Understanding Statistical Error: A Primer for Biologists

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 author’s 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 Biologists can 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.

Table of Contents:
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 Student’s 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 Pearson’s 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