Finite Population Correction and Sampling Fractions

Standard-error formulas assume you're drawing a tiny sample from an effectively infinite population — each draw barely changes what's left to draw from. That assumption breaks when you sample, say, 400 accounts out of a firm with only 1,000 total. Once you've looked at 40% of everyone, there's a lot less left to be surprised by, and treating the sample as if it came from an infinite pool overstates how uncertain your estimate really is.

The finite population correction (FPC) fixes this by multiplying the standard error by $\sqrt{\frac{N-n}{N-1}}$, where $N$ is the population size and $n$ is the sample size. As $n$ grows relative to $N$, this factor shrinks toward zero, squeezing the standard error down — right up to $n = N$, a full census, where uncertainty about the mean is exactly zero, since you've counted everyone.

Worked example. A pension fund wants the average balance across its $N = 2{,}000$ accounts and samples $n = 500$ of them, finding a sample standard error (uncorrected) of $120. The FPC factor is $\sqrt{(2000-500)/(2000-1)} = \sqrt{1500/1999} \approx 0.866$. Corrected standard error: 120 \times 0.866 \approx \104$ — about 13% tighter than the naive formula would claim, simply because 25% of the whole population was already in hand.