Finite population correction for the variance

Question

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?

Answer 1

The first thing you need to ask yourself to clear up your confusion is this: When constructing a variance estimator from the sample data, what is it you are estimating the variance of? Do you want an estimate of the variance parameter for an infinite superpopulation? Do you want an estimator for the mean of a finite population? Do you want to estimate some other variance quantity? The variance of what?

Depending on your answer to this question, the necessary adjustments in your variance estimator will then follow accordingly. The purpose of Bessel's correction is to adjust the variance estimator (of the variance parameter for an infinite superpopulation) to be unbiased. The purpose of FPC is to adjust for estimation of the variance of the mean of a finite population. If you are constructing a confidence interval for the mean of a finite population you would apply both of these adjustments, so you don't need a different version of Bessel's correction.

Answer 2

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.