= Finite population correction
{title2=$f=\sqrt{\frac{n-r}{n-1}}$}
For uniform <sampling without replacement> from a population of $n>1$ objects, the <finite population correction> accounts for the reduced spread caused by drawing distinct objects. For the marked-object count $K$ in a <hypergeometric distribution>, put $p=c/n$. Then
$$
\mathbb E K=rp,\qquad
\operatorname{Var}(K)=rp(1-p)\frac{n-r}{n-1}.
$$
Relative to the corresponding <binomial distribution>, the <variance> is multiplied by $f^2=(n-r)/(n-1)$ and the <standard deviation> by $f$. The square-root convention for the correction is used in Philip Stark's discussion of sample sums.https://www.stat.berkeley.edu/~stark/SticiGui/Text/standardError.htm[Stark, SticiGui, Standard Error: sample sums without replacement]{ref}
The factor has a direct interpretation. For $r=1$, replacement makes no difference and $f=1$. For $r=n$, the entire population is observed and the marked count is exactly $c$, so $f=0$. In the ten-object, five-draw example, $f^2=5/9$ and $(6/5)(5/9)=2/3$. The exceptional population size $n=1$ is deterministic and should be handled directly rather than inserting it into a zero denominator.
A short explanation uses an <indicator random variable> $I_j$ for a marked object at draw $j$. Each has <expected value> $p$ and <variance> $p(1-p)$. For distinct draw positions $i,j$,
$$
\mathbb E[I_iI_j]=\frac{c(c-1)}{n(n-1)},\qquad
\operatorname{Cov}(I_i,I_j)=-\frac{p(1-p)}{n-1}.
$$
Since $K=\sum_{j=1}^r I_j$, the <variance of a sum> includes $r$ individual terms and $r(r-1)$ ordered <covariance> terms, giving the displayed correction. <Linearity of expectation> still gives $rp$ despite this dependence.
For arbitrary fixed real population values, use $N>1$ for population size and $1\leq n\leq N$ for <sample size>. Write the <finite population mean> $\bar Y=N^{-1}\sum_i y_i$, the <finite population variance> $S^2=(N-1)^{-1}\sum_i(y_i-\bar Y)^2$ and $\sigma_N^2=N^{-1}\sum_i(y_i-\bar Y)^2$. Under <simple random sampling without replacement>, the <sample mean> satisfies
$$
\operatorname{Var}(\bar y_A)=\left(1-\frac nN\right)\frac{S^2}{n}=\frac{N-n}{N-1}\frac{\sigma_N^2}{n}.
$$
Thus $1-n/N$ is the <variance> factor relative to $S^2/n$, while $(N-n)/(N-1)$ is the exact <variance> ratio to independent uniform <sampling with replacement>, whose <sample mean> has <variance> $\sigma_N^2/n$. The <standard error> uses the square root of the corresponding factor. These formulas agree because $\sigma_N^2=(N-1)S^2/N$; the denominator convention must be specified. The <sample mean under simple random sampling> gives a derivation and an exact six-value example.
Back to article page
