Confidence interval for a proportion (Wald, Wilson, Clopper–Pearson)

Intervals for a population proportion; the Wilson score interval is recommended, the Wald interval is simple but unreliable for small n or p near 0 or 1, and Clopper–Pearson is exact and conservative.

\phat \pm z_{1-\alpha/2}\sqrt{\frac{\phat(1-\phat)}{n}}

Wald interval (textbook, not recommended for small samples)

$\phat$: sample proportion $x/n$
$n$: number of trials
$z_{1-\alpha/2}$: standard normal quantile (1.96 for 95%)

\frac{\phat + \frac{z^2}{2n} \pm z\sqrt{\frac{\phat(1-\phat)}{n} + \frac{z^2}{4n^2}}}{1 + \frac{z^2}{n}}

Wilson score interval (recommended)

$z$: $z_{1-\alpha/2}$

\big[\,B_{\alpha/2}(x,\, n-x+1),\;\; B_{1-\alpha/2}(x+1,\, n-x)\,\big]

Clopper–Pearson (exact) interval

$B_q(a, b)$: the $q$ quantile of a Beta( $a$ , $b$ ) distribution; the lower bound is 0 when $x = 0$ and the upper bound 1 when $x = n$
$x$: number of successes

When to use

Estimating a rate: conversion, default, defect or prevalence rates from $x$ successes in $n$ independent trials.

Wilson (recommended): good coverage even for small $n$ or extreme $\phat$ ; always inside [0, 1].
Clopper–Pearson: guaranteed coverage of at least the nominal level; wider (conservative). Common in regulated settings.
Agresti–Coull: add $z^2/2$ successes and $z^2/2$ failures, then use the Wald formula; close to Wilson.
Wald: only for large samples with $n\phat$ and $n(1-\phat)$ both comfortably above 10.

Assumptions

Independent trials with a common probability (binomial model); a random sample.

How to interpret

The interval gives plausible values of the population proportion. The Wilson interval is not centred on $\phat$ : it is pulled towards 0.5, which corrects the Wald interval's undercoverage.

Common mistakes

Using Wald with few successes: it can extend below 0 or above 1 (the engine reports, but does not truncate, such bounds) and its real coverage can be far below 95%.
Reporting $\phat = 0$ with a zero-width Wald interval when $x = 0$ .
Mixing percent and proportion units.

Example

$x = 18$ of $n = 60$ ( $\phat = 0.30$ ), 95%: Wald $(0.184, 0.416)$ ; Wilson $(0.199, 0.425)$ ; Clopper–Pearson $(0.188, 0.432)$ ; Agresti–Coull $(0.198, 0.426)$ . With $x = 2$ of $n = 20$ : Wald gives $(-0.031, 0.231)$ , an impossible lower bound; Wilson gives $(0.028, 0.301)$ .