How do I Find a Margin of Error When the Sample Size Is Less Than 30?

The margin of error estimates the accuracy of an estimate derived from a sample. The larger the sample, the more accurate the estimate, but when the sample size is small, the standard formulas do not work well.

Standard Formula

The standard formula for the margin of error of a binomial proportion is (pq/n)^.5, where p is the proportion of the sample saying one thing, q = 1-p, and n is the sample size. So, if we had a sample of 10 people, and 1 of them had a disease, then p = .1, q = .9, n = 10 and the margin of error is .09.

Simulation

An alternative method is to simulate the data. You can use a statistical program such as SAS or R to simulate a large population with the same proportion as in your sample, and then repeatedly draw samples from it. Then you can find the margin of error from the collection of samples.

An Alternative Formula

Agresti and Coull suggest a simple alternative formula which they show to be more accurate when n is small. Simply add two successes and two failures (that is, two people in each group) to the original sample. So, in our example, instead of p = 1/10 and q = 9/10 we would use p = 3/14 and q = 11/14. The estimate of the proportion in the population is now .21 and the margin of error is now 0.11.

References

• Questia: Approximate Is Better Than "Exact" for Interval Estimation of Binomial Proportions