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: