finite-population correction

Also called: finite population correction, FPC

A factor √((N − n) ÷ (N − 1)) that shrinks the standard error when the sample is a large part of a small population. Surveying 400 of 1,120 learners multiplies the SE by about 0.80.

The usual standard error, s/√n, assumes the population is so big that sampling a few people doesn’t change what’s left. When you survey a large share of a small population, that isn’t true: if you asked everyone, there would be no sampling error at all. The finite-population correction accounts for this:

$SE_{FPC} = \frac{s}{\sqrt{n}} \times \sqrt{\frac{N - n}{N - 1}}$

N is the population size and n the sample size. When n is a tiny share of N, the factor is almost 1 and can be ignored; when n approaches N, it falls toward 0.

A practical rule: bother with it when the sample is more than about 5% of the population, such as a survey of a company’s own customers or employees.

Example

A school surveys 400 of its 1,120 active learners.

Factor: √((1,120 − 400) ÷ 1,119) = √0.643 ≈ 0.80.
The SE is about 20% smaller than s/√400 alone would say.
Surveying 400 of a million would give a factor of about 1.

Common mistakes

Applying it to a huge population. The factor is about 1, so it changes nothing.
Forgetting the square root. (N − n)/(N − 1) itself is too small a factor.
Using it to fix a biased sample. It corrects random error, not who answered.

Learn it in the course

Square roots and √n: SD and the standard error · The standard deviation as √s², the standard error s/√n and why 4× the data halves it, the standard error of a proportion, the minimum detectable effect, the paired t and the finite-population correction.