A finite population correction factor is needed in computing the standard deviation of the sampling distribution of sample means irrespective of the size of the sample. whenever the sample size is more than $$ 5 \% $$ of the population size. whenever the population is infinite. whenever the sample size is less than $$ 5 \% $$ of the population size.

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Suppose we have a finite population of size $$ N $$ and we draw a sample of size $$ n $$ without replacement. The standard deviation of the sampling distribution of the sample mean (also called the standard error) is given by

$$
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N - n}{N - 1}},
$$

where $$ \sigma $$ is the standard deviation of the population and $$ \sqrt{\frac{N - n}{N - 1}} $$ is the finite population correction (FPC) factor.

When the sample size $$ n $$ is a small fraction of the total population $$ N $$, typically less than $$ 5\% $$ of $$ N $$, then

$$
\sqrt{\frac{N - n}{N - 1}} \approx 1.
$$

This implies that the FPC can be ignored. However, when $$ n $$ is more than $$ 5\% $$ of $$ N $$, the FPC becomes significant and must be included in the calculation.

Thus, the finite population correction factor is needed in computing the standard deviation of the sampling distribution of sample means whenever the sample size is more than $$ 5\% $$ of the population size.
The finite population correction factor is needed when the sample size is more than 5% of the population size.

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