## Margin of Error

There's really no excuse for not calculating the margin of error in a survey, since the calculation is so simple and it's so important to understand the accuracy of your data.

The formula is MOE = 1/sqrt(N) where N is the number of survey responses. If you punch this into a calculator or spreadsheet, the result will be a fraction, which you can also express as a percent.

For example, with 100 survey responses, the margin of error is 1/sqrt(100) = 1/10 = .1 = 10%.

Similarly, with 400 survey responses, the margin of error is 5%. Quadrupling the number of responses cuts the margin of error in half.

This formula is used all the time in error analysis, but it's actually an approximation which doesn't work for answers which are given on only a very small fraction of the surveys.

When less than five percent of the people gave a particular answer, the margin of error *for that answer* is approximately 2*sqrt(P/N), where P is the fraction of participants who gave that answer (the exact formula in all cases is 2*sqrt[P(1-P)/N] ).

For example, if you surveyed 1,000 people about their religion, and 10 (1%) answered "Jedi," then the margin of error for the percentage of Jedi in the survey is 2*sqrt(.01/1000) = 0.6%. This is actually quite a bit smaller than the margin of error given by the more common formula (3.2% in this case). It turns out that the common 1/sqrt(N) formula is the *largest*margin of error possible for any response on the survey.