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I'm trying to recreate the graph from here using equations (2) and (3), but I've yet to do so successfully. I'm not sure that equation (2) is correct. I traced it back to the original source here (see equation 7.39) where an unbiasing constant $c_5$ is used.

Am I missing something with how I'm supposed to calculate/apply the unbiasing constant in this formula? I just don't see how they arrive at the graph using the formulas on the Mathworld page.

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    $\begingroup$ I believe there is a typographical error on that Wolfram page: "$\sigma^2 = Ns^2/(N-1)$" should instead read "$\sigma^2 = E[Ns^2/(N-1)].$" With that in mind, define a function $f[s\_,n\_,\sigma\_]$ as given in equation (2) and plot it as in Plot[Evaluate@Table[f[s, n, 1], {n, 2, 12, 2}], {s, 0, 2}]. $\endgroup$
    – whuber ♦
    Apr 24, 2019 at 19:52
  • $\begingroup$ @whuber If I'm looking to calculate $\sigma^2$ what effect will the expectation add? Say $s = 2$ and $N = 10$, how does my answer change? $\endgroup$
    – cjken
    Apr 24, 2019 at 21:09
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    $\begingroup$ You cannot calculate $\sigma^2$: it must be specified. It is a parameter of the underlying Normal distribution. $\endgroup$
    – whuber ♦
    Apr 24, 2019 at 21:21

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My apology for having to paste in a picture of equations, but I am still learning how to format equations properly using MathJax and I will replace the picture ASAP!

PDFs for s expressions

The "unbiasing constants" are $E[•]/\sigma$, as relevant.

Below is a plot of six $s'$ PDFs that I computed from the $p_{s'}(s')$ PDF given above. These are for N = 2, 4, ..., 12, with the same color coding as at the Wolfram link, and $\sigma = 1$. So $p_{s'}(s') = f_N(s)$ and $\nu = N-1$. N.B. In Wolfram's $f_N(s)$, the $s$ is what I call $s'$ above.

Plot of s' PDFs

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