Understanding Statistical Error

This accessible introductory textbook provides a straightforward, practical explanation of how statistical analysis and error measurements should be applied in biological research.

Understanding Statistical Error - A Primer for Biologists:

• Introduces the essential topic of error analysis to biologists
• Contains mathematics at a level that all biologists can grasp
• Presents the formulas required to calculate each confidence interval for use in practice
• Is based on a successful series of lectures from the authors established course

Assuming no prior knowledge of statistics, this book covers the central topics needed for efficient data analysis, ranging from probability distributions, statistical estimators, confidence intervals, error propagation and uncertainties in linear regression, to advice on how to use error bars in graphs properly. Using simple mathematics, all these topics are carefully explained and illustrated with figures and worked examples. The emphasis throughout is on visual representation and on helping the reader to approach the analysis of experimental data with confidence.

This useful guide explains how to evaluate uncertainties of key parameters, such as the mean, median, proportion and correlation coefficient. Crucially, the reader will also learn why confidence intervals are important and how they compare against other measures of uncertainty.

Understanding Statistical Error - A Primer for Biologistscan be used both by students and researchers to deepen their knowledge and find practical formulae to carry out error analysis calculations. It is a valuable guide for students, experimental biologists and professional researchers in biology, biostatistics, computational biology, cell and molecular biology, ecology, biological chemistry, drug discovery, biophysics, as well as wider subjects within life sciences and any field where error analysis is required.

Dr Marek Gierlinski is a bioinformatician at College of Life Science, University of Dundee, UK. He attained his PhD in astrophysics and studied X-ray emission from black holes and neutron stars for many years. In 2009 he started a new career in bioinformatics, bringing his knowledge and skills in statistics and data analysis to a biological institute. He works on a variety of topics, including proteomics, DNA and RNA sequencing, imaging and numerical modelling.

Introduction 1

Why would you read an introduction? 1

What is this book about? 1

Who is this book for? 2

About maths 2

Acknowledgements 3

Chapter 1 Why do we need to evaluate errors? 4

Chapter 2 Probability distributions 7

2.1 Random variables 8

2.2 What is a probability distribution? 9

Probability distribution of a discrete variable 9

Probability distribution of a continuous variable 10

Cumulative probability distribution 11

2.3 Mean, median, variance and standard deviation 11

2.4 Gaussian distribution 13

Example: estimate an outlier 15

2.5 Central limit theorem 16

2.6 Log-normal distribution 18

2.7 Binomial distribution 20

2.8 Poisson distribution 23

Classic example: horse kicks 25

Inter-arrival times 26

2.9 Students t-distribution 28

2.10 Exercises 30

Chapter 3 Measurement errors 32

3.1 Where do errors come from? 32

Systematic errors 33

Random errors 34

3.2 Simple model of random measurement errors 35

3.3 Intrinsic variability 38

3.4 Sampling error 39

Sampling in time 39

3.5 Simple measurement errors 41

Reading error 41

Counting error 43

3.6 Exercises 46

Chapter 4 Statistical estimators 47

4.1 Population and sample 47

4.2 What is a statistical estimator? 49

4.3 Estimator bias 52

4.4 Commonly used statistical estimators 53

Mean 53

Weighted mean 54

Geometric mean 55

Median 56

Standard deviation 57

Unbiased estimator of standard deviation 59

Mean deviation 62

Pearsons correlation coefficient 63

Proportion 65

4.5 Standard error 66

4.6 Standard error of the weighted mean 70

4.7 Error in the error 71

4.8 Degrees of freedom 72

4.9 Exercises 73

Chapter 5 Confidence intervals 74

5.1 Sampling distribution 75

5.2 Confidence interval: what does it really mean? 77

5.3 Why 95%? 79

5.4 Confidence interval of the mean 80

Example 83

5.5 Standard error versus confidence interval 84

How many standard errors are in a confidence interval? 84

What is the confidence of the standard error? 85

5.6 Confidence interval of the median 86

Simple approximation 89

Example 89

5.7 Confidence interval of the correlation coefficient 90

Significance of correlation 93

5.8 Confidence interval of a proportion 95

5.9 Confidence interval for count data 99

Simple approximation 102

Errors on count data are not integers 102

5.10 Bootstrapping 103

5.11 Replicates 105

Sample size to find the mean 108

5.12 Exercises 109

Chapter 6 Error bars 112

6.1 Designing a good plot 112

Elements of a good plot 113

Lines in plots 115

A digression on plot labels 116

Logarithmic plots 117

6.2 Error bars in plots 118

Various types of errors 119

How to draw error bars 120

Box plots 121

Bar plots 123

Pie charts 128

Overlapping error bars 128

6.3 When can you get away without error bars? 130

On a categorical variable 130

When presenting raw data 130

Large groups of data points 130

When errors are small and negligible 131

Where errors are not known 131

6.4 Quoting numbers and errors 132

Significant figures 132

Writing significant figures 133

Errors and significant figures 135

Error with no error 137

Computer-generated numbers 138

Summary 140

6.5 Exercises 140

Chapter 7 Propagation of errors 142

7.1 What is propagation of errors? 142

7.2 Single variable 143

Scaling 144

Logarithms 144

7.3 Multiple variables 146

Sum or difference 146

Ratio or product 147

7.4 Correlated variables 149

7.5 To use error propagation or not? 150

7.6 Example: distance between two dots 151

7.7 Derivation of the error propagation formula for one variable 153

7.8 Derivation of the error propagation formula for multiple variables 155

7.9 Exercises 157

Chapter 8 Errors in simple linear regression 158

8.1 Linear relation between two variables 158

Mean response 159

True response and noise 160

Data linearization 161

8.2 Straight line fit 161

8.3 Confidence intervals of linear fit parameters 164

Example 168

8.4 Linear fit prediction errors 170

8.5 Regression through the origin 173

Example 174

8.6 General curve fitting 175

8.7 Derivation of errors on fit parameters 178

8.8 Exercises 179

Chapter 9 Worked example 181

9.1 The experiment 181

9.2 Results 182

Sasha 183

Lyosha 186

Masha 189

9.3 Discussion 190

9.4 The final paragraph 192

Solutions to exercises 193

Appendix A 206

Bibliography 209

Index 211