Finite Population Correction

The finite population correction (FPC) is the factor $1-n/N$ applied to variance formulas for sampling without replacement from a finite population of size $N$ with sample size $n$, for example, the variance of the sample mean under simple random sampling is $\text{Var}(\bar{x})=(1-n/N)s^2/n$ rather than the infinite-population formula $s^2/n$. The FPC reflects that sampling without replacement removes each drawn unit from further consideration, so as the sampling fraction $n/N$ grows, less of the population's total variability remains unaccounted for by the sample; the FPC approaches 1 (negligible adjustment) as $n/N\to 0$, the standard situation for large populations with modest sample sizes, where the FPC is routinely dropped from practical calculations, and approaches 0 as $n/N\to 1$, correctly implying essentially zero remaining sampling variance for a sample approaching a full census of the population.

• The finite population correction is only meaningful, and typically only applied, when the sampling fraction n/N is non-negligible, a common rule of thumb in applied practice is to include the FPC when n/N exceeds roughly 5%, and to safely omit it (treating the population as effectively infinite) for smaller sampling fractions, where it has negligible practical effect.
• A full census, where n equals N, has an FPC of exactly 0, correctly implying zero sampling variance, since there is no remaining population left unobserved to be uncertain about, this is the mathematically consistent limiting case that anchors the FPC formula's intuition.
• The FPC does not apply to sampling with replacement, since a with-replacement draw is always made from the full, unreduced population regardless of how many units have already been selected, which is exactly why with-replacement variance formulas have no corresponding correction term, this is the key structural difference in the variance formulas for the two sampling schemes.