I know how to estimate population standard deviation using a chi square distribution, but I don't know why it works. I'd like to have an intuition for the why.

I've tried googling around. I always find descriptions of the how, not the why. My guess as to what's going on:

  • The distribution of sample SDs is normally distributed (sorta makes sense, but I don't really understand why).
  • If they're normally distributed, when you square it, it becomes a chi square distribution with 1 degree of freedom (this makes sense to me). Since SD squared is Var, we have a chi square distribution of Vars.
  • From there, once we have a distribution, we can say that X% of the sample Vars are within a range (U, L).
  • There's some sort of adjustment, because sample Var is a biased estimator of population Var.
  • Where do multiple degrees of freedom come in to play?
  • $\begingroup$ "I know how to estimate population standard deviation using a chi square distribution" — Exactly what procedure are you referring to? Are you asking a question about Bessel's correction? $\endgroup$ Nov 25 '16 at 1:34
  • $\begingroup$ Squareroot((n-1)(s^2)/chi value) $\endgroup$ Nov 25 '16 at 2:40
  • $\begingroup$ See here and a bit lower down $\endgroup$
    – Glen_b
    Nov 25 '16 at 4:07

No I don't believe your rationale is correct. If X has a normal distribution then its sample variance is proportional to a chi square distribution with n-1 degrees of freedom where n is the sample size. So the chi-square distribution can be used to draw inferences about the population variance and hence also the standard deviation. But if X is not normally distributed there is no reason to use the chi-square distribution.


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