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 vii vüi Contents

We have used a Monte Carlo simulation for investigating the output of a digital instrument, of resolution half-width a, when its analogue input is a Gaussian signal. The digitizing process converts an actual mean to an observed mean and an actual variance to an observed variance. The resulting relationships are plotted as two sets of graphs, in one of which the observed mean is the parameter, while in the other the observed variance is the parameter. Given an observed mean and an observed variance, it is then easy to infer from the graphs the corresponding actual mean and actual variance. The graphs illustrate, for example, that unless the observed variance is very low, the actual variance can be recovered from the observed variance by subtracting a 2 /3 from the observed variance.

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An artefact standard may be used in a comparison between two laboratories to measure the difference between their respective realizations of a physical unit. In the usual three-stage sequence, one laboratory, the "pilot" laboratory, measures the artefact before and after its measurement at the second laboratory. Three procedures are described for analysing the results of such a comparison: "deviation from fit to pilot" (DFP), "overall constant drift" (OCD, which yields similar results to DFP for any difference in the laboratories' realizations but which is statistically more rigorous and also gives quite different estimates of uncertainties), and "separately fitted lines" (SFL). If OCD is used, the uncertainty of the comparison is minimized if the measurements at the two laboratories have the same mean epoch. If SFL is used, the uncertainty is minimized if the intervals between successive epochs at the second laboratory are invariant under time-reversal. SFL allows transportation effects to be estimated as Type A uncertainties, separately from residual scatter, and can signal their presence through a sharp reduction in degrees of freedom. Computer simulations of comparisons indicate that SFL is at least as accurate as DFP and OCD, and is often more accurate. Two case studies of Zener-based voltage standard artefacts are given.

Tristimulus response functions are either calculated from spectral-irradiance data or measured directly and used to calculate parameters representing colour. The tristimulus responses are correlated because they depend on the same spectral information. In this paper we derive and confirm the correlation coefficients between the tristimulus response functions. These coefficients are important in the estimation of uncertainty for colour coordinates.