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

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

                                                                                    (1)

Each xi is a sample from a Normal(m,s) distribution, so Equation (1) can be rewritten:

       (2)

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

so Equation (2) becomes:

                                                                                          (3)

Using the pivotal method, we can rearrange Equation (3) to provide an estimate of s :

There is some possibility of getting  mixed up with the sample standard deviation when the mean is unknown, which has a denominator of (n-1), so the equation is better written:

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