probability provides an accessible and thorough overview of the most important descriptive statistics for nominal, ordinal and quantitative data with particular attention to graphical representations. The authors distinguish their approach from many modern textbooks on descriptive statistics and probability theory by offering a combination of theoretical and mathematical depth, and clear and detailed explanations of concepts. Throughout the book, the user-friendly, interactive statistical software package JMP is used for calculations, the computation of probabilities and the creation of figures. The examples are explained in detail, and accompanied by step-by-step instructions and screenshots. The reader will therefore develop an understanding of both the statistical theory and its applications.

Traditional graphs such as needle charts, histograms and pie charts are included, as well as the more modern mosaic plots, bubble plots and heat maps. The authors discuss probability theory, particularly discrete probability distributions and continuous probability densities, including the binomial and Poisson distributions, and the exponential, normal and lognormal densities. They use numerous examples throughout to illustrate these distributions and densities.

Key features:

• Introduces each concept with practical examples and demonstrations in JMP.
• Provides the statistical theory including detailed mathematical derivations.
• Presents illustrative examples in each chapter accompanied by step-by-step instructions and screenshots to help develop the reader’s understanding of both the statistical theory and its applications.
• A supporting website with data sets and other teaching materials.

This book is equally aimed at students in engineering, economics and natural sciences who take classes in statistics as well as at masters/advanced students in applied statistics and probability theory. For teachers of applied statistics, this book provides a rich resource of course material, examples and applications.

Preface xiii

Acknowledgments xvii

1 What is statistics? 1

1.1 Why statistics? 1

1.2 Definition of statistics 3

1.3 Examples 4

1.4 The subject of statistics 5

1.5 Probability 6

1.6 Software 7

2 Data and its representation 8

2.1 Types of data and measurement scales 8

2.1.1 Categorical or qualitative variables 8

2.1.2 Quantitative variables 9

2.1.3 Hierarchy of scales 10

2.1.4 Measurement scales in JMP 10

2.2 The data matrix 11

2.3 Representing univariate qualitative variables 12

2.4 Representing univariate quantitative variables 16

2.4.1 Stem and leaf diagram 16

2.4.2 Needle charts for univariate discrete quantitative variables 17

2.4.3 Histograms and frequency polygons for continuous variables 22

2.4.4 Empirical cumulative distribution functions 27

2.5 Representing bivariate data 30

2.5.1 Qualitative variables 30

2.5.2 Quantitative variables 34

2.6 Representing time series 38

2.7 The use of maps 39

2.8 More graphical capabilities 47

3 Descriptive statistics of sample data 54

3.1 Measures of central tendency or location 55

3.1.1 Median 56

3.1.2 Mode 57

3.1.3 Arithmetic mean 58

3.1.4 Geometric mean 61

3.2 Measures of relative location 63

3.2.1 Order statistics, quantiles, percentiles, deciles 63

3.2.2 Quartiles 64

3.3 Measures of variation or spread 64

3.3.1 Range 64

3.3.2 Interquartile range 65

3.3.3 Mean absolute deviation 65

3.3.4 Variance 65

3.3.5 Standard deviation 68

3.3.6 Coefficient of variation 69

3.3.7 Dispersion indices for nominal and ordinal variables 70

3.4 Measures of skewness 76

3.5 Kurtosis 78

3.6 Transformation and standardization of data 78

3.7 Box plots 79

3.8 Variability charts 84

3.9 Bivariate data 88

3.9.1 Covariance 89

3.9.2 Correlation 92

3.9.3 Rank correlation 94

3.10 Complementarity of statistics and graphics 98

3.11 Descriptive statistics using JMP 100

4 Probability 106

4.1 Random experiments 108

4.2 Definition of probability 110

4.3 Calculation rules 113

4.4 Conditional probability 114

4.5 Independent and dependent events 119

4.6 Total probability and Bayes’ rule 122

4.7 Simulating random experiments 127

5 Additional aspects of probability theory 129

5.1 Combinatorics 129

5.1.2 Multiplication principle 130

5.1.3 Permutations 130

5.1.4 Combinations 131

5.2 Number of possible orders 132

5.2.1 Two different objects 133

5.2.2 More than two different objects 133

5.3 Applications of probability theory 134

5.3.1 Sequences of independent random experiments 134

5.3.2 Euromillions 135

6 Univariate random variables 138

6.1 Random variables and distribution functions 138

6.2 Discrete random variables and probability distributions 140

6.3 Continuous random variables and probability densities 143

6.4 Functions of random variables 151

6.4.1 Functions of one discrete random variable 151

6.4.2 Functions of one continuous random variable 152

6.5 Families of probability distributions and probability densities 154

6.6 Simulation of random variables 155

7 Statistics of populations and processes 159

7.1 Expected value of a random variable 159

7.2 Expected value of a function of a random variable 161

7.3 Special cases 162

7.4 Variance and standard deviation of a random variable 163

7.5 Other statistics 166

7.6 Moment generating functions 169

8 Important discrete probability distributions 173

8.1 The uniform distribution 173

8.2 The Bernoulli distribution 175

8.3 The binomial distribution 176

8.3.1 Probability distribution 176

8.3.2 Expected value and variance 183

8.4 The hypergeometric distribution 184

8.5 The Poisson distribution 188

8.6 The geometric distribution 194

8.7 The negative binomial distribution 197

8.8 Probability distributions in JMP 200

8.8.1 Tables with probability distributions and cumulative distribution functions 200

8.8.2 Graphical representations 204

8.9 The simulation of discrete random variables with JMP 209

9 Important continuous probability densities 212

9.1 The continuous uniform density 213

9.2 The exponential density 215

9.2.1 Definition and statistics 215

9.2.2 Some interesting properties 216

9.3 The gamma density 220

9.4 The Weibull density 221

9.5 The beta density 223

9.6 Other densities 224

9.7 Graphical representations and probability calculations in JMP 226

9.8 Simulating continuous random variables in JMP 230

10 The normal distribution 232

10.1 The normal density 233

10.2 Calculation of probabilities for normally distributed variables 237

10.2.1 The standard normal distribution 237

10.2.2 General normally distributed variables 238

10.2.3 JMP 240

10.2.4 Examples 241

10.3 Lognormal probability density 247

11 Multivariate random variables 252

11.1 Introductory notions 252

11.2 Joint (discrete) probability distributions 254

11.3 Marginal or unconditional (discrete) probability distribution 256

11.4 Conditional (discrete) probability distribution 257

11.5 Examples of discrete bivariate random variables 258

11.6 The multinomial probability distribution 266

11.7 Joint (continuous) probability density 268

11.8 Marginal or unconditional (continuous) probability density 276

11.9 Conditional (continuous) probability density 279

12 Functions of several random variables 282

12.1 Functions of several random variables 282

12.2 Expected value of functions of several random variables 283

12.3 Conditional expected values 288

12.4 Probability distributions of functions of random variables 289

12.4.1 Discrete random variables 289

12.4.2 Continuous random variables 290

12.5 Functions of independent Poisson, normally, and lognormally distributed random variables 295

13 Covariance, correlation, and variance of linear functions 300

13.1 Covariance and correlation 300

13.2 Variance of linear functions of two random variables 305

13.3 Variance of linear functions of several random variables 306

13.4 Variance of linear functions of independent random variables 307

13.4.1 Two independent random variables 307

13.4.2 Several pairwise independent random variables 308

13.5 Linear functions of normally distributed random variables 308

13.6 Bivariate and multivariate normal density 310

13.6.1 Bivariate normal probability density 310

13.6.2 Graphical representations 310

13.6.3 Independence, marginal, and conditional densities 314

13.6.4 General multivariate normal density 318

14 The central limit theorem 319

14.1 Probability density of the sample mean from a normally distributed population 319

14.2 Probability distribution and density of the sample mean from a non-normally distributed population 320

14.2.1 Central limit theorem 320

14.2.2 Illustration of the central limit theorem 322

14.3 Applications 326

14.4 Normal approximation of the binomial distribution 328

Appendix A The Greek alphabet 330

Appendix B Binomial distribution 331

Appendix C Poisson distribution 336

Appendix D Exponential distribution 339

Appendix E Standard normal distribution 341

Index 343

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