Finite population correction (FPC)

When sampling without replacement a sizeable fraction of a finite population, the standard error is multiplied by the finite population correction √((N − n)/(N − 1)).

\text{FPC} = \sqrt{\frac{N-n}{N-1}}, \qquad \SE(\xbar) = \frac{\sigma}{\sqrt{n}}\cdot\text{FPC}

Correction factor for a mean or proportion under simple random sampling without replacement

$N$: population size
$n$: sample size
$n/N$: sampling fraction

When to use

Simple random samples drawn without replacement where the sampling fraction $n/N$ exceeds roughly 5–10%: auditing a batch of invoices, surveying a small organisation. Also used in sample-size planning for small populations.

How to interpret

FPC ≤ 1: knowing a larger share of the population leaves less uncertainty. If $n = N$ (a census) the FPC is 0: no sampling error.
When $n/N$ is small the FPC is close to 1 and can be ignored.

Common mistakes

Applying it to samples with replacement or to an effectively infinite population.
Believing a sample must be a fixed percentage of the population to be accurate: precision depends mainly on $n$ , not on $n/N$ .

Example

$n = 100$ from $N = 500$ : FPC $= \sqrt{400/499} = 0.8953$ , so the SE is about 10% smaller than the usual formula. From $N = 2000$ : FPC $= 0.9749$ , a 2.5% reduction.