*10*

Most formulas used to computeÂ **standard errorsÂ **are based on the idea that (1) samples are selected with replacement or that (2) samples are selected from an infinite population.

In actual research, neither of these ideas hold true. Luckily this doesnâ€™t tend to be a problem if the sample size is less than 5% of the total population size.

However, when the sample size is larger than 5% of the total population itâ€™s best to apply a **finite population correction **(often abbreviated *FPC*), which is calculated as:

**FPC = âˆš(N-n) / (N-1)**

where:

**N:Â**Population size**n:Â**Sample size

**How to Use the Finite Population Correction Factor**

To apply a finite population correction, simply multiply it by the standard error that you would have originally used.

For example, the standard error of a mean is calculated as:

**Standard error of mean:** s / âˆšn

By applying the finite population correction, the formula becomes:

**Standard error of mean:** s / âˆšnÂ * âˆš(N-n) / (N-1)

The following examples illustrate how to use the finite population correction in different scenarios.

**Example 1: Confidence Interval for a Proportion**

Researchers want to estimate the proportion of residents in a county of 1,300 people that are in favor of a certain law. They select a random sample of 100 residents and ask them about their stance on the law. Here are the results:

- Sample sizeÂ
**n = 100** - Proportion in favor of lawÂ
**p = 0.56**

Typically the formula to calculate a 95% confidence interval for a population proportion is:

**95% C.I. = pÂ +/-Â z*(âˆšp(1-p) / n)**

However, our sample size in this example is 100/1,300 = 7.7% of the population, which exceeds 5%. Thus, we need to apply a finite population correction to our formula for the confidence interval:

**95% C.I. = pÂ +/-Â z*(âˆšp(1-p)/n) * âˆš(N-n) / (N-1)**

Thus, our 95% confidence interval can be calculated as:

**95% C.I. = **0.56Â +/-Â 1.96*(âˆš.56(1-.56) / 100) * âˆš(1300-100) / (1300-1)Â = **[0.4665, 0.6535]**

**Example 2: Confidence Interval for a Mean**

Researchers want to estimate the mean weight of a certain species of 500 turtles so they select a random sample of 40 turtles and weight each of them. Here are the results:

- Sample sizeÂ
**n = 40** - Sample mean weightÂ
**xÂ = 300** - Sample standard deviationÂ
**s = 18.5**

Typically the formula to calculate a 95% confidence interval for a population mean is:

**95% C.I. = xÂ +/-Â t _{Î±/2}*(s/âˆšn)**

However, our sample size in this example is 40/500 = 8% of the population, which exceeds 5%. Thus, we need to apply a finite population correction to our formula for the confidence interval:

**95% C.I. = xÂ +/-Â t _{Î±/2}*(s/âˆšn) Â * âˆš(N-n) / (N-1)**

Thus, our 95% confidence interval can be calculated as:

**95% C.I. = **300Â +/-Â 2.0227*(18.5/âˆš40) * âˆš(500-40) / (500-1)Â = **[294.32, 305.69]**

**Additional Resources**

What are Confidence Intervals?

Margin of Error vs. Standard Error: Whatâ€™s the Difference?

Standard Deviation vs. Standard Error: Whatâ€™s the Difference?