How do you calculate the margin of error in a statistical sampling scheme?

Previously we have discussed how to calculate sample size for a survey. How do you calculate the margin of error given the sample size?
The Margin of Error (MOE) can be calculated using equation 1.

Equation 1. MOE = Z x √p x (1 - p) / √n

Where
n: the sample size
Z: is the Z-score and is derived from the confidence limit. This has been explained previously.
p: is the sample proportion. In this equation it is represented as a decimal.
This equation requires the addition of a correction factor to allow for finite-size when working with small populations. Most student projects will have to work using this finite correction factor. The modified equation (equation 2) is shown below.

Equation 2. MOE= Z x √p x (1 - p) / √(N - 1) x n / (N - n)

Note the only new variable in equation 2 is N which is the size of sample population e.g. the number of employees at a cheese processing company.

The Z-score for commonly used confidence levels is given in Table 1.

The calculator below has been programmed to determine the MOE using a finite population e.g. your sub sample the number of people in a school, town, football club) of the much larger population of e.g. a country, province. Note the number of samples required has been rounded up to the next highest number.

Recommended further reading

Alreck, P.L. and  Settle, R.B. (2003).The Survey Research Handbook (paperback). 3rd edition. New York: McGraw-Hill  Professional.

Daniel WW (1999). Biostatistics: A Foundation for Analysis in the Health Sciences. 7th edition. New York: John Wiley & Sons.

How to cite this article

Mullan, W.M.A. (2021) [On-line] Available from: Accessed: