1.2.3. Combined standard uncertainty

The combined standard uncertainty \(u_{c}(y)\) of the output quantity \(y\) is determined according to ISO GUM (ISO Guide on Uncertainty of Measurement (1995); see also EURACHEM / CITAC Guide “Quantifying Uncertainty in Analytical Measurement” (2000)) using the “Gaussian law of propagating uncertainties” and taking covariances between individual input quantities.

For a type A standard uncertainty of a quantity which must be derived from a set of repeated measurements a necessary small statistical evaluation of mean and standard deviation has to be performed outside this program; only mean and standard deviation will be used in UncertRadio. Usually, with measurements of activities the greater part of uncertainties of quantities belongs to Type B. Within the framework of the Bayesian theory of measurement (Weise & Wöger, 1999; Weise et al., 2006) which is underlying the basics of this program an explicit differentiation between quantities of type A and B is not necessary. This is also the reason that degrees of freedom are not considered in this program.

The analytical derivation of formulae for the combined uncertainty for instance of a mass or volume dependent activity using the law of propagating uncertainties may easily yield a certain number of less or more complex formulae the correctness of which is often not easily being controlled. Therefore, a numerical procedure is applied. The first step consists in transferring all quantities/parameters being required for the calculations, these may easily become more than 20, into a program array MeasdValue(i) = p(i). Then, in a subprogram RESULT the value of the output quantity is calculated from the values of the array elements p(i) by using the function parser.

Similarly, the known measurement uncertainties of the individual quantities/parameters are transferred to an array StdUnc(i) = u(i). For the calculation of the combined uncertainty a subroutine Uncpropa is used to which the two arrays p(i) and u(i) are transferred. Covariances are considered in this subroutine. The sensitivity coefficients, i.e. the partial derivates of the function calculated with RESULT with respect to the array elements p(i), are numerically approximated by differential quotients.

Further details: see Uncertainty propagation.