So for example an unbiased estimate of the population variance ( the second central moment ) is given by
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For such small samples, a test of equality between the two population variances would not be very powerful.
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Conversely, the sample variance is an unbiased estimate of the population variance, but is not Fisher consistent.
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Mo et al . refined the idea by noting that the population variance of the feature should be taken into account.
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The computational formula for the population variance follows in a straightforward manner from the linearity of expected values and the definition of variance:
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The true distribution of the test statistic actually depends ( slightly ) on the two unknown population variances ( see Behrens Fisher problem ).
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Under the assumption of equal population variances, the pooled sample variance provides a higher precision estimate of variance than the individual sample variances.
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However, in the Behrens Fisher problem, the two population variances are not known to be equal, nor is their ratio known.
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Estimating the population variance by taking the sample's variance is close to optimal in general, but can be improved in two ways.
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The former can give an unbiased s _ p ^ 2 to estimate \ sigma ^ 2 when the two groups share an equal population variance.
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