Essential Mathematics and Statistics for Forensic Science

Craig Adam

• Publisher: Wiley
• Year: 2011
• Print ISBN: 9780470742532, 9780470742525
• eBook ISBN: 9781119964186
• Edition: 1
• Topic: Medicine
• Subtopic: Forensic Medicine

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.

1

Getting the basics right

Introduction: Why forensic science is a quantitative science

This is the first page of a whole book devoted to mathematical and statistical applications within forensic science. As it is the start of a journey of discovery, this is also a good point at which to look ahead and discuss why skills in quantitative methods are essential for the forensic scientist. Forensic investigation is about the examination of physical evidence related to criminal activity. In carrying out such work what are we hoping to achieve?

For a start, the identification of materials may be necessary. This is achieved by physicochemical techniques, often methods of chemical analysis using spectroscopy or chromatography, to characterize the components or impurities in a mixture such as a paint chip, a suspected drug sample or a fragment of soil. Alternatively, physical methods such as microscopy may prove invaluable in identify pollen grains, hairs or the composition of gunshot residues. The planning and execution of experiments as well as the analysis and interpretation of the data requires knowledge of units of measurement and experimental uncertainties, proficiency in basic chemical calculations and confidence in carrying out numerical calculations correctly and accurately. Quantitative analysis may require an understanding of calibration methods and the use of standards as well as the construction and interpretation of graphs using spreadsheets and other computer-based tools.

More sophisticated methods of data analysis are needed for the interpretation of toxicological measurements on drug metabolites in the body, determining time since death, reconstructing bullet trajectories or blood-spatter patterns. All of these are based on an understanding of mathematical functions including trigonometry, and a good grasp of algebraic manipulation skills.

Samples from a crime scene may need to be compared with reference materials, often from suspects or other crime scenes. Quantitative tools, based on statistical methods, are used to compare sets of experimental measurements with a view to deciding whether they are similar or distinguishable: for example, fibres, DNA profiles, drug seizures or glass fragments. A prerequisite to using these tools correctly and to fully understanding their implication is the study of basic statistics, statistical distributions and probability.

The courts ask about the significance of evidence in the context of the crime and, as an expert witness, the forensic scientist should be able to respond appropriately to such a challenge. Methods based on Bayesian statistics utilizing probabilistic arguments may facilitate both the comparison of the significance of different evidence types and the weight that should be attached to each by the court. These calculations rely on experimental databases as well as a quantitative understanding of effects such as the persistence of fibres, hair or glass fragments on clothing, which may be successfully modelled using mathematical functions. Further, the discussion and presentation of any quantitative data within the report submitted to the court by the expert witness must be prepared with a rigour and clarity that can only come from a sound understanding of the essential mathematical and statistical methods applied within forensic science.

This first chapter is the first step forward on this journey. Here, we shall examine how numbers and measurements should be correctly represented and appropriate units displayed. Experimental uncertainties will be introduced and ways to deal with them will be discussed. Finally, the core chemical calculations required for the successful execution of a variety of chemical analytical investigations will be explored and illustrated with appropriate examples from the discipline.

1.1 Numbers, their representation and meaning

1.1.1 Representation and significance of numbers

Numbers may be expressed in three basic ways that are mathematically completely equivalent. First, we shall define and comment on each of these.

(a) Decimal representation is the most straightforward and suits quantities that are either a bit larger or a bit smaller than 1, e.g.

These numbers are clear and easy to understand, but

are less so, as the magnitude or power of 10 in each is hard to assimilate quickly due to difficulty in counting long sequences of zeros.

(b) Representation in scientific notation (sometimes called standard notation) overcomes this problem by separating the magnitude as a power of ten from the significant figures expressed as a number between 1 and 10 e.g. for the examples given in (a), we get:

This is the best notation for numbers that are significantly large or small. The power of 10 (the exponent) tells us the order of magnitude of the number. For example, using the calibrated graticule on a microscope, the diameter of a human hair might be measured as 65 μm (micrometres or microns). In scientific notation and using standard units (see Table 1.1), this becomes 6.5 × 10⁻⁵ m (metres) and so the order of magnitude is 10⁻⁵ m. Note that when using a calculator the ‘exp’ key or equivalent allows you to enter the exponent in the power of ten.

(c) An alternative that is widely used when the number represents a physical measurement, for example such as distance, speed or mass, is to attach a prefix to the appropriate unit of measurement, which directly indicates the power of ten. These are available ...

Table of contents

• Cover
• Title
• Copyright
• Preface
• 1: Getting the basics right
• 2: Functions, formulae and equations
• 3: The exponential and logarithmic functions and their applications
• 4: Trigonometric methods in forensic science
• 5: Graphs - their construction and interpretation
• 6: The statistical analysis of data
• 7: Probability in forensic science
• 8: Probability and infrequent events
• 9: Statistics in the evaluation of experimental data: comparison and confidence
• 10: Statistics in the evaluation of experimental data: computation and calibration
• 11: Statistics and the significance of evidence
• 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
• Index