Math · Statistics
Proportion confidence interval calculator.
Enter the number of successes and the sample size. You get the Wilson score, exact Clopper-Pearson and normal approximation intervals side by side. Everything runs in your browser.
If 120 of 400 people say yes (30%), the 95% Wilson score interval is 25.7% to 34.7%. The exact Clopper-Pearson interval is 25.5% to 34.8% and the simple normal approximation is 25.5% to 34.5%.
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The three intervals
With p̂ = x / n and z = z1−α/2 (1.960 at 95%):
Wilson score: ( p̂ + z²/2n ± (z/2n) √( 4n p̂(1−p̂) + z² ) ) / ( 1 + z²/n ).
Normal approximation (Wald): p̂ ± z √( p̂(1−p̂)/n ).
Clopper-Pearson (exact): the lower limit is the α/2 quantile of a Beta(x, n − x + 1) distribution and the upper limit the 1 − α/2 quantile of Beta(x + 1, n − x), with 0 as the lower limit when x = 0 and 1 as the upper limit when x = n.
Worked example
120 successes in 400: p̂ = 0.30. Wilson 95%: 25.72% to 34.66%. Exact: 25.55% to 34.75%. Normal approximation: 0.30 ± 1.960 × 0.02291 = 25.51% to 34.49%. With plenty of data they nearly agree. With little data they do not: for 2 successes in 50 (4%) the normal approximation gives −1.43% to 9.43%, an impossible negative lower limit, while Wilson gives 1.10% to 13.46% and the exact interval 0.49% to 13.71%. For 8 of 10, Wilson is 49.02% to 94.33%, exact is 44.39% to 97.48%, and the normal approximation runs from 55.21% to 104.79%, past 100%.
Which should I use?
Wikipedia describes the Wilson interval as asymmetric, free of the overshoot and zero-width problems of the normal approximation, and safe for small samples and skewed observations. It describes the Clopper-Pearson interval as exact, never under the nominal coverage but usually conservative (wider than needed). Wilson is therefore a good default for conversion rates and survey percentages; use the exact interval when you need a guarantee of at least the stated coverage. The intervals assume a simple random sample of independent yes/no trials.
Sources, opened 8 October 2026: Wikipedia, Binomial proportion confidence interval (Wilson score interval formula, normal approximation interval, overshoot and zero-width problems, Clopper-Pearson as an exact and conservative interval). The Beta-quantile form of the Clopper-Pearson limits is checked in the tests against the binomial tail definition (each tail equals α/2).
Frequently asked questions
What is the best confidence interval for a proportion?
For most conversion-rate and survey work the Wilson score interval is a good default: it stays inside 0% to 100% and behaves well for small samples. The exact Clopper-Pearson interval is wider but never has less than the stated coverage.
Why can the normal approximation give impossible values?
It is symmetric around the observed rate with a width based on the observed rate, so with few successes it can go below 0% or above 100%, or collapse to zero width at 0 successes. For 2 of 50 it gives -1.43% to 9.43%.
How do I find the confidence interval for a conversion rate?
Enter the number of conversions as successes and the visitors as the sample size. 120 conversions from 400 visitors gives a 95% Wilson interval of 25.7% to 34.7%.
How is this different from margin of error?
Margin of error is the half-width of the normal-approximation interval. The sample size calculator uses it to plan a survey. Wilson and exact intervals are asymmetric, so they are reported as a range rather than a plus-or-minus figure.
What if I have zero successes?
The exact interval then starts at 0 and the Wilson interval at 0 too, with a positive upper limit that shrinks as the sample grows. The normal approximation collapses to zero width, which is misleading.
Does 95% mean there is a 95% chance the true rate is inside?
Strictly it is a statement about the method: in the long run about 95% of intervals built this way from repeated samples contain the true proportion.
Planning and learning aid
Educational calculator. Assumes independent yes/no observations from a simple random sample; it does not adjust for clustering, weighting or survey design effects.