# Population Variance vs Sample Variance [duplicate]

I'm particularly interested why N appears in the denominator for the formula of population variance, but (n-1) appears in the denominator for the sample variance, where N is the total number of elements in the population and n is the total number of elements in the simple random sample?

Basically, the (n-1) seems less intuitive to me.

I'd love to know the reasoning behind this, as well as seeing any mathematical derivations that back up that reasoning.