5.13. Calculation of the weighted mean and its standard uncertainty

The weighted mean \(\overline{A}\) of activities \(A_{i}\) of the individual gamma lines is calculated according to the following equation:

\(\overline{A} = \frac{\sum_{i = 1}^{n}{\ \frac{A_{i}}{u^{2}\left( A_{i} \right)}}}{\sum_{i = 1}^{n}{\ \frac{1}{u^{2}\left( A_{i} \right)}}}\) (1)

The standard uncertainty of the weighted mean is calculated as follows:

\(u\left( \overline{A} \right) = \sqrt{\frac{1}{\sum_{i = 1}^{n}{\ \frac{1}{u^{2}\left( A_{i} \right)}}}} = u_{int}\left( \overline{A} \right)\) (2)

This is termed as “internal standard” deviation. It only considers the uncertainties of the individual activities and is Bayes compliant.

It may, however, happen, that the individual values \(A_{i}\) show deviations are larger than would be expected from the uncertainties \(u\left( A_{i} \right)\). In order to consider also an additional uncertainty component due to these “external” influences, the so-called “external standard deviation” is often used being defined as follows:

\(u_{ext}\left( \overline{A} \right) = \sqrt{\frac{\sum_{i = 1}^{n}{\ \frac{\left( A_{i} - \overline{A} \right)^{2}}{u^{2}\left( A_{i} \right)}}}{(n - 1)\sum_{i = 1}^{n}{\ \frac{1}{u^{2}\left( A_{i} \right)}}}}\) (3)

Note, that this type of standard deviation is, however, no longer Bayes compliant.

The factors \(1/u^{2}\left( A_{i} \right)\) within the sums contained in the equations (1-3) represent statistical weights. These must be considered as being constant. Nevertheless, before they are applied, they are calculated from other variable values \(x_{in}(j)\), which also contribute to the \(A_{i}\). If, after having calculated Eq. (1), followed by a numerical uncertainty propagation for \(\overline{A}\) with respect to the \(x_{in}(j)\) by using differences quotients follows, the values \(u^{2}\left( A_{i} \right)\) must not be modified. Under this constraint the uncertainty propagation for Eq. (1) directly yields the uncertainty given by Eq. (2). This condition is considered since version 2.4.05. The values \(u^{2}\left( A_{i} \right)\) may only be recalculated within the iterations for calculating the decision threshold and the detection limit, once per single iteration step.

UncertRadio is calculating both, internal and external standard uncertainty but for further internal calculations it makes only use of the internal standard uncertainty. The external standard deviation is shown only for information. The reason for doing that is that for the user it might be an useful information that the external standard deviation is considerably larger than the internal standard deviation. The latter can give advice about possible sources of errors which should be considered.

Covariances between peak efficiency values, taken from the same efficiency curve, are part of the covariance matrix Ux of the input values x:

\(cov\left( A_{m},A_{k} \right) = \frac{A_{m}}{\varepsilon_{m}}\frac{A_{k}}{\varepsilon_{k}} \cdot cov\left( \epsilon_{m},\epsilon_{k} \right)\)

Important note: According to Cox et al. (2006b), the equations for the weighted mean and its uncertainty given here can only be considered as “good” approximations, if covariances exist between the individual activity values. In such a case, instead, a least-squares procedure for the mean is to be applied. The corresponding procedure is described in Section 5.14.