Finite Population Corrections for Sample Means

When we sample $n$ observations without replacement from a finite population of size $N,$ the observations are not independent, so the usual formula $\sigma/\sqrt{n}$ overstates the standard error of the sample mean.

The corrected standard error is

$$\sigma_{\bar X} \;=\; \dfrac{\sigma}{\sqrt{n}}\,\sqrt{\dfrac{N-n}{N-1}}.$$

The factor $\sqrt{\dfrac{N-n}{N-1}}$ is called the finite population correction (FPC).

Notice two extreme cases:

• If $n=1,$ then the FPC equals $1$ and there is no correction.
• If $n=N$ (we sample the entire population), then the FPC equals $0$ and $\sigma_{\bar X}=0,$ because $\bar X$ exactly equals $\mu.$

As an illustration, suppose $N=150,$ $\sigma=8,$ and $n=30.$ Then

$$\begin{align*} \sigma_{\bar X} &= \dfrac{8}{\sqrt{30}}\,\sqrt{\dfrac{150-30}{150-1}} \\[4pt] &= 1.461 \cdot \sqrt{0.8054} \\[4pt] &= 1.461 \cdot 0.8974 \\[4pt] &\approx 1.311. \end{align*}$$

Rule of thumb. The FPC is often omitted when $n/N \le 0.05,$ since its value is then close to $1.$ Whenever the sample is more than $5\%$ of the population, the FPC should be applied.