MEASUREMENT UNCERTAINTIES IN SCIENCE AND TECHNOLOGY

Michael Grabe

Springer  Textbook, Physik & Astronomie, 10. März 2005, 269 p. 56 Illus. ISBN: 3-540-20944-1

At the turn of the 19th century, Carl Friedrich Gauß founded error calculus by predicting the then unknown position of the planet Ceres. Ever since, error calculus has occupied a place at the heart of science.

In this book, Grabe illustrates the breakdown of traditional error calculus in the face of modern measurement techniques.  Revising Gauß' error calculus ab initio, he treats random and unknown systematic errors on an equal footing from the outset. Furthermore, Grabe also proposes what may be called well defined measurement conditions, a prerequisite for defining confidence intervals that are consistent with basic statistical concepts. The resulting measurement uncertainties  are as robust and reliable as required by modern-day science, engineering and technology.

CONTENTS

CHARACTERIZATION, COMBINATION AND PROPAGATION OF ERRORS

1. PROPERTIES OF RESULTS OF MEASUREMENT
1.1 Basic ideas of measurement
1.2 Quotation of numerical results and rounding

2. FORMALISATION OF PROCESSES OF MEASUREMENT
2.1 Fundamental equation of measurement
2.2 Random variables, probability densities and parent distributions
2.3 Elements of evaluation

3. DENSITIES DERIVED FROM NORMAL PARENT DISTRIBUTIONS
3.1 One dimensional normal density
3.2 Multidimensional density
3.3 Chi-Square density and F-density
3.4 Student's (Gosset's) density
3.5 Fisher's density
3.6 Hotelling's density

4. ESTIMATORS AND EXPECTATIONS
4.1 Statistical ensemble
4.2 Variances and Covariances
4.3 Elementary model of analysis of variance

5. COMBINATION OF MEASUREMENT ERRORS
5.1 Expectation of the arithmetic mean
5.2 Uncertainty of the arithmetic mean
5.3 Uncertainty of a function of one variable
5.4 Systematic errors of measurands

6. PROPAGATION OF MEASUREMENT ERRORS
6.1 Taylor's series expansion
6.2 Expectation of the arithmetic mean
6.3 Uncertainty of the arithmetic mean
6.4 Uncertainties of functional relationships
6.5 Error propagation covering several stages

7. LEAST SQUARES FORMALISM

8. CONSEQUENCES OF SYSTEMATIC ERRORS
8.1 Structure of solution vector
8.2 Vector of input data
8.3 Gauß-Markoff Theorem
8.4 Choice of the weights
8.5 Consistency of the input data

9. UNCERTAINTIES OF LEAST SQUARES ESTIMATORS
9.1 Empirical variance-covariance matrix
9.2 Propagation of systematic errors
9.3 Overall uncertainties
9.4 Uncertainty iof a function
9.5 Uncertainty spaces

LINEAR AND LINEARIZED SYSTEMS

10. SYSTEMS WITH TWO PARAMETERS
10.1 Straight lines
10.2 Linearisation
10.3 Transformation

11. SYSTEMS WITH THREE PARAMETERS
11.1 Planes
11.2 Parabolae
11.3 Approximation by piecewise smooth functions
11.4 Circles

12. SPECIAL METROLOGICAL SYSTEMS
12.1 Dissemination of units
12.3 Pairwise comparisons
12.4 Fundamental constants of physics
12.5 Empirical function

APPENDIX

A. On the rank of some matrices
B. Variance-Covariance matrices
C. Linear funktions of normally distributed variables
D. Orthogonal projections
F. Expansion of solution vectors
G. Student's density
H. Quantiles of  Hotelling's density

REFERENCES

INDEX