A Practical Guide to Data Analysis for

It is usually straightforward to calculate the result of a practical experiment in the laboratory. Estimating the accuracy of that result is often regarded by students as an obscure and tedious routine, involving much arithmetic. An estimate of the error is, however, an integral part of the presentation of the results of experiments. This textbook is intended for undergraduates who are carrying out laboratory experiments in the physical sciences for the first time. It is a practical guide on how to analyse data and estimate errors. The necessary formulas for performing calculations are given, and the ideas behind them are explained, although this is not a formal text on statistics. Specific examples are worked through step by step in the text. Emphasis is placed on the need to think about whether a calculated error is sensible. At first students should take this book with them to the laboratory, and the format is intended to make this convenient. The book will provide the necessary understanding of what is involved, should inspire confidence in the method of estimating errors, and enable numerical calculations without too much effort. The author's aim is to make practical classes more enjoyable. Students who use this book will be able to complete their calculations quickly and confidently, leaving time to appreciate the basic physical ideas involved in the experiments.

Preface

xi

(2)

Glossary and Conventions

xiii

1 Experimental errors

1

(43)

1.1 Why estimate errors?

1

(3)

1.2 Random and systematic errors

4

(5)

1.2.1 What they are

4

(3)

1.2.2 Estimating random errors

7

(1)

1.2.3 Worrying about systematic errors

8

(1)

1.3 Distributions

9

(3)

1.4 Mean and variance

12

(4)

1.5 Gaussian distribution

16

(2)

1.6 The meaning of Sigma

18

(3)

1.7 Combining errors

21

(6)

1.7.1 Linear situations

22

(3)

1.7.2 Products

25

(1)

1.7.3 The general case

26

(1)

1.8 Systematic errors

27

(2)

1.9 An example including random and systematic errors

29

(2)

1.10 Combining results of different experiments

31

(2)

1.11 Worked examples

33

(4)

1.11.1 Mean and variance

33

(1)

1.11.2 Using a Gaussian distribution

34

(1)

1.11.3 Central Limit Theorem

34

(1)

1.11.4 Combining errors

35

(1)

1.11.5 Combining results

36

(1)

1.12 Does it feel right?

37

(3)

Problems

40

(4)

2 Least squares fitting

44

(29)

2.1 What are we trying to do?

44

(2)

2.2 Weighted sum of squares

46

(3)

2.3 Determining the parameters

49

(2)

2.4 The error on the gradient and intercept

51

(4)

2.4.1 Using the errors Sigma(i)

51

(2)

2.4.2 Using the scatter of points

53

(2)

2.5 Other examples

55

(1)

2.6 y^(obs) as numbers

56

(1)

2.7 Parameter testing

57

(1)

2.8 Distribution testing

58

(5)

2.9 Worked example of straight line fit

63

(6)

2.10 Summary of straight line fitting

69

(2)

Problems

71

(2)

Appendix 1 Useful formulae

73

(2)

Appendix 2 Partial differentiation

75

(3)

Appendix 3 The binomial distribution

78

(4)

Appendix 4 The Poisson distribution

82

(4)

Appendix 5 Student's distribution

86

(3)

Appendix 6 Statistical tables

89

(3)

Appendix 7 Random numbers

92

(2)

Index

94

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A Practical Guide to Data Analysis for Physical Science Students

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Table of Contents

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