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,andandfinding the square root of the result. The equation is

We can easily show thatisan unbiased estimator for if

sinceandareboth unbiased estimators for

so that

and