# Why is a symmetric distribution sufficient for the sample mean and variance to be uncorrelated?

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

• Perhaps you can tell us where you found the statement about the sample mean and variance being independent in symmetric distributions? Feb 25, 2014 at 23:54
• It is on page 425 of Hogg and Craig's Introduction to Mathematical Statistics, 7th edition. Feb 25, 2014 at 23:58
• Relevant threads on Math.SE: math.stackexchange.com/a/111129/321264, math.stackexchange.com/q/235634/321264. Jun 25, 2020 at 15:02

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.

• I take it that you looked up the book? Feb 26, 2014 at 0:28
• No - just what you have said in the question Feb 26, 2014 at 0:32
• Thank you, I think that the conditional expectation you outline more or less sums it. Feb 26, 2014 at 0:33
• Part of this answer is unfortunately not correct. In the first part you talk about independence but the question doesn't ask about independence, it asks about zero correlation. It's not sufficient to show they're not independent when proving zero correlation, since variables can be dependent but have zero correlation. Jun 11, 2015 at 16:40
• @Glen_b Henry clearly stated the assumption you are picking on: "if they have a correlation." I suspect you may have overlooked this parenthetical remark when you asserted the second part of the answer is incorrect.
– whuber
Jun 11, 2015 at 16:45