Why small populations need smaller samples
Textbook standard errors pretend the population is endless: each draw leaves the pool unchanged. Sample without replacement from a finite list and the pool shrinks informatively; by the time you have sampled half the list, you know far more than the textbook formula credits. The finite population correction prices that knowledge: multiply standard errors by √((N − n)/(N − 1)), a factor under 1, or equivalently shrink a computed sample size to n = n0 ÷ (1 + (n0 − 1)/N).
A standard calculation asks for n0 = 385 from a population of N = 2,000. Corrected: 385 ÷ (1 + 384/2000) = 385 ÷ 1.192 = 323. Sixty-two interviews saved, same margin of error. Shrink the population to N = 500 and the correction deepens: 385 ÷ 1.768 = 218, because the sample now covers over 40% of everyone. As N grows toward infinity the correction fades to nothing, which is the correct behaviour: national surveys of millions can ignore it safely.
When it matters and when it does not
The working rule in survey practice: apply the correction when the sampling fraction n/N passes about 5%. Below that, √((N−n)/(N−1)) sits above 0.975 and the adjustment is smaller than the rounding in the rest of the design. Above it, ignoring the correction wastes fieldwork budget on precision you already own. Institutional research lives in the correction's home territory: one school's students, one firm's customers, one city's registered voters.
Applying the correction to sampling with replacement, or to samples drawn from an effectively open population (website visitors, a production stream). The correction is a statement about a fixed finite list sampled without replacement; outside that design it does not apply.
Frequently asked questions
What is the finite population correction factor?
sqrt((N - n)/(N - 1)) for standard errors. It shrinks standard errors (or required sample sizes) when a substantial fraction of a finite population is sampled without replacement.
At what sampling fraction should I bother?
About 5% of the population. Below it the factor exceeds 0.975 and changes little; above it the savings grow quickly.
Does the correction change the point estimate?
No. Means and proportions are computed as usual. The correction only adjusts standard errors and required sample sizes: precision statements, not estimates.