Estimating confidence intervals for binomial proportions

Wald

The simplest approximation of the confidence interval for a binomial probability estimate. It uses the bounds of the normal distribution to approximate the uncertainty.

where: Z is the inverse cumulative density of the standard normal distribution (or probit), evaluated at the confidence interval , is the estimated probability (generally the proportion of successes or failures in the sample (n)).

At high confidence ranges, especially with small sample sizes, the Wald Interval estimate can produce unintuitive results: probabilities of less than zero or greater than 1. This is corrected in the Wilson Score Interval.

Wilson

The Wilson Score Interval is an asymmetric approximation, still based on the normal distribution. It explicitly recognises the minimum of 0 and maximum of 1 constraints to the estimated probability and this leads to asymmetric intervals at the extremes.

Jeffreys

Jeffreys Interval is derived from Bayesian statistics in which the observed estimate (successes and failures) can be combined with the prior estimate. For binomial data, the prior is derived from the Beta distribution, so the probability interval is simply:

where B is inverse cumulative density function of the Beta distribution.

All

The differences between the methods is visible in the following comparison chart.