Introduction to Statistics for Forensic Scientists	3
	Contents	7
	Preface	11
	List of figures	13
	List of tables	15
	1 A short history of statistics in the law	17
	1.1 History	17
	1.2 Some recent uses of statistics in forensic science	19
	1.3 What is probability?	20
	2 Data types, location and dispersion	23
	2.1 Types of data	23
	2.2 Populations and samples	25
	2.3 Distributions	25
	2.4 Location	27
	2.5 Dispersion	29
	2.6 Hierarchies of variation	30
	3 Probability	33
	3.1 Aleatory probability	33
	One throw of a six-sided die	33
	A single throw with more than one outcome of interest	34
	Two six-sided dice	35
	3.2 Binomial probability	37
	3.3 Poisson probability	40
	3.4 Empirical probability	41
	Modelled empirical probabilities	41
	Truly empirical probabilities	43
	4 The normal distribution	45
	4.1 The normal distribution	45
	4.2 Standard deviation and standard error of the mean	46
	4.3 Percentage points of the normal distribution	48
	4.4 The t-distribution and the standard error of the mean	50
	4.5 t-testing between two independent samples	52
	4.6 Testing between paired observations	56
	4.7 Confidence, significance and p-values	58
	5 Measures of nominal and ordinal association	61
	5.1 Association between discrete variables	61
	5.2 c(2) test for a 2 × 2 table	62
	5.3 Yules Q	64
	5.4 c(2) tests for greater than 2 × 2 tables	65
	5.5 f(2) and Cramers V(2)	66
	5.6 The limitations of c(2) testing	67
	5.7 Interpretation and conclusions	68
	6 Correlation	71
	6.1 Significance tests for correlation coefficients	75
	6.2 Correlation coefficients for non-linear data	76
	6.3 The coefficient of determination	79
	6.4 Partial correlation	79
	6.5 Partial correlation controlling for two or more covariates	85
	7 Regression and calibration	91
	7.1 Linear models	91
	7.2 Calculation of a linear regression model	94
	7.3 Testing ‘goodness of fit’	96
	7.4 Testing coefficients a and b	97
	7.5 Residuals	99
	7.6 Calibration	101
	A linear calibration model	102
	Calculation of a confidence interval for a point	105
	7.7 Points to remember	107
	8 Evidence evaluation	111
	8.1 Verbal statements of evidential value	111
	8.2 Evidence types	112
	8.3 The value of evidence	113
	8.4 Significance testing and evidence evaluation	118
	9 Conditional probability and Bayes’ theorem	121
	9.1 Conditional probability	121
	9.2 Bayes’ theorem	124
	9.3 The value of evidence	128
	10 Relevance and the formulation of propositions	133
	10.1 Relevance	133
	10.2 Hierarchy of propositions	134
	10.3 Likelihood ratios and relevance	136
	10.4 The logic of relevance	138
	10.5 The formulation of propositions	139
	10.6 What kind of propositions can we not evaluate?	140
	11 Evaluation of evidence in practice	145
	11.1 Which database to use	145
	Type and geographic factors	145
	DNA and database selection	147
	11.2 Verbal equivalence of the likelihood ratio	149
	11.3 Some common criticisms of statistical approaches	152
	12 Evidence evaluation examples	155
	12.1 Blood group frequencies	155
	12.2 Trouser fibres	157
	12.3 Shoe types	160
	12.4 Airweapon projectiles	164
	12.5 Height description from eyewitness	166
	13 Errors in interpretation	171
	13.1 Statistically based errors of interpretation	171
	Transposed conditional	171
	Defender’s fallacy	172
	Another match error	173
	Numerical conversion error	173
	13.2 Methodological errors of interpretation	174
	Different level error	174
	Defendant’s database fallacy	175
	Independence assumption	175
	14 DNA I	177
	14.1 Loci and alleles	177
	14.2 Simple case genotypic frequencies	178
	14.3 Hardy-Weinberg equilibrium	180
	14.4 Simple case allelic frequencies	182
	14.5 Accounting for sub-populations	184
	15 DNA II	187
	15.1 Paternity – mother and father unrelated	187
	15.2 Database searches and value of evidence	190
	15.3 Discussion	192
	16 Sampling and sample size estimation	195
	16.1 Estimation of a mean	195
	16.2 Sample sizes for t-tests	197
	Two sample t-test	197
	One sample t-test	199
	16.3 How many drugs to sample	200
	16.4 Concluding comments	204
	17 Epilogue	207
	17.1 Graphical models and Bayesian Networks	208
	Graphical models	208
	Bayesian networks	210
	17.2 Kernel density estimation	211
	17.3 Multivariate continuous matching	212
	Appendices	215
	A Worked solutions to questions	215
	B Percentage points of the standard normal distribution	241
	C Percentage points of t-distributions	243
	D Percentage points of c(2)-distributions	245
	E Percentage points of beta-beta distributions	247
	F Percentage points of F-distributions	249
	G Calculating partial correlations using Excel software	251
	H Further algebra using the “third law”	255
	References	259
	Index	265