Preface ..................................................... xi
Part I Data and error analysis .............................. 1
1 Introduction ................................................. 3
2 The presentation of physical quantities with their
inaccuracies ................................................. 3
2.1 How to report a series of measurements .................. 5
2.2 How to represent numbers ................................ 9
2.3 How to express inaccuracies ............................ 10
2.4 Reporting units ........................................ 13
2.5 Graphical presentation of experimental data ............ 14
3 Errors: classification and propagation ...................... 18
3.1 Classification of errors ............................... 18
3.2 Error propagation ...................................... 19
4 Probability distributions ................................... 27
4.1 Introduction ........................................... 27
4.2 Properties of probability distributions ................ 29
4.3 The binomial distribution .............................. 32
4.4 The Poisson distribution ............................... 36
4.5 The normal distribution ................................ 37
4.6 The central limit theorem .............................. 41
4.7 Other distributions .................................... 42
5 Processing of experimental data ............................. 53
5.1 The distribution function of a data series ............. 54
5.2 The average and the mean squared deviation of a data
series ................................................. 57
5.3 Estimates for mean and variance ........................ 58
5.4 Accuracy of mean and Student's t-distribution .......... 59
5.5 Accuracy of variance ................................... 60
5.6 Handling data with unequal weights ..................... 61
5.7 Robust estimates ....................................... 63
6 Graphical handling of data with errors ...................... 71
6.1 Introduction ........................................... 71
6.2 Linearization of functions ............................. 73
6.3 Graphical estimates of the accuracy of parameters ...... 77
6.4 Using calibration ...................................... 78
7 Fitting functions to data ................................... 84
7.1 Introduction ........................................... 84
7.2 Linear regression ...................................... 87
7.3 General least-squares fit .............................. 92
7.4 The chi-squared test ................................... 95
7.5 Accuracy of the parameters ............................. 98
7.6 F-test on significance of the fit ..................... 106
8 Back to Bayes: knowledge as a probability distribution ..... 111
8.1 Direct and inverse probabilities ...................... 111
8.2 Enter Bayes ........................................... 112
8.3 Choosing the prior .................................... 114
8.4 Three examples of Bayesian inference .................. 114
8.5 Conclusion ............................................ 121
References ................................................. 123
Answers to exercises ....................................... 125
Part II Appendices ........................................ 133
Al Combining uncertainties .................................... 135
A2 Systematic deviations due to random errors ................. 138
A3 Characteristic function .................................... 141
A4 From binomial to normal distributions ...................... 143
A4.1 The binomial distribution ............................ 143
A4.2 The multinomial distribution ......................... 144
A4.3 The Poisson distribution ............................. 145
A4.4 The normal distribution .............................. 146
A5 Central limit theorem ...................................... 148
A6 Estimation of the variance ................................. 151
A7 Standard deviation of the mean ............................. 154
A8 Weight factors when variances are not equal ................ 158
A9 Least-squares fitting ...................................... 160
A9.1 How do you find the best parameters α and b
in y ≈ αx + b? ....................................... 160
A9.2 General linear regression ............................ 161
A9.3 SSQ as a function of the parameters .................. 162
A9.4 Covariances of the parameters ........................ 163
Part III Python codes ..................................... 167
Part IV Scientific data ................................... 197
Chi-squared distribution ................................... 199
F-distribution ............................................. 201
Least-squares fitting ...................................... 203
Normal distribution ........................................ 205
Physical constants ......................................... 209
Probability distributions .................................. 211
Student's t-distribution ................................... 213
Units ...................................................... 215
Index ...................................................... 220
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