Intuitive Explanation of Bessel's Correction When calculating a sample variance a factor of $N-1$ appears instead of $N$ (see this link ).  Does anybody have an intuitive way of explaining this to students who need to use this fact but maybe haven't taken a statistics course?
http://en.wikipedia.org/wiki/Bessel%27s_correction

The Wikipedia article linked to above has a section (written by me) titled "The source of the bias".  It explains it via a concrete example.

But note also that correcting for bias, when it can be done, is not always a good idea.  I wrote this paper about that: http://arxiv.org/pdf/math/0206006.pdf
Here's a really simple example:

Say that you have a list of the heights of all the players in a sports league.  Call that your "population" and calculate the variance.  Now pick a sample of three players at random and calculate the variance of your sample.  Chances are that your sample won't pick up the most extreme values, thereby causing the variance of the sample to be lower than the variance of the overall population.

In order to correct for that, you need to adjust your sample variance upward.  That's why you use 1/(n-1) instead of 1/n.  At small values of n (such as 3) the difference between these two will be pretty big.  As your sample size increases, the difference gets smaller.  This is because your sample is more likely to capture the extreme values in the population and therefore your sample variance will approach the population variance.