Preferred Label : pooled standard deviation;
IUPAC definition : 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 precision
although their means may differ, the pooled standard deviations s p from k series
of measurements can be calculated as \[s_{{p}} \sqrt{\frac{(n_{1}- 1)\ s_{1} {2} (n_{2}-
1)\ s_{2} {2} ... (n_{k}- 1)\ s_{k} {2}}{n_{1} n_{2} ... n_{k}- k}}\] The suffices
1, 2, ..., k refer to the different series of measurements. In this case it is assumed
that there exists a single underlying standard deviation σ of which the pooled standard
deviation s 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 \[s_{{p}} \sqrt{\frac{\sum (x_{i1}- x_{i2})
{2}}{2\ k}}\] Results from various series of measurements can be combined in the following
way to give a pooled relative standard deviation s r,p: \[s_{{r,p}} \sqrt{\frac{\sum
(n_{i}-1)\ s_{{r,}i} {2}}{\sum n_{i}- 1}} \sqrt{\frac{\sum (n_{i}-1)\ s_{i} {2}\
x_{i} {-2}}{\sum n_{i}-1}}\];
Origin ID : P04758;
See also
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 precision
although their means may differ, the pooled standard deviations s p from k series
of measurements can be calculated as \[s_{{p}} \sqrt{\frac{(n_{1}- 1)\ s_{1} {2} (n_{2}-
1)\ s_{2} {2} ... (n_{k}- 1)\ s_{k} {2}}{n_{1} n_{2} ... n_{k}- k}}\] The suffices
1, 2, ..., k refer to the different series of measurements. In this case it is assumed
that there exists a single underlying standard deviation σ of which the pooled standard
deviation s 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 \[s_{{p}} \sqrt{\frac{\sum (x_{i1}- x_{i2})
{2}}{2\ k}}\] Results from various series of measurements can be combined in the following
way to give a pooled relative standard deviation s r,p: \[s_{{r,p}} \sqrt{\frac{\sum
(n_{i}-1)\ s_{{r,}i} {2}}{\sum n_{i}- 1}} \sqrt{\frac{\sum (n_{i}-1)\ s_{i} {2}\
x_{i} {-2}}{\sum n_{i}-1}}\]
