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Overview of chapters
Chapter 1 Quantities and units 19
Chapter 2 Fundamentals of measurement theory 39
Chapter 3 Dispersion and uncertainty 65
Chapter 4 Quantities in the measurement model. Evaluation of the value and uncertainty of a quantity 81
Chapter 5 Evaluation of the value of the output quantity and propagation of uncertainty of measurement in the measurement model 107
Chapter 6 Variance and regression analysis of results of measurement 139
Chapter 7 The result of measurement and decisions based on it 167
Chapter 8 Measuring instruments, their categories and general characteristics 191
Chapter 9 Calibration of measuring instruments 217
Chapter 10 International and European metrology organisations 239
Contents
Abstract 11
Preface 13
Abbreviations 15
Chapter 1 Quantities and units 19
1.1 Measurable quantities 19
1.2 Base quantities and derived quantities 21
1.3 Dimension of a quantity 23
1.4 Relations between quantities 25
1.5 The value of a quantity 27
1.6 Units of measurement 30
1.7 The International System of Units SI 31
1.8 Orthography of SI units and their symbols 37
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Chapter 2 Fundamentals of measurement theory 39
2.1 The basic assertion in measurement 39
2.2 Distributions of random variables and their parameters 48
2.3 Types of distributions generally used in measurement processes 56
2.3.1 Rectangular distribution 56
2.3.2 Symmetric triangular distribution 57
2.3.3 Symmetric trapezoidal distribution 59
2.3.4 Normal distribution 61
2.3.5 Inverse sine distribution 63
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Chapter 3 Dispersion and uncertainty 65
3.1 From the definition of a measurand to the measurement procedure 65
3.2 Dispersion of results of measurement 67
3.2.1 Causes of dispersion 67
3.2.2 Deviations 67
3.2.3 Random effects 70
3.2.4 Systematic effects and deviations caused by them 71
3.2.5 Graphical illustration of deviations 73
3.3 Uncertainty of measurement 75
3.3.1 General characterisation 75
3.3.2 Sources of uncertainty of measurement 77
3.3.3 Treatment of uncertainty of measurement 78
3.3.4 Evaluation methods of uncertainty of measurement 78
3.4 Parameters that characterise uncertainty of measurement 79
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Chapter 4 Quantities in the measurement model. Evaluation of the value and uncertainty of a quantity 81
4.1 Measurement model 81
4.2 Evaluation of the value of an input quantity and its uncertainty based on actual measurements 83
4.2.1 Single and repeated measurement of an input quantity 83
4.2.1.1 Evaluation of the quantity value 83
4.2.1.2 Evaluation of the standard uncertainty of an arithmetic mean 85
4.2.1.3 Pooling of standard deviations 87
4.2.1.4 Evaluation of the standard uncertainty of a single measurement 89
4.2.2 Establishment of outliers in a series of measurement 89
4.2.2.1 Grubbs test 90
4.2.2.2 z-score method 92
4.2.3 Repeatability and reproducibility of measurement results 94
4.3 Input quantities and their uncertainties evaluated by other methods 97
4.3.1 Introductory remarks 97
4.3.2 Uncertainty caused by the finite resolution of an indicating device 99
4.3.3 Uncertainty caused by the dead band of a measuring instrument 99
4.3.4 Uncertainty caused by rounding of results of measurement 100
4.3.5 Uncertainty of an input quantity imported to the measurement model 100
4.3.6 Uncertainty of the value of a quantity expressed by upper and lower limits 101
4.3.7 Asymmetrical distribution of possible values of an input quantity 103
4.3.8 Uncertainty associated with the indication of a calibrated measuring instrument 103
4.3.9 Uncertainty associated with an indication of a verified measuring instrument 104
4.3.10 Uncertainty of a controlled quantity 104
4.3.11 Uncertainty due to imperfections of the measurement model and method 105
4.3.12 Uncertainty derived from the object 105
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Chapter 5 Evaluation of the value of the output quantity and propagation of uncertainty of measurement in the measurement model 107
5.1 Requirements on the model for propagation of uncertainty of measurement 107
5.2 Evaluation of the value of the output quantity 108
5.3 Combined uncertainty of a result of measurement 109
5.3.1 Combined uncertainty in the case of uncorrelated input quantities 110
5.3.1.1 Law of propagation of uncertainty 110
5.3.1.2 Propagation law for a sum or difference 112
5.3.1.3 Propagation law for a product or quotient 113
5.3.2 Justification of the law of propagation of uncertainty 114
5.3.3 Combined uncertainty in the case of correlated input quantities 115
5.3.4 Evaluation of covariance 118
5.3.5 Combined uncertainty in the case of a product of two input quantities with zero estimate 121
5.3.6 Negligibility criterion of a component of uncertainty 122
5.3.7 The central limit theorem and joint distributions 123
5.4 Expanded uncertainty of a result of measurement 126
5.4.1 The need to report an expanded uncertainty 126
5.4.2 The interval defined by the expanded uncertainty 127
5.4.3 The value of the coverage factor 128
5.4.4 Students distribution and degrees of freedom 130
5.4.5 The effective degrees of freedom 132
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Chapter 6 Variance and regression analysis of results of measurement 139
6.1 Analysis of variance 139
6.1.1 One-factor analysis of variance based on the F-test 140
6.1.2 One-factor analysis of variance using Cochrans C test 146
6.1.3 Treatment of results of measurement with differing uncertainty 150
6.1.4 Other considerations 152
6.1.5 Two-factor analysis of variance 152
6.2 Analysis of regression 153
6.2.1 First-order regression task 153
6.2.2 Polynomial regression task 159
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Chapter 7 The result of measurement and decisions based on it 167
7.1 Presentation of numerical results 167
7.2 Rounding of numerical values 168
7.3 Presentation and documentation of a result of measurement 169
7.4 Consistency of results of measurement 172
7.4.1 Comparison of results obtained under repeatability conditions 172
7.4.1.1 t-test of consistency of two arithmetic means 172
7.4.1.2 Repeatability limit test of consistency of two means 173
7.4.2 Comparison of results obtained under reproducibility conditions 175
7.4.2.1 Test of consistency of two results based on their expanded uncertainty 175
7.4.2.2 Reproducibility limit test of the consistency of two means 177
7.5 Assessment of compliance with specifications 178
7.6 Interlaboratory comparisons 182
7.6.1 General remarks 182
7.6.2 Establishment of the reference value 184
7.6.3 Evaluation of data 185
7.6.3.1 General considerations 185
7.6.3.2 Evaluations based on a consensus value 185
7.6.3.3 Evaluation of results based on a reference value 188
7.6.4 Concluding remarks on interlaboratory comparisons 189
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Chapter 8 Measuring instruments, their categories and general characteristics 191
8.1 Measuring instruments 191
8.1.1 Measuring transducers 193
8.1.2 Material measures 194
8.1.3 Displaying measuring instruments 194
8.1.4 Measuring systems 196
8.1.5 Measurement standards 197
8.1.6 Reference materials 200
8.1.7 Comparators 200
8.2 Output devices of measuring instruments and their general characteristics 202
8.2.1 Analogue scales 202
8.2.2 Digital scale s204
8.3 Conditions of use of a measuring instrument 204
8.4 Metrological characteristics of a measuring instrument 206
8.4.1 Response characteristic, sensitivity, discrimination, resolution, dead band, response time and transparency 206
8.4.2 Accuracy of a measuring instrumen t207
8.4.3 Stability and drift 209
8.4.4 Repeatability of a measuring instrument 209
8.4.5 Maximum permissible errors of a measuring instrument 210
8.5 Types of metrological control of measuring instruments 212
8.5.1 Calibration 212
8.5.2 Adjustment 213
8.5.3 Type approval 213
8.5.4 Verification 214
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Chapter 9 Calibration of measuring instruments 217
9.1 General aspects 217
9.2 Simplified calibration model 220
9.3 Correction and its uncertainty 222
9.4 Correction curve 229
9.5 Presentation of results of calibration 233
9.6 Statement of compliance 234
9.7 Best measurement capability 235
9.8 Drafting of calibration methods 236
9.9 Traceability 236
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Chapter 10 International and European metrology organisations 239
10.1 The Metre Convention and organisations established through it 239
10.1.1 The General Conference on Weights and Measures 241
10.1.2 The International Committee for Weights and Measures 242
10.1.3 The International Bureau of Weights and Measures 244
10.2 The International Organization for Standardization 245
10.3 The International Electrotechnical Commission 246
10.4 The International Union of Pure and Applied Physics 247
10.5 The International Organization of Legal Metrology 247
10.6 The International Measurement Confederation 248
10.7 EUROMET a European Collaboration in Measurement Standards 248
10.8 EURACHEM a Focus for Analytical Chemistry in Europe 249
10.9 The European co-operation for Accreditation 250
10.10 The European Cooperation in Legal Metrology 251
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Concluding remarks 253
References 257
Index 261
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