Why there are several methods
The textbook Wald formula, p̂ ± z√(p̂(1 − p̂)/n), is simple but misbehaves with small samples or proportions near 0% or 100%. Other methods fix this. Wilson is usually the best general choice. See how to calculate a confidence interval for a proportion.
Worked examples
| Data | Wald | Wilson | Exact |
|---|---|---|---|
| 52 of 80 | 54.5% to 75.5% | 54.1% to 74.5% | 53.5% to 75.3% |
| 3 of 50 | 0% to 12.6% | 2.1% to 16.2% | 1.3% to 16.5% |
| 0 of 20 | 0% to 0% | 0% to 16.1% | 0% to 16.8% |
The last row shows the failure clearly: zero successes in 20 does not mean the true rate is exactly 0%, yet Wald says so.
Accuracy
All four match statsmodels’ proportion_confint, including its truncation at 0% and 100%. To compare two proportions, use the two-proportions calculator. For a survey-style figure, see the margin of error calculator.
Frequently asked questions
Which confidence interval is best for a proportion?
Wilson is the usual recommendation. Clopper-Pearson guarantees at least the stated coverage but tends to be wide.
What if I have zero successes?
Use Wilson or the exact method. A quick approximation is the rule of three: the one-sided 95% upper limit is about 3/n, or 15% for n = 20.
What is wrong with the Wald interval?
Its actual coverage can fall well below the stated level with small samples or extreme proportions, and it can run past 0% or 100%.