Preface ........................................................ xi
1 The importance of uncertainty in science and technology ...... 1
1.1 Measurement matters ..................................... 3
1.2 Review ................................................. 13
2 Measurement fundamentals .................................... 15
2.1 The system of units of measurement ..................... 15
2.2 Scientific and engineering notations ................... 21
2.3 Rounding and significant figures ....................... 22
2.4 Another way of expressing proportional uncertainty ..... 26
2.5 Review ................................................. 26
3 Terms used in measurement ................................... 27
3.1 Measurement and related terms .......................... 27
3.2 Review ................................................. 34
4 Introduction to uncertainty in measurement .................. 35
4.1 Measurement and error .................................. 35
4.2 Uncertainty is a parameter that characterises the
dispersion of values ................................... 43
4.3 Standard deviation as a basic measure of uncertainty ... 45
4.4 The uncertainty in the estimate of uncertainty ......... 49
4.5 Combining standard uncertainties ....................... 50
4.6 Review ................................................. 52
5 Some statistical concepts ................................... 53
5.1 Sampling from a population ............................. 53
5.2 The least-squares model and least-squares fitting ...... 59
5.3 Covariance and correlation ............................. 77
5.4 Review ................................................. 82
6 Systematic errors ........................................... 83
6.1 Systematic error revealed by specific information ...... 83
6.2 Systematic error revealed by changed conditions ........ 92
6.3 Review ................................................. 96
7 Calculation of uncertainties ................................ 97
7.1 The measurand model and propagation of uncertainties
from inputs to measurand ............................... 97
7.2 Correlated inputs ..................................... 109
7.3 Review ................................................ 125
8 Probability density, the Gaussian distribution and
central limit theorem ...................................... 126
8.1 Distribution of scores when tossing coins or dice ..... 126
8.2 General properties of probability density ............. 128
8.3 The uniform or rectangular distribution ............... 133
8.4 The Gaussian distribution ............................. 135
8.5 Experimentally observed non-Gaussian distributions .... 139
8.6 The central limit theorem ............................. 143
8.7 Review ................................................ 153
9 Sampling a Gaussian distribution ........................... 154
9.1 Sampling the distribution of the mean of a sample
of size n, from a Gaussian population ................. 154
9.2 Sampling the distribution of the variance of
a sample of size n, from a Gaussian population ........ 155
9.3 Sampling the distribution of the standard deviation
of a sample of size n, from a Gaussian population ..... 159
9.4 Review ................................................ 161
10 The t-distribution and Welch-Satterthwaite formula ......... 162
10.1 The coverage interval for a Gaussian distribution ..... 163
10.2 The coverage interval using a t-distribution .......... 169
10.3 The Welch-Satterthwaite formula ....................... 174
10.4 Review ................................................ 185
11 Case studies in measurement uncertainty .................... 191
11.1 Reporting measurement results ......................... 191
11.2 Determination of the coefficient of static friction
for glass on glass .................................... 192
11.3 A crater-formation experiment ......................... 197
11.4 Determination of the density of steel ................. 203
11.5 The rate of evaporation of water from an open
container ............................................. 210
11.6 Review ................................................ 217
Appendix A Solutions to exercises ............................ 218
Appendix В 95% Coverage factors, k, as a function of the
number of degrees of freedom, ν ............................ 222
Appendix С Further discussion following from the Welch-
Satterthwaite formula ...................................... 223
References .................................................... 226
Index ......................................................... 229
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