Essential Mathematics and Statistics for Forensic Science

Craig Adam has over twenty years experience in teaching mathematics within the context of science at degree level. Initially this was within the physics discipline, but more recently he has developed and taught courses in mathematics and statistics for students in forensic science. As head of natural sciences at Staffordshire University in 1998, he led the initial development of forensic science degrees at that institution. Once at Keele University he worked within physics before committing himself principally to forensic science from 2004. His current research interests are focused on the use of chemometrics in the interpretation and evaluation of data from the analysis of forensic materials, particularly those acquired from spectroscopy. His teaching expertise areas within forensic science, apart from mathematics and statistics, include blood dynamics and pattern analysis, enhancement of marks and impressions, all aspects of document analysis, trace evidence analysis and evidence evaluation.

Preface. 1 Getting the basics right. Introduction: Why forensic science is a quantitativescience. 1.1 Numbers, their representation and meaning. Self-assessment exercises and problems. 1.2 Units of measurement and their conversion. Self-assessment problems. 1.3 Uncertainties in measurement and how to deal with them. Self-assessment problems. 1.4 Basic chemical calculations. Self-assessment exercises and problems. Chapter summary. 2 Functions, formulae and equations. Introduction: Understanding and using functions, formulae andequations. 2.1 Algebraic manipulation of equations. Self-assessment exercises. 2.2 Applications involving the manipulation of formulae. Self-assessment exercises and problems. 2.3 Polynomial functions. Self-assessment exercises and problems. 2.4 The solution of linear simultaneous equations. Self-assessment exercises and problems. 2.5 Quadratic functions. Self-assessment problems. 2.6 Powers and indices. Self-assessment problems. Chapter summary. 3 The exponential and logarithmic functions and theirapplications. Introduction: Two special functions in forensic science. 3.1 Origin and definition of the exponential function. Self-assessment exercises. 3.2 Origin and definition of the logarithmic function. Self-assessment exercises and problems. Self-assessment exercises. 3.3 Application: the pH scale. Self-assessment exercises. 3.4 The "decaying" exponential. Self-assessment problems. 3.5 Application: post-mortem body cooling. Self-assessment problems. 3.6 Application: forensic pharmacokinetics. Self-assessment problems. Chapter summary. 4 Trigonometric methods in forensic science. Introduction: Why trigonometry is needed in forensicscience. 4.1 Pythagoras s theorem. Self-assessment exercises and problems. 4.2 The trigonometric functions. Self-assessment exercises and problems. 4.3 Trigonometric rules. Self-assessment exercises. 4.4 Application: heights and distances. Self-assessment problems. 4.5 Application: ricochet analysis. Self-assessment problems. 4.6 Application: aspects of ballistics. Self-assessment problems. 4.7 Suicide, accident or murder? Self-assessment problems. 4.8 Application: bloodstain shape. Self-assessment problems. 4.9 Bloodstain pattern analysis. Self-assessment problems. Chapter summary. 5 Graphs - their construction and interpretation. Introduction: Why graphs are important in forensic science. 5.1 Representing data using graphs. 5.2 Linearizing equations. Self-assessment exercises. 5.3 Linear regression. Self-assessment exercises. 5.4 Application: shotgun pellet patterns in firearmsincidents. Self-assessment problem. 5.5 Application: bloodstain formation. Self-assessment problem. 5.6 Application: the persistence of hair, fibres and flints onclothing. Self-assessment problem. 5.7 Application: determining the time since death by fly egghatching. 5.8 Application: determining age from bone or tooth material Self-assessment problem. 5.9 Application: kinetics of chemical reactions. Self-assessment problems. 5.10 Graphs for calibration. Self-assessment problems. 5.11 Excel and the construction of graphs. Chapter summary. 6 The statistical analysis of data. Introduction: Statistics and forensic science. 6.1 Describing a set of data. Self-assessment problems. 6.2 Frequency statistics. Self-assessment problems. 6.3 Probability density functions. Self-assessment problems. 6.4 Excel and basic statistics. Chapter summary. 7 Probability in forensic science. Introduction: Theoretical and empirical probabilities. 7.1 Calculating probabilities. Self-assessment problems. 7.2 Application: the matching of hair evidence. Self-assessment problems. 7.3 Conditional probability. Self-assessment problems. 7.4 Probability tree diagrams. Self-assessment problems. 7.5 Permutations and combinations. Self-assessment problems. 7.6 The binomial probability distribution. Self-assessment problems. Chapter summary. 8 Probability and infrequent events. Introduction: Dealing with infrequent events. 8.1 The Poisson probability distribution. Self-assessment exercises. 8.2 Probability and the uniqueness of fingerprints. Self-assessment problems. 8.3 Probability and human teeth marks. Self-assessment problems. 8.4 Probability and forensic genetics. 8.5 Worked problems of genotype and allele calculations. Self-assessment problems. 8.6 Genotype frequencies and subpopulations. Self-assessment problems. Chapter summary. 9 Statistics in the evaluation of experimental data:comparison and confidence. How can statistics help in the interpretation of experimentaldata? 9.1 The normal distribution. Self-assessment problems. 9.2 The normal distribution and frequency histograms. 9.3 The standard error in the mean. Self-assessment problems. 9.4 The t-distribution. Self-assessment exercises and problems. 9.5 Hypothesis testing. Self-assessment problems. 9.6 Comparing two datasets using the t-test. Self-assessment problems. 9.7 The t -test applied to paired measurements. Self-assessment problems. 9.8 Pearson's 2 test. Self-assessment problems. Chapter summary. 10 Statistics in the evaluation of experimental data:computation and calibration. Introduction: What more can we do with statistics anduncertainty? 10.1 The propagation of uncertainty in calculations. Self-assessment exercises and problems. Self-assessment exercises and problems. 10.2 Application: physicochemical measurements. Self-assessment problems. 10.3 Measurement of density by Archimedes' upthrust. Self-assessment problems. 10.4 Application: bloodstain impact angle. Self-assessment problems. 10.5 Application: bloodstain formation. Self-assessment problems. 10.6 Statistical approaches to outliers. Self-assessment problems. 10.7 Introduction to robust statistics. Self-assessment problems. 10.8 Statistics and linear regression. Self-assessment problems. 10.9 Using linear calibration graphs and the calculation ofstandard error. Self-assessment problems. Chapter summary. 11 Statistics and the significance of evidence. Introduction: Where do we go from here? - Interpretation andsignificance. 11.1 A case study in the interpretation and significance offorensic evidence. 11.2 A probabilistic basis for interpreting evidence. Self-assessment problems. 11.3 Likelihood ratio, Bayes' rule and weight of evidence. Self-assessment problems. 11.4 Population data and interpretive databases. Self-assessment problems. 11.5 The probability of accepting the prosecution case - giventhe evidence. Self-assessment problems. 11.6 Likelihood ratios from continuous data. Self-assessment problems. 11.7 Likelihood ratio and transfer evidence. Self-assessment problems. 11.8 Application: double cot-death or double murder? Self-assessment problems. Chapter summary. References. Bibliography. Answers to self-assessment exercises and problems. Appendix I: The definitions of non-SI units and theirrelationship to the equivalent SI units. Appendix II: Constructing graphs using MicrosoftExcel. Appendix III: Using Microsoft Excel for statisticscalculations. Appendix IV: Cumulative z -probability table for thestandard normal distribution. Appendix V: Student's t -test: tables of criticalvalues for the t -statistic. Appendix VI: Chi squared 2 test: table ofcritical values. Appendix VII: Some values of Qcrit forDixon's Q test. Some values for Gcrit for Grubbs two-tailed test. Index.