Type A evaluation of standard
2.1 In most cases, the best available estimate of the expectation or expected value μq of a quantity q that varies randomly [a random variable (A.1)], and for which n independent observations qk have been obtained under the same conditions of measurement (see A.2), is the arithmetic mean or average /q (A.3) of the n observations:
Thus,for an input quantity Xi estimated from n independent repeated observations Xik the arithmetic mean /X obtained from Equation (3) is used as the input estimate Xi in Equation (2) to determine the measurement result y;that is Xi =/x. Those input estimates not evaluated from repeated observations must be obtained by other methods, such as those indicated in the second category of 1.3.
2.2 The individual observations qk differ in value because of random variations in the influence quantities, or random effects. The experimental variance of the observations, which estimates the variance σ2 of the probability distribution of q, is given by
This estimate of variance and its positive square root s(qk), termed the experimental standard deviation (A.4), characterize the variability of the observed values qk, or more specifically, their dispersion about their mean /q.
The best estimate of
, the variance of the mean, is given by
The experimental variance of the mean s2(/q) and the experimental standard deviation of the mean (A.4, Note 2), equal to the positive square root of s2(/q), quantify how well /q estimates the expectation μq of q, and either may be used as a measure of the uncertainty of /q.
Thus,for an input quantity Xi determined from n independent repeated observations Xik,
the standard uncertainty
of its estimate
calculated according to Equation (5). For convenience,
are sometimes called a Type A variance and a Type A standard uncertainty, respectively.
The number of observations n should be large enough to ensure that /q provides a reliable estimate of the expectation μq of the random variable q and that provides a reliable estimate of the variance (see 3.2, note).The difference between s2(/q) and σ2(/q) must be considered when one constructs confidence intervals. In this case, if the probability distribution of q is a normal distribution (see 3.4), the difference is taken into account through the t-distribution.
Although the variance s2(/q) is the more fundamental quantity, the standard deviation s(/q) is more convenient in practice because it has the same dimension as q and a more easily comprehended value than that of the variance.
2.4 For a well-characterized measurement under statistical control, a combined or pooled estimate of variance s2p (or a pooled experimental standard deviation Sp) that characterizes the measurement may be available. In such cases, when the value of a measured q is determined from n independent observations, the experimental variance of the arithmetic mean /q of the observations is estimated better by than by and the standard uncertainty is .
2.5 Often an estimate Xi of an input quantity Xi is obtained from a curve that has been fitted to experimental data by the method of least squares. The estimated variances and resulting standard uncertainties of the fitted parameters characterizing the curve and of any predicted points can usually be calculated by well-known statistical procedures.
2.6 The degrees of freedom (A.5) Vi of , equal to n - 1 in the simple case where and are calculated from n independent observations as in 2.1 and 2.3, should always be given when Type A evaluations of uncertainty components are documented.
2.7 If the random variations in the observations of an input quantity are correlated, for example, in time, the mean and experimental standard deviation of the mean as given in 2.1 and 2.3 may be inappropriate estimators (A.6) of the desired statistics (A.7). In such cases, the observations should be analysed by statistical methods specially designed to treat a series of correlated, randomly-varying measurements.
Such specialized methods are used to treat measurements of frequency standards. However, it is possible that as one goes from short-term measurements to long-term measurements of other metrological quantities, the assumption of uncorrelated random variations may no longer be valid and the specialized methods could be used to treat these measurements as well.
2.8 The discussion of Type A evaluation of standard uncertainty in 2.1 to 2.7 is not meant to be exhaustive; there are many situations, some rather complex, that can be treated by statistical methods. An important example is the use of calibration designs, often based on the method of least squares, to evaluate the uncertainties arising from both short- and long-term random variations in the results of comparisons of material artefacts of unknown values, such as gauge blocks and standards of mass, with reference standards of known values. In such comparatively simple measurement situations, components of uncertainty can frequently be evaluated by the statistical analysis of data obtained from designs consisting of nested sequences of measurements of the measured for a number of different values of the quantities upon which it depends — a so-called analysis of variance.
At lower levels of the calibration chain, where reference standards are often assumed to be exactly known because they have been calibrated by a national or primary standards laboratory, the uncertainty of a calibration result may be a single Type A standard uncertainty evaluated from the pooled experimental standard deviation that characterizes the measurement.