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?