# Pooled standard deviation

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Pooled standard deviation is a way to find a better estimate of the true standard deviation given several different samples taken in different circumstances where the mean may vary between samples but the true standard deviation (precision) is assumed to remain the same. It is calculated by

$s_p=\sqrt{\frac{\sum_{i=1}^k((n_i - 1)s_i^2)}{\sum_{i=1}^k(n_i - 1)}}$

or with simpler notation,

$s_p=\sqrt{\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2+\cdots+(n_k - 1)s_k^2)}{n_1+n_2+\cdots+n_k - k}}$

where sp is the pooled standard deviation, ni is the sample size of the i'th sample, si is the standard deviation of the i'th sample, and k is the number of samples being combined. n−1 is used instead of n for the same reason it may be used in calculating standard deviations from samples.