The formula for variance (and standard dev) uses n. Why is n-1 used when estimating pop variance from a sample? I realise that using n-1 (in the denominator) results in a larger value. But why n-1 and not say, n-2? Is there a simple explanation?

Also, doesn't our estimate also depend on how large n (sample size) is compared to pop size N? Should we use a "correction factor" similar to that used for confidence intervals?