TITLE: Intuitive explanation for dividing by n-1 when calculating sample variance?
QUESTION [7 upvotes]: I understand how to mathematically show that the sample variance (that involves dividing by n-1) is an unbiased estimator of the population variance (which divides by n), and the mathematics has been shown many times here on Math.SE.
I am wondering however if there is an intuitive way to understand this result that I can use to easily explain why this is done to layman.  So far I have seen many derivations but I haven't seen an elegant intuitve explanation for the result.

REPLY [3 votes]: The sample variance is computed using deviations from the sample mean $\bar x$ instead of the population mean $\mu$, and this is the source of the bias.
When the squared deviations are accumulated, there is a deficit of $\bar x-\mu$ every time, so that the computed variance is too small.

Hint:
$$(x_i-\mu)^2=((x_i-\bar x)+(\bar x-\mu))^2=(x_i-\bar x)^2+2(x_i-\bar x)(\bar x-\mu)+(\bar x-\mu)^2.$$
When averaging over $i$, the double product in the middle vanishes due to the factor $(\bar x-\bar x)$, and you get
$$\sigma^2=\overline{(x_i-\mu)^2}=\overline{(x_i-\bar x)^2}+\overline{(\bar x-\mu)^2}=s^2+\frac{\sigma^2}N.$$