Once the answers are in
Margin of error calculator
Calculate the margin of error for a survey percentage from your sample size, confidence level, and population. See the confidence interval around your result.
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Margin of error
±4.9
percentage points, at 95% confidence
A finding of 50.0% is consistent with anything from 45.1% to 54.9% in the group you are describing.
This estimates sampling error under the stated assumptions. It does not account for nonresponse or selection bias. Subgroups need their own margin of error; smaller samples generally have wider margins.
How this tool works
The margin is z × √(p × (1 − p) ÷ n), expressed in percentage points, where n is your number of completed responses and p the proportion you found. The z values are approximately 1.645, 1.960, and 2.576 for 90%, 95%, and 99% confidence. Leaving p at 50% gives the widest margin, which is the one to quote for a survey as a whole, because every other proportion in it is narrower.
When you give a population size N and your sample is a meaningful share of it, the margin is multiplied by the finite-population correction √((N − n) ÷ (N − 1)). Sampling 400 of 1,000 people narrows the margin by about a quarter; sampling 400 of 4,000,000 changes nothing worth printing. When n equals N you have a census, and the margin is zero; a population smaller than the sample is refused.
Using the conservative 50% proportion, 400 responses at 95% confidence give a margin of about ±4.9 percentage points. At 800 responses it is about ±3.5. Precision improves with the square root of the sample size.
The margin describes sampling variation under the model assumptions. It does not account for nonresponse, gaps in the sampling frame, or misunderstood questions. Larger samples do not remove these biases. Calculate subgroup margins using the number of responses in each subgroup.
Method reference: NIST: confidence intervals for proportions.
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A useful next step
Start from the precision you need and work out the responses it takes.
Sampling error, subgroup precision, and the biases a margin does not cover.
Overlapping intervals are not a test; run the test.
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Put it into practice in your next survey.
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