Statistics for Scientists and Engineers

Ramalingam Shanmugam is the Editor-in-Chief for the journals: Advances in Life Sciences and Health, International Journal of Research in Medical Sciences, and Global Journal of Research and Review. He is the Book-Review Editor of the Journal of Statistical Computation and Simulation. He directed Statistics Consulting Center in the Mississippi State University. He served the Argonne National Lab., University of Colorado, University of South Alabama and the Indian Statistical Institute. He has published 120 research articles and is a fellow of the International Statistical Institute. Currently, he is a professor in the School of Health Administration, Texas State University. He is a recipient of several research awards from the Texas State University.

Rajan Chattamvelli has worked as an Analyst Specialist at Denver Public Health and was a visiting professor at the Indian Institute of Management. He was Chair of the Department of Computer Applications at Presidency College and Periyar Maniammai University, India.

This book provides the theoretical framework needed to build, analyze and interpret various statistical models. It helps readers choose the correct model, distinguish among various choices that best captures the data, or solve the problem at hand.

This is an introductory textbook on probability and statistics. The authors explain theoretical concepts in a step-by-step manner and provide practical examples. The introductory chapter in this book presents the basic concepts. Next, the authors discuss the measures of location, popular measures of spread, and measures of skewness and kurtosis. Probability theory, discrete distributions, and important continuous distributions that are often encountered in practical applications are analyzed. Mathematical Expectation is covered, along with Generating Functions and Functions of Random Variables. It discusses joint distributions, and novel methods to find the mean deviation of discrete and continuous statistical distributions.
Provides insight on coding complex algorithms using the 'loop unrolling technique'
Covers illuminating discussions on Poisson limit theorem, central limit theorem, mean deviation generating functions, CDF generating function and extensive summary tables
Contains extensive exercises at the end of each chapter and examples from interdisciplinary fields

Statistics for Scientists and Engineers is a great resource for students in engineering, physical sciences, and management, and also practicing engineers who require skill sets to model practical problems in a statistical setting.

Ramalingam Shanmugam is the Editor-in-Chief for the journals: Advances in Life Sciences and Health, International Journal of Research in Medical Sciences, and Global Journal of Research and Review. He is the Book-Review Editor of the Journal of Statistical Computation and Simulation. He directed Statistics Consulting Center in the Mississippi State University. He served the Argonne National Lab., University of Colorado, University of South Alabama and the Indian Statistical Institute. He has published 120 research articles and is a fellow of the International Statistical Institute. Currently, he is a professor in the School of Health Administration, Texas State University. He is a recipient of several research awards from the Texas State University.

Rajan Chattamvelli has worked as an Analyst Specialist at Denver Public Health and was a visiting professor at the Indian Institute of Management. He was Chair of the Department of Computer Applications at Presidency College and Periyar Maniammai University, India.

This book provides the theoretical framework needed to build, analyze and interpret various statistical models. It helps readers choose the correct model, distinguish among various choices that best captures the data, or solve the problem at hand.

This is an introductory textbook on probability and statistics.

Preface xv

About The Companion Website xxi

1 Descriptive Statistics 1

1.1 Introduction 1

1.2 Statistics as A Scientific Discipline 3

1.2.1 Scales of Measurement 4

1.3 The NOIR Scale 6

1.3.1 The Nominal Scale 6

1.3.2 The Ordinal Scale 6

1.3.3 The Interval Scale 7

1.3.4 The Ratio Scale 7

1.4 Population Versus Sample 8

1.4.1 Parameter Versus Statistic 9

1.5 Combination Notation 11

1.6 Summation Notation 11

1.6.1 Nested Sums 12

1.6.2 Increment Step Sizes 14

1.7 Product Notation 19

1.7.1 Evaluating Large Powers 20

1.8 Rising and Falling Factorials 22

1.9 Moments and Cumulants 22

1.10 Data Transformations 23

1.10.1 Change of Origin 23

1.10.2 Change of Scale 24

1.10.3 Change of Origin and Scale 24

1.10.4 Min-Max Transformation 25

1.10.5 Nonlinear Transformations 27

1.10.6 Standard Normalization 27

1.11 Data Discretization 28

1.12 Categorization of Data Discretization 28

1.12.1 Equal Interval Binning (EIB) 28

1.12.2 Equal Frequency Binning (EFB) 30

1.12.3 Entropy-Based Discretization (EBD) 31

1.12.4 Error in Discretization 34

1.13 Testing for Normality 34

1.13.1 Graphical Methods for Normality Checking 34

1.13.2 Ogive Plots 35

1.13.3 P-P and Q-Q Plots 35

1.13.4 Stem-and-Leaf Plots 36

1.13.5 Numerical Methods for Normality Testing 37

1.14 Summary 38

2 Measures of Location 43

2.1 Meaning of Location Measure 43

2.1.1 Categorization of Location Measures 44

2.2 Measures of Central Tendency 44

2.3 Arithmetic Mean 45

2.3.1 Updating Formula For Sample Mean 46

2.3.2 Sample Mean Using Change of Origin and Scale 49

2.3.3 Trimmed Mean 50

2.3.4 Weighted Mean 50

2.3.5 Mean of Grouped Data 51

2.3.6 Updating Formula for Weighted Sample Mean 51

2.3.7 Advantages of Mean 53

2.3.8 Properties of The Mean 53

2.4 Median 54

2.4.1 Median of Grouped Data 55

2.5 Quartiles and Percentiles 57

2.6 MODE 58

2.6.1 Advantages of Mode 58

2.7 Geometric Mean 59

2.7.1 Updating Formula For Geometric Mean 60

2.8 Harmonic Mean 61

2.8.1 Updating Formula For Harmonic Mean 61

2.9 Which Measure to Use? 63

2.10 Summary 63

3 Measures of Spread 67

3.1 Need For a Spread Measure 67

3.1.1 Categorization of Dispersion Measures 69

3.2 RANGE 71

3.2.1 Advantages of Range 72

3.2.2 Disadvantage of Range 72

3.2.3 Applications of Range 73

3.3 Inter-Quartile Range (IQR) 73

3.3.1 Change of Origin and Scale Transformation for Range 74

3.4 The Concept of Degrees of Freedom 74

3.5 Averaged Absolute Deviation (AAD) 76

3.5.1 Advantages of Averaged Absolute Deviation 77

3.5.2 Disadvantages of Averaged Absolute Deviation 77

3.5.3 Change of Origin and Scale Transformation for AAD 77

3.6 Variance and Standard Deviation 77

3.6.1 Advantages of Variance 78

3.6.2 Change of Origin and Scale Transformation for Variance 81

3.6.3 Disadvantages of Variance 81

3.6.4 A Bound for Sample Standard Deviation 82

3.7 Coefficient of Variation 82

3.7.1 Advantages of Coefficient of Variation 82

3.7.2 Disadvantages of Coefficient of Variation 83

3.7.3 An Interpretation of Coefficient of Variation 83

3.7.4 Change of Origin and Scale for CV 83

3.8 Gini Coefficient 84

3.9 Summary 84

4 Skewness and Kurtosis 89

4.1 Meaning of Skewness 89

4.1.1 Absolute Versus Relative Measures of Skewness 91

4.2 Categorization of Skewness Measures 93

4.3 Measures of Skewness 94

4.3.1 Bowley's Skewness Measure 94

4.3.2 Pearson's Skewness Measure 97

4.3.3 Coefficient of Quartile Deviation 98

4.3.4 Other Skewness Measures 99

4.4 Concept of Kurtosis 99

4.4.1 An Interpretation of Kurtosis 99

4.4.2 Categorization of Kurtosis Measures 101

4.5 Measures of Kurtosis 102

4.5.1 Pearson's Kurtosis Measure 102

4.5.2 Skewness-Kurtosis Bounds 104

4.5.3 L-kurtosis 104

4.5.4 Spectral Kurtosis (SK) 105

4.5.5 Detecting Faults Using SK 106

4.5.6

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