Why does the variance of the sampling distribution of the statistic do not disapear when sampling the whole population?

Question

This might sound a stupid question but it is indeed a real one.

I'm trying to figure a Confidence interval for the average age of my population.

Given i have a population of 100 individual, and i sample 3 of them. From CLT, i can say that $Var[\bar{x}_3] = \frac{s^2}{3} $. Alright.

I want to increase precision of my confidence interval and thus i sample more. I head toward a 100 individual sample.

From CLT, i can say that $Var[\bar{x}_{100}] = \frac{s^2}{100} $, which clearly is not zero. Small but not Zero.

Where did i do something wrong?

Thanks a lot!

Answer 1

You are sampling in an i.i.d. fashion, i.e. with replacement. A sample of size 100 does not necessarily include all 100 individuals; most likely, it will not (you have both people sampled several times and people that are not sampled). So you still have uncertainty, and variance in your estimator.

In short: if you get independent samples (which matters to apply the statistical results you are using) you are not (necessarily) sampling the whole population.

Answer 2

You are sampling from a finite population so you samples are not in fact independent - once you sampled a person, they are out and you are sampling from a smaller pool of people. Once your sample size gets compatible (order of magnitude) with population size, these effects start to matter more and more.