Contents

Preface

1 Maths in Biology

1.1 What can go wrong?

1.2 Estimating

1.3 How to use this book

1.4 Mathematical conventions used in this book

2 Manipulating Numbers

2.1 Manipulating numbers

2.2 Solving equations

2.3 Why do you need to know all this?

2.4 Fractions

2.5 The number 1

2.6 Lowest common multiple and greatest common factor

2.7 Adding and subtracting fractions

2.8 Multiplying fractions

2.9 Dividing fractions

2.10 Fractions, decimals and percentages

2.11 Ratios and proportions

3 Units and Conversions

3.1 The SI system of units

3.2 SI prefixes

3.3 SI usage

3.4 Measuring energy

3.5 Temperature

4 Molarities and Dilutions

4.1 Avogadro’s number

4.2 Molecular weight

4.3 Solutions

4.4 Spectroscopy

4.5 Dilutions

5 Areas and Volumes

5.1 Geometry

5.2 Calculating areas and volumes

6 Exponents and Logs

6.1 Exponents

6.2 Exponential functions

6.3 Logarithms

7 Introduction to Statistics

7.1 What is statistics?

7.2 Statistical variables

7.3 Statistical methods

7.4 Frequency distributions

7.5 Frequency distribution graphs

8 Descriptive Statistics

8.1 Populations and samples

8.2 The central tendency

8.3 Variability

8.4 Standard error

8.5 Confidence intervals

8.6 Parametric and non-parametric statistics

8.7 Choosing an appropriate statistical test

8.8 Exploratory data analysis

9 Probability

9.1 Probability theory

9.2 Replacing vs not replacing selections

9.3 Calculating the probability of multiple events

9.4 The binomial distribution

9.5 Coincidences

10 Inferential Statistics

10.1 Statistical inference

10.2 Procedure for hypothesis testing

10.3 Standard scores (z-scores)

10.4 Student’s t-test (t-test)

10.5 Analysis of variance (ANOVA)

10.6 χ²-test

10.7 Fisher’s exact test

11 Correlation and Regression

11.1 Regression or correlation?

11.2 Correlation

11.3 Regression

Appendix 1 Answers to Problems

Appendix 2 Software for Biologists

E-mail

Word processors

Presentation and graphics

Internet resources

Statistics software

Appendix 3 Statistical Formulae and Tables

Critical values of the χ² distribution

Critical values of student’s t-Test

Table of critical values of the F-statistic

Table of critical values of the correlation coefficient, r

Table of binomial probabilities

Appendix 4 Glossary

Index

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Library of Congress Cataloging-in-Publication Data

Cann, Alan.
Maths in biology/Alan J. Cann.
p. cm.
Includes bibliographical references (p.).
ISBN 0-471-49834-3 (cased) — ISBN 0-471-49835-1 (pbk.)
1. Biomathematics. I. Title.
QH323.5.C363 2002
570′.1′51 — dc21
2002028068

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0 471 49834 3 Hardback
0 471 49835 1 Paperback

Preface

This book arose from my own need for a text that I would be happy to recommend to my students. Although there is no particular shortage of volumes claiming to help biologists with mathematics, all those I am familiar with have one of two flaws. Either they are written by well-meaning mathematicians and pay scant attention to biology, or they are not appropriate for the level at which most of the problems lie – new college students who do not have much confidence in approaching mathematical problems, in spite of extensive prior exposure to mathematics in school.

I make no claim to be a mathematical genius. Indeed, I believe my struggle to explain the material in an easily accessible form is one of the strengths of this book, bringing me closer to the students I am trying to communicate with. I reject any charges of ‘dumbing down’ – anyone who has ever tried to help a panic-stricken student in the grip of maths phobia will know that a calming but not patronizing voice is an essential attribute in these circumstances. Throughout, my intention is to provide a highly accessible text for students who, with or without formal mathematics qualifications, are frightened by the perceived ‘difficulty’ of mathematics and unwilling, inept or inexperienced in applying mathematical skills. To accommodate these students, many of whom opt to undertake studies in biology in the belief (conscious or unconscious) that this is a way of pursuing a scientific career while avoiding mathematics, the ethos of the book is consciously informal and intended to be confidence-building.

The maths in this volume has been checked vigorously, but I cannot guarantee that the text is entirely free of numerical errors. In addition there may be some passages where the subject matter is not expressed as clearly as I would have hoped. I rely on readers to point these out to me – as I am sure they will.

Alan J. Cann
University of Leicester, UK.
alan.cann@le.ac.uk

1

Maths in Biology

Mathematics, from the Greek, manthano, ‘to learn’

Some people opt to undertake studies in biology in the belief (conscious or unconscious) that this is a way of pursuing a scientific career while avoiding maths. This book is designed to be accessible to students who, with or without formal mathematical qualifications, are frightened by the perceived ‘difficulty’ of maths and hence are unwilling to apply what mathematical skills they might have. Have you ever noticed when you have been taught how to solve a mathematical problem, that you still don’t know why you need to do a particular step? This is the root of many problems with maths, so this book will try explain the why of maths, in addition to the how. Sometimes, these explanations may seem unnecessary, but I urge you not to skip them – understanding why you need to do something is the key to remembering how to do it. The intention is to be informal and confidence-building to ensure that all readers will gain a general appreciation of basic mathematical, statistical and data handling techniques appropriate to biology. I will try to explain the jargon which confuses the non-numerically minded.

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!