Measurement Errors and Uncertainties (eBook)

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 5
Contents 9
General Information About Measurements 13
1.1. Basic Concepts and Terms 13
1.2. Metrology and the Basic Metrological Problems 15
1.3. Initial Points of the Theory of Measurements 22
1.4. Classi.cation of Measurements 27
1.5. Classi.cation of Measurements Errors 32
1.6. Principles of Estimation of Measurement Errors and Uncertainties 34
1.7. Presentation of Results of Measurements Rules for Rounding Off
1.8. Basic Conventional Notations 39
Measuring Instruments and Their Properties 41
2.1. Types of Measuring Instruments 41
2.2. The Concept of an Ideal Instrument: Metrological Characteristics of Measuring Instruments 44
2.3. Standardization of the Metrological Characteristics of Measuring Instruments 48
2.4. Some Suggestions for Changing Methods of Standardization of Errors of Measuring Instruments and Their Analysis 60
2.5. Dynamic Characteristics of Measuring Instruments and Their Standardization 64
2.6. Statistical Analysis of the Errors of Measuring Instruments Based on Data Provided by Calibration Laboratories 69
Statistical Methods for Experimental Data Processing 103
4.1. Requirements for Statistical Estimations 103
4.2. Estimation of the Parameters of the Normal Distribution 104
4.3. Outlying Results 107
4.4. Construction of Con.dence Intervals 109
4.5. Methods for Testing Hypotheses About the Form of the Distribution Function of a Random Quantity 113
4.6. Methods for Testing Sample Homogeneity 115
4.7. Trends in Applied Statistics and Experimental Data Processing 121
4.8. Example: Analysis of Measurement Results in Comparisons of Measures of Mass 124
Direct Measurements 127
5.1. Relation Between Single and Multiple Measurements 127
5.2. Identi.cation and Elimination of Systematic Errors 130
5.3. Estimation of Elementary Errors 136
5.4. Method for Calculating the Errors and Uncertainties of Single Measurements 140
5.5. Example: Calculation of Uncertainty in Voltage Measurements Performed with a Pointer- Type Voltmeter 144
5.6. Methods for Calculating the Uncertainty in Multiple Measurements 150
5.7. Comparison of Different Methods for Combining Systematic and Random Errors 161
5.8. Essential Aspects of the Estimation of Measurement Errors when the Number of Measurements Is Small 165
5.9. General Plan for Estimating Measurement Uncertainty 167
Indirect Measurements 170
6.1. Basic Terms and Classi.cation 170
6.2. Correlation Coef .cient and its Calculation 171
6.3. The Traditional Method of Experimental Data Processing 173
6.4. Shortcomings of the Traditional Method 177
6.5. The Method of Reduction 179
6.6. The Method of Transformation 180
6.7. Errors and Uncertainty of Indirect Measurement Results 185
Examples of Measurements and Measurement Data Processing 190
7.1. An Indirect Measurement of the Electrical Resistance of a Resistor 190
7.2. The Measurement of the Density of a Solid Body 193
7.3. The Measurement of Ionization Current by the Compensation Method 200
7.4. The Measurement of Power at High Frequency 203
7.5. The Measurement of Voltage with the Help of a Potentiometer and a Voltage Divider 204
7.6. Calculation of the Uncertainty of the Value of a Compound Resistor 208
Combined Measurements 211
8.1. General Remarks About the Method of Least Squares 211
8.2. Measurements with Linear Equally Accurate Conditional Equations 213
8.3. Reduction of Linear Unequally Accurate Conditional Equations to Equally Accurate Conditional Equations 215
8.4. Linearization of Nonlinear Conditional Equations 216
8.5. Examples of the Applications of the Method of Least Squares 218
8.6. Determination of the Parameters in Formulas from Empirical Data and Construction of Calibration Curves 223
Combining the Results of Measurements 229
9.1. Introductory Remarks 229
9.2. Theoretical Principles 229
9.3. Effect of the Error of the Weights on the Error of the Weighted Mean 233
9.4. Combining the Results of Measurements in Which the Random Errors Predominate 235
9.5. Combining the Results of Measurements Containing both Systematic and Random Errors 236
9.6. Example: Measurement of the Activity of Nuclides in a Source 243
Calculation of the Errors of Measuring Instruments 246
10.1. The Problems of Calculating Measuring Instrument Errors 246
10.2. Methods for Calculating Instrument Errors 247
10.3. Calculation of the Errors of Electric Balances ( Unique Instrument) 258
10.4. Calculation of the Error of ac Voltmeters ( Mass- Produced Instrument) 260
10.5. Calculation of the Error of Digital Thermometers ( Mass- Produced Instrument) 267
Problems in the Theory of Calibration 271
11.1. Types of Calibration 271
11.2. Estimation of the Errors of Measuring Instruments in Veri . cation 273
11.3. Rejects of Veri.cation and Ways to Reduce Their Number 277
11.4. Calculation of a Necessary Number of Standards 283
Conclusion 290
12.1. Measurement Data Processing: Past, Present, and Future 290
12.2. Remarks on the “International Vocabulary of Basic and General Terms in Metrology” 292
12.3. Drawbacks of the “Guide to the Expression of Uncertainty in Measurement” 293
Appendix 295
Glossary 301
References 305
Index 308

5 Direct Measurements (p. 115-116)

5.1. Relation Between Single and Multiple Measurements

The classical theory of measurement errors is constructed based on the welldeveloped statistical methods and pertains to multiple measurements. In practice, however, the overwhelming majority of measurements are single measurements, and however strange it may seem, for this class of measurements, there is no accepted method for estimating errors.

In searching for a solid method for estimating errors in single measurements, it is first necessary to establish the relation between single and multiple measurements. At first glance, it seems natural to regard single measurements as a particular case of multiple measurements, when the number of measurements is equal to 1. Formally this is correct, but it does not give anything, because statistical methods do not work when n = 1. In addition, the question of when one measurement is suf.cient remains open. In the approach examined, to answer this question—and this is the fundamental question—it is first necessary to perform a multiple measurement, and then, analyzing the results, to decide whether a single measurement was possible. But such an answer is in general meaningless: A multiple measurement has already been performed, and nothing is gained by knowing, for example, that in a given case, one measurement would have suf.ced. Admittedly, it can be objected that such an analysis will make it possible not to make multiple measurements when future such measurements are performed. Indeed, that is what is done, but only when preliminary measurements are performed, i.e., in scientific investigations when some new object is studied. This is not done in practical measurements.

When it is necessary to measure, for example, the voltage of some source with a given accuracy, a voltmeter with suitable accuracy is chosen and the measurement is performed. If, however, the numbers on the voltmeter indicator dance about, then it is impossible to perform a measurement with the prescribed accuracy, and the measurement problem must be reexamined rather than performing a multiple measurement. For practical applications, we can state the opinion that single measurements are well founded by experience, concentrated in the construction of the corresponding measuring instruments, and that measuring instruments are manufactured so that single measurements could be performed.

From the foregoing assertion a completely different point of view follows regarding the relationship between single and multiple measurements, namely, that single measurements are the primary, basic form of measurement, whereas multiple measurements are derived from single measurements. Multiple measurements are performed when necessary, based on the formulation of the measuring problem. It is interesting that these problems are known beforehand, they can even be enumerated. Namely, multiple measurements are performed as follows: (a) When investigating a new phenomenon or a new object and relationships between the quantities characterizing the object, as well as their connection with other physical quantities, are being determined, or briefiy, when preliminary measurements, according to the classification given in Chapter 1, are performed.

(b) When measuring the average value of some parameter, corresponding to the goal of the measuring problem.
(c) When the effect of random errors of measuring instruments must be reduced.
(d) In measurements for which measuring instruments have not yet been developed.

Of the four cases presented above, the first is typical for investigations in science and the third is typical for calibration practice. There is another point of view, namely, that any measurement must be a multiple measurement, because otherwise it is impossible to judge the measurement process and its stability and to estimate its inaccuracy.