There are quite a few comparisons of sample and population variance, just google and see which is the most readable for you.
I always think about what happens when you have one or two points in a sample.
When you have a single point N=1,
* Sample variance: is not defined because you divide by N-1=0. This is because you use the single point to estimate the mean and there is nothing left to calculate the variance - the "error" from the mean (x-mu) is zero because the mean and the point are the same
* Population variance: you know the mean, so the single point can be used to calculate (x-mu)^2, and as only a single point is used in the summation, you divide by 1.
When you have two points N=2
* Sample variance. The mean is the average of the two points. The two "errors" from the mean are always the same, so really you have just one bit of information when you estimate the variance, so you divide by N-1=1.
* Population variance: you know the mean, the two points are independent, as is the squared error (x_i-mu)^2 so to estimate the variance, add them up and divide by 2.
Not necessarily rigorous but it is easy to remember and sounds plausible.
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