Measurement errors and uncertainty : theory and practice

The major objective of this book is to give methods for estimating errors and uncertainties of real measurements: measurements that are performed in industry, commerce, and experimental research. This book is needed because the existing theory of measurement errors was historically developed as an abstract mathematical discipline. As a result, this theory allows estimation of uncertainties of some ideal measurements only and is not applicable to most practical cases. In particular, it is not applicable to single measurements. This situation did not bother mathematicians, whereas engineers, not being bold enough to assert that the mathematical theory of errors cannot satisfy their needs, solved their particular problems in one or another ad hoc manner. Actually, any measurement of a physical quantity is not abstract, but it involves an entirely concrete procedure that is always implemented with concrete te- nical devices--measuring instruments--under concrete conditions. Therefore, to obtain realistic estimates of measurement uncertainties, mathematical methods must be supplemented with methods that make it possible to take into account data on properties of measuring instruments, the conditions under which measu- ments are performed, the measurement procedure, and other features of measu- ments. The importance of the methods of estimating measurement inaccuracies for practice can scarcely be exaggerated. Indeed, in another stage of planning a m- surement or using a measurement result, one must know its error limits or unc- tainty. Inaccuracy of a measurement determines its quality and is related to its cost.

Author Notes

Rabinovich was head of the Laboratory of Theoretical Metrology at the All-Union State Research Institute of Metrology in the former Soviet Union. His theory of galvanometrical self-balancing instruments led to the creation of numerous new measuring devices.

Preface	p. v
Chapter 1 General Information About Measurements	p. 1
1.1 Basic Concepts and Terms	p. 1
1.2 Metrology and the Basic Metrological Problems	p. 3
1.3 Initial Points of the Theory of Measurements	p. 10
1.4 Classification of Measurements	p. 15
1.5 Classification of Measurement Errors	p. 20
1.6 Principles of Estimation of Measurement Errors and Uncertainties	p. 22
1.7 Presentation of Results of Measurements; Rules for Rounding Off	p. 24
1.8 Basic Conventional Notations	p. 28
Chapter 2 Measuring Instruments and Their Properties	p. 29
2.1 Types of Measuring Instruments	p. 29
2.2 The Concept of an Ideal Instrument: Metrological Characteristics of Measuring Instruments	p. 32
2.3 Standardization of the Metrological Characteristics of Measuring Instruments	p. 36
2.4 Some Suggestions for Changing Methods of Standardization of Errors of Measuring Instruments and Their Analysis	p. 48
2.5 Dynamic Characteristics of Measuring Instruments and Their Standardization	p. 52
2.6 Statistical Analysis of the Errors of Measuring Instruments Based on Data Provided by Calibration Laboratories	p. 57
Chapter 3 Prerequisites for the Analysis of the Inaccuracy of Measurements and for Synthesis of Their Components	p. 61
3.1 Relationship Between Error and Uncertainty	p. 61
3.2 Classification of Elementary Errors	p. 62
3.3 Mathematical Models of Elementary Errors	p. 64
3.4 Methods for Describing Random Quantities	p. 66
3.5 Construction of the Composition of Uniform Distributions	p. 70
3.6 Universal Method for Constructing the Composition of Distributions	p. 74
3.7 Natural Limits of Measurements	p. 81
Chapter 4 Statistical Methods for Experimental Data Processing	p. 91
4.1 Requirements for Statistical Estimations	p. 91
4.2 Estimation of the Parameters of the Normal Distribution	p. 92
4.3 Outlying Results	p. 95
4.4 Construction of Confidence Intervals	p. 97
4.5 Methods for Testing Hypotheses About the Form of the Distribution Function of a Random Quantity	p. 101
4.6 Methods for Testing Sample Homogeneity	p. 103
4.7 Trends in Applied Statistics and Experimental Data Processing	p. 109
4.8 Example: Analysis of Measurement Results in Comparisons of Measures of Mass	p. 112
Chapter 5 Direct Measurements	p. 115
5.1 Relation Between Single and Multiple Measurements	p. 115
5.2 Identification and Elimination of Systematic Errors	p. 118
5.3 Estimation of Elementary Errors	p. 124
5.4 Method for Calculating the Errors and Uncertainties of Single Measurements	p. 128
5.5 Example: Calculation of Uncertainty in Voltage Measurements Performed with a Pointer-Type Voltmeter	p. 132
5.6 Methods for Calculating the Uncertainty in Multiple Measurements	p. 138
5.7 Comparison of Different Methods for Combining Systematic and Random Errors	p. 149
5.8 Essential Aspects of the Estimation of Measurement Errors when the Number of Measurements Is Small	p. 153
5.9 General Plan for Estimating Measurement Uncertainty	p. 155
Chapter 6 Indirect Measurements	p. 159
6.1 Basic Terms and Classification	p. 159
6.2 Correlation Coefficient and its Calculation	p. 160
6.3 The Traditional Method of Experimental Data Processing	p. 162
6.4 Shortcomings of the Traditional Method	p. 166
6.5 The Method of Reduction	p. 168
6.6 The Method of Transformation	p. 169
6.7 Errors and Uncertainty of Indirect Measurement Results	p. 174
Chapter 7 Examples of Measurements and Measurement Data Processing	p. 179
7.1 An Indirect Measurement of the Electrical Resistance of a Resistor	p. 179
7.2 The Measurement of the Density of a Solid Body	p. 182
7.3 The Measurement of Ionization Current by the Compensation Method	p. 189
7.4 The Measurement of Power at High Frequency	p. 192
7.5 The Measurement of Voltage with the Help of a Potentiometer and a Voltage Divider	p. 193
7.6 Calculation of the Uncertainty of the Value of a Compound Resistor	p. 197
Chapter 8 Combined Measurements	p. 201
8.1 General Remarks About the Method of Least Squares	p. 201
8.2 Measurements with Linear Equally Accurate Conditional Equations	p. 203
8.3 Reduction of Linear Unequally Accurate Conditional Equations to Equally Accurate Conditional Equations	p. 205
8.4 Linearization of Nonlinear Conditional Equations	p. 206
8.5 Examples of the Applications of the Method of Least Squares	p. 208
8.6 Determination of the Parameters in Formulas from Empirical Data and Construction of Calibration Curves	p. 213
Chapter 9 Combining the Results of Measurements	p. 219
9.1 Introductory Remarks	p. 219
9.2 Theoretical Principles	p. 219
9.3 Effect of the Error of the Weights on the Error of the Weighted Mean	p. 223
9.4 Combining the Results of Measurements in Which the Random Errors Predominate	p. 225
9.5 Combining the Results of Measurements Containing both Systematic and Random Errors	p. 226
9.6 Example: Measurement of the Activity of Nuclides in a Source	p. 233
Chapter 10 Calculation of the Errors of Measuring Instruments	p. 237
10.1 The Problems of Calculating Measuring Instrument Errors	p. 237
10.2 Methods for Calculating Instrument Errors	p. 238
10.3 Calculation of the Errors of Electric Balances (Unique Instrument)	p. 249
10.4 Calculation of the Error of ac Voltmeters (Mass-Produced Instrument)	p. 251
10.5 Calculation of the Error of Digital Thermometers (Mass-Produced Instrument)	p. 258
Chapter 11 Problems in the Theory of Calibration	p. 263
11.1 Types of Calibration	p. 263
11.2 Estimation of the Errors of Measuring Instruments in Verification	p. 265
11.3 Rejects of Verification and Ways to Reduce Their Number	p. 269
11.4 Calculation of a Necessary Number of Standards	p. 275
Chapter 12 Conclusion	p. 283
12.1 Measurement Data Processing: Past, Present, and Future	p. 283
12.2 Remarks on the "International Vocabulary of Basic and General Terms in Metrology"	p. 285
12.3 Drawbacks of the "Guide to the Expression of Uncertainty in Measurement"	p. 286
Appendix	p. 289
Glossary	p. 295
References	p. 299
Index	p. 303

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