I'm an energy engineer, so my knowledge on the argument is rather limited, so forgive me in case it's a stupid question.
This question is very linked to this: How can I find the standard deviation of the sample standard deviation from a normal distribution? but instead of drawing samples from one unique normal distribution now there are N normal distributions with null mean and each one with its own variance.
I have about 8000 samples (N=8000) drawn out of N normal distribution with 0 mean and each one its own standard deviation (I exactly know).
Given Y equal to the sample standard deviation, I would like to know its distribution (is it a chi distribution, what's the mean? the standard deviation?).
It is rather easy to compute the mean and the standard deviation of Y^2, which are respectively the average of the N variances and the square root of the double of the summation of (sigma_i)^4 divided by N^2.
In the link above it is possible to read how to find the bias of sample standard deviation if the samples were drawn from one unique normal distribution, is it possible to generalise it?