Wilson confidence interval calculator for pass rates

A pass rate measured on n tasks is an estimate, and its interval says how far from the true rate it could be. Enter the number of tasks and the observed pass rate to get the Wilson interval, or switch modes and set a target half-width to see how many independent tasks that precision needs.

What it computes#

The observed pass rate $\hat p = k/n$ estimates the true rate $p$ of the population the claim is about. The Wilson interval around it has

$$\begin{aligned} \text{center} &= \frac{\hat p + z^2/(2n)}{1 + z^2/n} \\ \text{half-width} &= \frac{z}{1 + z^2/n}\sqrt{\frac{\hat p(1 - \hat p)}{n} + \frac{z^2}{4n^2}} \end{aligned}$$

where $z$ is the standard normal quantile, 1.96 for 95%. The half-width shrinks roughly with $1/\sqrt{n}$, so halving it takes about four times as many tasks.

When it applies#

The Wilson interval is the default for independent binary outcomes at moderate n. Unlike the Wald form, it keeps a nonzero width when every task passes or every task fails.

• The tasks must represent the population the claim is about. A suite drawn from another distribution estimates a different estimand, however narrow its interval.
• When trials repeat within tasks, for example several attempts per task, the outcomes are not independent: resample whole tasks instead.
• The sample size it suggests counts independent tasks, before clustering or stratification.

Report every pass rate with its interval and its n. The derivation and worked examples are in Estimands, pass rates, and intervals.