# ANOVA - why are the mean squares chi-squared distributed?

I recently came across Analysis of Variance (ANOVA), and it seems that the total sum of squares of the residuals (or "within groups / treatments") divided by the degrees of freedom is Chi-squared distributed. I've also been told that this is the case for "between groups / treatments" under the null hypothesis.

I was wondering why this is the case, and roughly how we might go about showing this (I'm not after a rigorous proof, though).

One thing I'm wondering behind this question is are they both genuinely Chi-squared distributed, or is it an asymptotic result? (i.e. it holds in the limit as our sample size tends to infinity).

Thanks

Lastly, I was able to figure out why they're (proportional to) Chi-squared for the case of equal replication (slightly non-rigorously, but good enough for me), and this was a super-useful result / link. en.wikipedia.org/wiki/Cochran%27s_theorem#Sample_mean_and_sample_variance