Sampling > Finite Population Correction Factor

## What is the Finite Population Correction Factor?

The Finite Population Correction Factor (FPC) is used when you **sample without replacement from more than 5% of a finite population**. It’s needed because under these circumstances, the Central Limit Theorem doesn’t hold and the standard error of the estimate (e.g. the mean or proportion) will be too big. **In basic terms, the FPC captures the difference between sampling with replacement and sampling without replacement.**

Most real-life surveys involve finite populations sampled without replacement. For example, you might perform a telephone survey of 10,000 people; once a person has been called, they won’t be called again.

**Note**: A downside of using the FPC is that it can cause uncertainty when applying the results to a larger population, so you should be careful when making inferences.

## Formula

The general formula is:

**FPC = ((N-n)/(N-1))**

^{1/2}Where:

- N = population size,
- n = sample size.

If the calculated value for the FPC is close to 1, it can be ignored. As the sample size falls under 5%, the value becomes somewhat insignificant (an FPC is .998 for a sample of 50).

The following table of values shows how the FPC decreases for a population of 10,000 as the sample size gets larger:

## How to Use the Formula

Basically, place the correction at the end of the formula you want to use. For example, the standard error of the mean formula is:

And with the correction, the formula is:

Or, for a confidence interval for a mean and unknown population standard deviation, the formula (with FPC) is:

## Example

Thirty people from a population of 300 were asked how much they had in savings. The sample mean (x̄) was $1,500, with a sample standard deviation of $89.55. Construct a 95% confidence interval estimate for the population mean.

**Solution **(using degrees of freedom = n – 1 = 29) and t_{α/2} = 2.0452 for a 95% confidence level):

= $1,500 ± 33.44(0.9503)

= $1,500 ± 31.776

= $1,468.22 ≤ μ ≤ $1,531.78.

**References**:

Kandethody, M. et, al. Mathematical Statistics with Applications. Elsevier India (2012). p.187.

good stuff