Proof That the Pooled Standard Deviation is an Unbiased Estimator For the Standard Deviation if The Population Variances are Equal

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The pooled standard deviation is found by taking a weighted average of the variances of two samples,andand finding the square root of the result. The equation is

We can easily show thatis an unbiased estimator for if

sinceandare both unbiased estimators for

so that

and

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