Single proportion confidence interval calculator

A single proportion confidence interval answers a simple question with a surprisingly subtle method: given that you saw x events in n observations, what is the plausible range for the true proportion or prevalence behind that count? It is the right tool whenever your statistic is a share of a single group: the proportion of patients who responded, the fraction of records that met an inclusion rule, the prevalence of a condition in one cohort. It is the wrong tool when you are comparing two groups, where you want a difference, a risk ratio, or an odds ratio instead, a choice our odds ratio and risk ratio calculator handles directly.

The reason this calculator offers three intervals is that the obvious method is the weakest. The normal approximation, or Wald interval, centres a symmetric margin on the observed proportion. It is fine with a large sample and a proportion well away from the extremes, but near 0 or 1, or with a small sample, it produces limits that are too narrow, undercover the true value, and can even fall below 0 or above 1. The Wilson score interval is the recommended default: it recentres toward one half, never leaves the valid 0-to-1 range, and behaves well across the whole spectrum of counts. The Clopper-Pearson exact interval is built straight from the binomial distribution and guarantees that true coverage is at least the stated level, at the cost of being a little wider than necessary, which makes it the conservative choice for very small samples.

Suppose 7 of 40 screened participants meet a criterion, a proportion of 0.175. The Wald interval gives roughly 0.057 to 0.293, the Wilson score interval gives roughly 0.087 to 0.323, and the Clopper-Pearson exact interval is wider still, around 0.073 to 0.336. The Wald interval looks tightest, but that tightness is false economy; its coverage is poor at this sample size, so the Wilson interval is the one to report. Push the example to an extreme, 0 events in 40, and the contrast is stark: the Wald interval collapses to a single point at 0, plainly wrong, while the Wilson and Clopper-Pearson intervals still return a sensible upper limit near 0.09, matching the rough rule of three. This is exactly why a single proportion is the building block for a prevalence meta-analysis, where each study’s proportion and its variance feed the pool; the full pooling workflow is covered in our guide to conducting a meta-analysis.

Common mistakes researchers make

The first mistake is defaulting to the Wald interval out of habit, which is the version taught first and trusted least; prefer Wilson, and reserve Clopper-Pearson for guaranteed coverage on tiny samples. The second is pooling raw proportions across studies without a variance-stabilising transformation; proportions near 0 or 1 have unstable variances, so a logit or Freeman-Tukey transformation keeps the pooled estimate inside the valid range, and ignoring that distorts the heterogeneity statistics you read afterwards. The third is reporting a proportion with no interval at all, which hides how few events the figure rests on. If you need a single-arm interval for any other measure, the general-purpose confidence interval calculator covers ratios and mean differences too, and pooling prevalence correctly across studies is part of our systematic review statistics service.