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$$
s_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2+...+(n_k-1)s_k^2}{n_1+n_2+...+n_k-k}}
$$
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 $\sigma$ 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
$$
s_p=\sqrt{\frac{\sum(x_{i1}-x_{i2})^2}{2k}}
$$
Results from various series of measurements can be combined in the following way to give a pooled relative standard deviation $s_{r,p}$:
$$
s_p=\sqrt{\frac{\sum(n_i-1)s_{r,i}^2}{\sum n_i -1}}
$$
== See also ==
[[Category:Analysis]]