Proof That the Pooled Standard Deviation is an Unbiased Estimator for the Standard Deviation if the Variances are Equal
The pooled standarddeviation is found by taking a weighted average of the variances oftwo samples,
and
andfinding the square root of the result. The equation is
We can easily show that
isan unbiased estimator for 
if
sinceandareboth unbiased estimators for
so that
and