pooled standard deviation

https://doi.org/10.1351/goldbook.P04758
A problem often arises when the combination of several series of measurements performed under similar conditions is desired to achieve an improved estimate of the imprecision of the process. If it can be assumed that all the series are of the same @P04799@ although their means may differ, the pooled standard deviations \(s_{\text{p}}\) from \(k\) series of measurements can be calculated as \[s_{\text{p}}=\sqrt{\frac{(n_{1}- 1)\ s_{1}^{2}+(n_{2}- 1)\ s_{2}^{2}+...+(n_{k}- 1)\ s_{k}^{2}}{n_{1}+n_{2}+...+n_{k}- k}}\] The suffices \(1,2,...,k\) refer to the different series of measurements. In this case it is assumed that there exists a single underlying @S05911@ \(\sigma \) of which the pooled @S05911@ \(s_{\text{p}}\) is a better estimate than the individual calculated standard deviations \(s_{1},s_{2},...,s_{k}\), For the special case where \(k\) sets of duplicate measurements are available, the above equation reduces to \[s_{\text{p}}=\sqrt{\frac{\sum (x_{i1}- x_{i2})^{2}}{2\ k}}\] Results from various series of measurements can be combined in the following way to give a @P04757@ \(s_{\text{r,p}}\): \[s_{\text{r,p}} = \sqrt{\frac{\sum (n_{i}-1)\ s_{\text{r,}i}^{2}}{\sum n_{i}- 1}} = \sqrt{\frac{\sum (n_{i}-1)\ s_{i}^{2}\ x_{i}^{-2}}{\sum n_{i}-1}}\]
Source:
PAC, 1981, 53, 1805. (Assignment and Presentation of Uncertainties of the Numerical Results of Thermodynamic Measurements) on page 1821 [Terms] [Paper]