Wilson Score Interval Calculator
Confidence interval for a proportion that behaves at small n and near 0 or 100% — Wilson score with and without continuity correction, versus the Wald interval.
How it works
centre = (p̂ + z²/2n) ÷ (1 + z²/n); half-width = z·√[p̂(1−p̂)/n + z²/4n²] ÷ (1 + z²/n)
The familiar Wald interval p̂ ± z√(p̂(1−p̂)/n) collapses to zero width at 0 or 100% and under-covers badly for small samples. Wilson’s score interval inverts the score test instead: it pulls the centre toward 50%, never leaves 0–1 and keeps close to nominal coverage even with a handful of trials. It is what you should use for click-through rates, defect rates and any proportion from fewer than a few hundred observations.
Worked example
7 successes in 20: Wilson 95% interval 18.1%–57.1%, versus the Wald interval 14.1%–55.9% which is both narrower and mis-centred.
Frequently asked questions
When does Wald become acceptable?
When np̂ and n(1−p̂) both exceed about 10 — and even then Wilson costs nothing.
Wilson or Clopper–Pearson?
Clopper–Pearson guarantees at least nominal coverage but is conservative (too wide); Wilson is closer to nominal on average and is the usual recommendation.
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This calculator is for educational purposes. Double-check important results before acting on them.