Multiplying the standard sample variance as computed in that fashion by ( equivalently, using instead of in the estimator's formula ) corrects for this, and gives an unbiased estimator of the population variance.
32.
Now a question arises : is the estimate of the population variance that arises in this way using the sample mean " always " smaller than what we would get if we used the population mean?
33.
That is why the sum of squares of the deviations from the " sample " mean is too small to give an unbiased estimate of the population variance when the average of those squares is found.
34.
For example, consider again the estimation of an unknown population variance & sigma; 2 of a Normal distribution with unknown mean, where it is desired to optimise " c " in the expected loss function
35.
However, if the population variances are equal, or if the samples are rather small and the population variances can be assumed to be approximately equal, it is more accurate to use Student's t-test,.
36.
However, if the population variances are equal, or if the samples are rather small and the population variances can be assumed to be approximately equal, it is more accurate to use Student's t-test,.
37.
In other situations, it makes more sense to think of a robust measure of scale as an estimator of its own expected value, interpreted as an alternative to the population variance or standard deviation as a measure of scale.
38.
It is true that on average it underestimates the population variance, but to call it a serious error is nonsense : sometimes biased estimators perform better-- indeed in some cases " far " better-- than unbiased ones.
39.
If the " biased sample variance " ( the second central moment of the sample, which is a downward-biased estimate of the population variance ) is used to compute an estimate of the population's standard deviation, the result is
40.
For example, the square root of the unbiased estimator of the population variance is " not " a mean-unbiased estimator of the population standard deviation : the square root of the unbiased sample variance, the corrected sample standard deviation, is biased.
