Essential Mathematics and Statistics for Forensic Science

This text is an accessible, student-friendly introduction to the wide range of mathematical and statistical tools needed by the forensic scientist in the analysis, interpretation and presentation of experimental measurements. From a basis of high school mathematics, the book develops essential quantitative analysis techniques within the context of a broad range of forensic applications. This clearly structured text focuses on developing core mathematical skills together with an understanding of the calculations associated with the analysis of experimental work, including an emphasis on the use of graphs and the evaluation of uncertainties. Through a broad study of probability and statistics, the reader is led ultimately to the use of Bayesian approaches to the evaluation of evidence within the court. In every section, forensic applications such as ballistics trajectories, post-mortem cooling, aspects of forensic pharmacokinetics, the matching of glass evidence, the formation of bloodstains and the interpretation of DNA profiles are discussed and examples of calculations are worked through. In every chapter there are numerous self-assessment problems to aid student learning. Its broad scope and forensically focused coverage make this book an essential text for students embarking on any degree course in forensic science or forensic analysis, as well as an invaluable reference for post-graduate students and forensic professionals. Key features: Offers a unique mix of mathematics and statistics topics, specifically tailored to a forensic science undergraduate degree. All topics illustrated with examples from the forensic science discipline. Written in an accessible, student-friendly way to engage interest and enhance learning and confidence. Assumes only a basic high-school level prior mathematical knowledge.

Table of Contents:
Preface. 1 Getting the basics right. Introduction: Why forensic science is a quantitative science. 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 and equations. 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 their applications. 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 forensic science. 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 firearms incidents. Self-assessment problem. 5.5 Application: bloodstain formation. Self-assessment problem. 5.6 Application: the persistence of hair, fibres and flints on clothing. Self-assessment problem. 5.7 Application: determining the time since death by fly egg hatching. 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 experimental data? 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 χ2 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 and uncertainty? 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 of standard error. Self-assessment problems. Chapter summary. 11 Statistics and the significance of evidence. Introduction: Where do we go from here? - Interpretation and significance. 11.1 A case study in the interpretation and significance of forensic 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 - given the 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 their relationship to the equivalent SI units. Appendix II: Constructing graphs using Microsoft Excel. Appendix III: Using Microsoft Excel for statistics calculations. Appendix IV: Cumulative z -probability table for the standard normal distribution. Appendix V: Student's t -test: tables of critical values for the t -statistic. Appendix VI: Chi squared χ2 test: table of critical values. Appendix VII: Some values of Qcrit for Dixon's Q test. Some values for Gcrit for Grubbs’ two-tailed test. Index.