While reading, I came across the puzzling statement that the sample mean and variance are uncorrelated only in symmetric distributions and there is strong correlation if the distribution is heavily skewed.
First of all, is it true? I already know this holds for the case of a normal population, but can that proved for every symmetric distribution? Although I think the result is counter-intuitive, I would appreciate simple arguments.
Thank you
Too long for a comment:
It is not true that sample mean and variance are always independent if the distribution is symmetric. For example, take a sample from a distribution which takes values $\pm1$ with equal probability: if the sample mean is $\pm1$ then the sample variance will be $0$, while if the sample mean is not $\pm1$ then the sample variance will be positive.
It is true that the distributions of the sample mean and variance have zero correlation (if they have a correlation) if the distribution is symmetric. This is because $E(s_X^2|\bar{X}-\mu=k)=E(s_X^2|\bar{X}-\mu=-k)$ by symmetry.
Neither of these points deal with the statement in the book, which says only if but not if.
For an example of the final statement, if most of a distribution is closely clustered but there can be the occasional particular very large value, then the sample mean will be largely determined by the number of very large values in the sample, and the more of them there are, the higher the sample variance will be too, leading to high correlation.
