Finite population correction for the variance Just when I thought I was starting to understand Bessel's correction, I noticed that it is not valid when the sample size equals the population size and so likely not valid for sample sizes close to the population size.
I know that Bessel's correction is for small samples, but my question is: What can we do if we sample, say, 20% of the population or 50%?
Is there an alternative to Bessel's correction that takes into account the size of the sample relative to the population, analogous to the FPC?

If you have the whole population, you know what the variance of the population is. You don't need to estimate it. If you only have a sample of the population, then you have to estimate the variance of the population. Bessel's correction gives you an unbiased estimate of the variance of the population. Done.
I think what you're asking for is the variance of the sample average. For an infinite population, this is simply the estimate of the variance of the population (using Bessel's correction) divided by n. For a finite sample, it's the estimate of the variance of the population (using Bessel's correction) times the finite population correction. So it's both.
I should note that the above applies to questions of descriptive statistics. Causal inference under Rubin's potential outcomes framework is somewhat more complicated.
