I need a bit of clarification here. In the list of length 100, each entry in that list is a binomial random variable X~Binom(10, .5). The variance of each of these binomials is Var(X) = 10(.5)(.5) = 2.5.

But I don't think that's the variance you're talking about. Do you mean the sample variance of the list of length 100? I'm sure that we can figure that out too (or at least what it should look like) but I want to confirm that's the variance you are interested in. Off the top of my head, I think this sample variance should have a roughly Chi-Square distribution, since Binom(10, .5) is close to normal (sort of). In the Wikipedia article, it explains how Chi-Square can be expressed a sum of normals squared (https://en.wikipedia.org/wiki/Chi-squared_distribution). I could maybe figure out the actual distribution if you really need it.

Then you ask what should the distribution look like? Is this answered by the answer above, or did you mean something related to the 50 samples in particular?

Thanks, I'm wondering how to calculate the distribution of sample variances. Yes, each variance is calculated from the list of 100 counts of #tails. It's just something that has bothered me for awhile.

Okay. I'll see what I can do. No promises though - I may have taught probability recently, but I left all the hard problems to my students ;)

I ended up using monte carlo, but it seems like something that should be known.

Approximately normal. See stuff like : http://onlinestatbook.com/2/sampling_distributions/samp_dist...

I am asking how to calculate the expected distribution of sample variances, not sample means.