# pooled variance

In statistics, pooled variance is a method for estimating variance of several different populations when the mean of each population may be different, but one may assume that the variance of each population is the same. If the populations are indexed $i = 1,\ldots,k$, then the pooled variance $s^2_p$ can be estimated by the weighted average of the sample variances $s^2_i$

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

where $n_i$ is the sample size of population $i$. Use of $(n_i-1)$ weighting factors instead of $n_i$ comes from Bessel’s correction.

Under the assumption of equal population variances, the pooled sample variance provides a higher precision estimate of variance than the individual sample variances. This higher precision can lead to increased statistical power when used in statistical tests that compare the populations, such as the t-test.

The square-root of a pooled variance estimator is known as a pooled standard deviation.