Classical statistics estimation of the Normal distribution standard deviation when the mean is unknown

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The equation for the sample standard deviation (the unbiased estimate of the population standard deviation when the mean is unknown), is:

                               (1)

where x is the sample mean.  is provided by the Excel function STDEV. From the variance identities for uncorrelated variables, we have:

From Central Limit Theorem, we know , so:

In other words, the summation in Equation (1) has a variance equal to the sum of (n-1) Normal distributions, so the equation can be rewritten:

A Chisq(n) distribution is the sum of n squared unit Normal distributions:

so we get:

Using the pivotal method, we can rearrange the equation to provide an estimate of s :

This equation can be directly used in an ModelRisk example model to simulate uncertainty about s.