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# Type A evaluation of standard uncertainty

**Type A evaluation of standard
uncertainty**

Thus,for an input quantity *X*_{i} estimated from n independent repeated observations *X*_{ik} the arithmetic mean /X obtained from Equation (3) is used as the input estimate *X*_{i} in Equation (2) to determine the measurement result y;that is *X*_{i} =/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.

This estimate of variance and its positive square root s(*q*_{k}), termed the experimental standard deviation (A.4), characterize the variability of the observed values *q*_{k}, or more specifically, their dispersion about their mean /q.

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* X*_{ik,} the standard uncertainty of its estimate is with *S*^{2}*(/X)* calculated according to Equation (5). For convenience, and 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 s^{2}(/q) and σ^{2}(/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 s^{2}(/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 .

**2.6** 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.

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.

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.