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If sample standard deviation is biased, why do we use it in typical mean tests?

Why do we not use an unbiased estimator by dividing the sample standard deviation by C4?

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marked as duplicate by Scortchi May 11 at 20:44

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Some reasons (in no particular order):

  1. Not all variables are normal. Dividing the empirical standard deviation by C4 makes the estimator unbiased under the assumption of normality but not necessarily in other cases.
  2. Unbiased estimators are not necessarily precise. E.g. they may have high variance and thus high mean squared error.
  3. "Typical mean tests" have well known properties and the distributions of test statistics under the null (and even under some alternatives) are easy to work with.
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    $\begingroup$ Perhaps it's worth explicitly pointing out that merely rescaling a test statistic won't affect the results of a test one iota. $\endgroup$ – Scortchi May 11 at 20:46

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