A Student's Guide to Data and Error Analysis

Cover......Page 1
A Student’s Guide to Data and Error Analysis......Page 5
Contents......Page 9
Preface......Page 13
1. Introduction......Page 17
List all data, a histogram or percentiles......Page 19
Example: 30 observations......Page 20
Significan figues......Page 23
Examples......Page 24
Examples......Page 25
Non-SI units......Page 27
2.5 Graphical presentation of experimental data......Page 28
Exercises......Page 31
Systematic errors......Page 32
Propagation through functions......Page 33
Combination of independent terms......Page 34
Example 1......Page 35
Systematic errors due to random deviations......Page 36
Monte Carlo methods......Page 37
Exercises......Page 39
4.1 Introduction......Page 41
Expectation, mean and variance......Page 43
Moments and central moments......Page 44
Characteristic function......Page 45
Definitio and properties......Page 46
Examples......Page 47
4.4 The Poisson distribution......Page 50
The Gauss function......Page 51
Probability scales......Page 53
Significan deviations......Page 54
4.6 The central limit theorem......Page 55
Log-normal distribution......Page 56
The Lorentz distribution: undefine variance......Page 57
Lifetime and exponential distributions......Page 58
F-distribution......Page 61
Example......Page 63
Exercises......Page 64
5. Processing of experimental data......Page 67
5.1 The distribution function of a data series......Page 68
5.2 The average and the mean squared deviation of a data series......Page 71
5.3 Estimates for mean and variance......Page 72
5.4 Accuracy of mean and Student’s t-distribution......Page 73
5.5 Accuracy of variance......Page 74
5.6 Handling data with unequal weights......Page 75
Accuracy of the estimated mean......Page 76
Elimination of outliers......Page 77
Rank-based estimates......Page 78
Exercises......Page 82
6.1 Introduction......Page 85
6.2 Linearization of functions......Page 87
Example: urease kinetics......Page 90
6.3 Graphical estimates of the accuracy of parameters......Page 91
6.4 Using calibration......Page 92
Exercises......Page 97
7.1 Introduction......Page 98
7.2 Linear regression......Page 101
The best parameter estimates......Page 102
Uncertainties in the parameters......Page 103
?......Page 104
Correlation coefficien between x and y values of a data series......Page 105
7.3 General least-squares t......Page 106
Example: nonlinear fi......Page 107
7.4 The chi-squared test......Page 109
Example 2......Page 111
7.5 Accuracy of the parameters......Page 112
Covariances of the parameters......Page 113
Relation between . 2 and 1-D parameter distribution......Page 114
Relation between . 2 and 2-D parameter distribution......Page 115
Example......Page 117
7.6 F-test on signi cance of the t......Page 120
Example......Page 121
Exercises......Page 122
8.1 Direct and inverse probabilities......Page 125
8.2 Enter Bayes......Page 126
Updating knowledge: Avogadro’s number......Page 128
Inference from a series of normally distributed samples......Page 129
Infer a rate constant from a few events......Page 134
8.5 Conclusion......Page 135
References......Page 137
Why do squared uncertainties add up in sums?......Page 149
A special case: sampling exponential functions......Page 152
A3. Characteristic function......Page 155
A4.1 The binomial distribution......Page 157
A4.2 The multinomial distribution......Page 158
From binomial to Poisson......Page 159
From Poisson to normal......Page 160
A5. Central limit theorem......Page 162
Uncorrelated data points......Page 165
Correlated data points......Page 166
How is this result influence when the data are correlated?......Page 168
Example......Page 169
How accurate is the estimated standard deviation?......Page 171
with the same expectations µ but with unequal standard deviations si?......Page 172
How large is the variance in x ?......Page 173
A9.1 How do you nd the best parameters a and b in y  ax + b?......Page 174
A9.2 General linear regression......Page 175
A9.3 SSQ as a function of the parameters......Page 176
A9.4 Covariances of the parameters......Page 177
Why is the s.d. of a parameter given by the projection of the ellipsoid . 2 = 1?......Page 178
Nonlinear least-squares fi......Page 179
Cumulative . 2-distribution......Page 213
Values of . 2 for 1%, 10%, 50%, 90%, and 99%......Page 214
Use in ANOVA (analysis of variance) in regression......Page 215
F-distribution, percentage points 95% and 99%......Page 216
General least-squares ftting......Page 217
Correlation coefficien r of x and y:......Page 218
One-dimensional Gauss function......Page 219
Multivariate Gauss functions......Page 220
Probability that =1 . n samples exceeds an interval......Page 221
Probability that =1 . n samples exceeds a value......Page 222
Physical constants......Page 223
Accuracies of derived quantities......Page 224
Continuous one-dimensional probability functions......Page 225
Continuous two-dimensional probability functions......Page 226
Properties and moments......Page 227
Values of t at 75%, 90%, 95%, 99%, and 99.5%......Page 228
Definition SI basic units......Page 229
Derived SI units......Page 230
Non-SI units (incl. British, US) (see also atomic units on p. 5)......Page 231
Units......Page 232
Molecular units......Page 233
Index......Page 234