Essential Mathematics and Statistics for Forensic Science E-Book

Adam Craig

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 568

Veröffentlichungsjahr: 2011

Beschreibung

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.

Leseprobe

Contents

Preface

1 Getting the basics right

1.1 Numbers, their representation and meaning

1.2 Units of measurement and their conversion

1.3 Uncertainties in measurement and how to deal with them

1.4 Basic chemical calculations

2 Functions, formulae and equations

2.1 Algebraic manipulation of equations

2.2 Applications involving the manipulation of formulae

2.3 Polynomial functions

2.4 The solution of linear simultaneous equations

2.5 Quadratic functions

2.6 Powers and indices

3 The exponential and logarithmic functions and their applications

3.1 Origin and definition of the exponential function

3.2 Origin and definition of the logarithmic function

3.3 Application: the pH scale

3.4 The “decaying” exponential

3.5 Application: post-mortem body cooling

3.6 Application: forensic pharmacokinetics

4 Trigonometric methods in forensic science

4.1 Pythagoras’s theorem

4.2 The trigonometric functions

4.3 Trigonometric rules

4.4 Application: heights and distances

4.5 Application: ricochet analysis

4.6 Application: aspects of ballistics

4.7 Suicide, accident or murder?

4.8 Application: bloodstain shape

4.9 Bloodstain pattern analysis

5 Graphs - their construction and interpretation

5.1 Representing data using graphs

5.2 Linearizing equations

5.3 Linear regression

5.4 Application: shotgun pellet patterns in firearms incidents

5.5 Application: bloodstain formation

5.6 Application: the persistence of hair, fibres and flints on clothing

5.7 Application: determining the time since death by fly egg hatching

5.8 Application: determining age from bone or tooth material

5.9 Application: kinetics of chemical reactions

5.10 Graphs for calibration

5.11 Excel and the construction of graphs

6 The statistical analysis of data

6.1 Describing a set of data

6.2 Frequency statistics

6.3 Probability density functions

6.4 Excel and basic statistics

7 Probability in forensic science

7.1 Calculating probabilities

7.2 Application: the matching of hair evidence

7.3 Conditional probability

7.4 Probability tree diagrams

7.5 Permutations and combinations

7.6 The binomial probability distribution

8 Probability and infrequent events

8.1 The Poisson probability distribution

8.2 Probability and the uniqueness of fingerprints

8.3 Probability and human teeth marks

8.4 Probability and forensic genetics

8.5 Worked problems of genotype and allele calculations

8.6 Genotype frequencies and subpopulations

9 Statistics in the evaluation of experimental data: comparison and confidence

9.1 The normal distribution

9.2 The normal distribution and frequency histograms

9.3 The standard error in the mean

9.4 The t-distribution

9.5 Hypothesis testing

9.6 Comparing two datasets using the t-test

9.7 The t-test applied to paired measurements

9.8 Pearson’s χ2 test

10 Statistics in the evaluation of experimental data: computation and calibration

10.1 The propagation of uncertainty in calculations

10.2 Application: physicochemical measurements

10.3 Measurement of density by Archimedes’ upthrust

10.4 Application: bloodstain impact angle

10.5 Application: bloodstain formation

10.6 Statistical approaches to outliers

10.7 Introduction to robust statistics

10.8 Statistics and linear regression

10.9 Using linear calibration graphs and the calculation of standard error

11 Statistics and the significance of evidence

11.1 A case study in the interpretation and significance of forensic evidence

11.2 A probabilistic basis for interpreting evidence

11.3 Likelihood ratio, Bayes’ rule and weight of evidence

11.4 Population data and interpretive databases

11.5 The probability of accepting the prosecution case - given the evidence

11.6 Likelihood ratios from continuous data

11.7 Likelihood ratio and transfer evidence

11.8 Application: double cot-death or double murder?

References

Bibliography

Answers to self-assessment exercises and problems

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

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

This edition first published 2010, © 2010 by John Wiley & Sons Ltd.

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Library of Congress Cataloguing-in-Publication Data

Adam, Craig.

Essential mathematics and statistics for forensic science/Craig Adam.

p. cm.

Includes index.

ISBN 978-0-470-74252-5 – ISBN 978-0-470-74253-2

1. Mathematical statistics. 2. Forensic statistics. 3. Mathematics. I. Title.

QA276.A264 2010

510 – dc22

2009044658

ISBN: 9780470742525 (HB)

ISBN: 9780470742532 (PB)

Preface

It is the view of most scientists that the mathematics required by their discipline is best taught within the context of the subject matter of the science itself. Hence, we find the presence of texts such as Mathematics for Chemists in university campus bookshops. However, for the discipline of forensic science that has more recently emerged in distinct undergraduate programmes, no such broadly based, yet subject-focused, text exists. It is therefore the primary aim of this book to fill a gap on the bookshelves by embracing the distinctive body of mathematics that underpins forensic science and delivering it at an introductory level within the context of forensic problems and applications.

What then is this distinctive curriculum? For a start, forensic scientists need to be competent in the core mathematics that is common across the physical sciences and have a good grasp of the quantitative aspects of experimental work, in general. They also require some of the mathematical skills of the chemist, particularly in basic chemical calculations and the data analysis associated with analytical work. In addition, expertise in probability and statistics is essential for the evaluation and interpretation of much forensic experimental work. Finally, within the context of the court, the forensic scientist needs to provide an interpretation of the evidence that includes its significance to the court’s deliberations. This requires an understanding of the methods of Bayesian statistics and their application across a wide range of evidence types and case scenarios.

To work with this book a level of understanding no higher than the core curriculum of high school mathematics (GCSE or equivalent in the UK) together with only some elementary knowledge of statistics is assumed. It is recommended that if students believe they have some deficiency in their knowledge at this level then they should seek support from their tutor or from an alternative introductory text. Some examples of these are indicated within the bibliography. In a similar fashion, to pursue further study of the subject in greater depth, the reader is referred again to the books listed in the bibliography of this text. Note that this book has been devised deliberately without the inclusion of calculus. The vast majority of mathematical manipulations within forensic science may be understood and applied without the use of calculus and by omitting this topic the burden on the student of acquiring expertise in a new and quite complex topic in mathematics has been avoided.

Although the focus overall is on forensic applications, the structure of this book is governed by the mathematical topics. This is inevitable in a linear subject where one’s understanding in one area is dependent on competence in others. Thus, the exploration of the exponential, logarithmic and trigonometric functions in Chapters 3 and 4 requires a sound understanding of functions and equations together with an expertise in manipulative algebra, which is developed in Chapter 2. Chapter 1 covers a range of basic topics in quantitative science such as units, experimental measurements and chemical calculations. In each topic students may test and extend their understanding through exercises and problems set in a forensic context.

As an experimental discipline, forensic science uses graphs extensively to display, analyse and quantitatively interpret measurements. This topic forms the basis of Chapter 5, which also includes the development of techniques for linearizing equations in order to fit a mathematical model to experimental data through linear regression. The basic principles of statistics, leading to a discussion of probability density functions, form the substance of Chapter 6. Development of these topics is strongly dependent on the student’s expertise in probability and its applications, which is the subject of Chapter 7. Within the forensic discipline the uniqueness or otherwise of biometrics, such as fingerprints and DNA profiles, is sufficiently important that a separate chapter (8) has been included that is devoted to a discussion of the statistical issues specific to dealing with infrequent events.

The focus of the final three chapters is the analysis, interpretation and evaluation of experimental data in its widest sense. Chapter 9 has a strong statistical thrust, as it introduces some of the key statistical tests both for the comparison of data and for establishing the methodology of confidence limits in drawing conclusions from our measurements. To complement this, Chapter 10 deals with the propagation of uncertainties through experimental work and shows how these may be deal with, including extracting uncertainties from the results of linear regression calculations. The final chapter is based around the interpretation by the court of scientific testimony and how the quantitative work of the forensic scientist may be conveyed in a rigorous fashion, thereby contributing properly to the legal debate.

In summary, this book aims to provide

(1) the core mathematics and statistics needed to support a typical undergraduate study programme and some postgraduate programmes in forensic science

(2) many examples of specific applications of these techniques within the discipline

(3) links to examples from the published research literature to enable the reader to explore further work at the forefront of the subject

(4) a body of mathematical skills on which to build a deeper understanding of the subject either in the research context or at professional level.

In compiling this book I am grateful to Dr Andrew Jackson, Dr Sheila Hope and Dr Vladimir Zholobenko for reading and commenting on draft sections. In addition, comments from anonymous reviewers have been welcomed and have often led to corrections and refinements in the presentation of the material. I hope this final published copy has benefited from their perceptive comments, though I retain the responsibility for errors of any kind. I also wish to thank Dr Rob Jackson for permission to use his experimental data and Professor Peter Haycock for originating some of the problems I have used in Chapter 7. I should also like to thank my family - Alison, Nicol and Sibyl - for their patience and understanding during the preparation of this book. Without their support this book would not have been possible.

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.