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For a given set of data randomly sampled from a Normal distribution, whose mean m is unknown and unknown standard deviation s, the distribution of uncertainty of the true standard deviation is calculated from the formula:



\sigma =\widehat{\sigma}\sqrt{\frac{n-1}{\chi^{2}(n-1)}}


where \chi^{2}(n-1) is a chi-squared distribution with n-1 degrees of freedom. It has a mean of (n-1) so the term in the square root is centered on 1, which means that the uncertainty distribution for s is centered around \widehat{\sigma}.


[This page provides an explanation of the derivation]


The models below show how to sample from the uncertainty distributions of each parameter:



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