Clopper–Pearson Interval

The Clopper–Pearson interval is a confidence interval for a proportion computed directly from the binomial distribution rather than from a normal approximation. The NIST handbook introduces it for exactly the situation the approximations handle badly: “if the number of failures is very small or if the sample size \(N\) is very small, symmetrical confidence limits that are approximated using the normal distribution may not be accurate enough for some applications. An exact method based on the binomial distribution is shown next.”

How it is constructed

Rather than computing a center and a margin, you solve for the two proportions that would make your observation sit at the edge of plausibility. With \(N_d\) events in \(N\) trials, the upper limit solves

\[\sum_{k=0}^{N_d} \binom{N}{k} p_U^k (1-p_U)^{N-k} = \alpha/2\]

and the lower limit solves the analogous equation with the sum running to \(N_d - 1\) and the right-hand side \(1 - \alpha/2\). The handbook’s characterization of the result is the important part: “the interval \((p_L, p_U)\) is an exact \(100(1-\alpha)\%\) confidence interval for \(p\). However, it is not symmetric about the observed proportion.”

Its worked example is a useful sanity anchor: twenty units sampled, four defective, \(\hat{p} = 0.20\), and a 90% interval computed by solving those equations. Here only four events are observed, so the symmetric normal approximation deserves particular caution; adequacy depends on event and nonevent counts, not on twenty trials alone.

What “exact” means, and what it costs

Exact refers to the coverage guarantee, not to the precision of the answer. Under independent binomial trials with a common event probability, the method guarantees at least the nominal coverage for every true \(p\), where the approximations aim for the nominal level only approximately and can fall short of it at particular values of \(p\) and sample sizes. That guarantee is bought by being conservative: because a binomial count is discrete, the interval usually covers more than the stated level, and it is therefore wider than it strictly needs to be.

Two practical consequences. If you are sizing a study around a target precision, a Clopper–Pearson interval will need more observations than a Wilson interval to reach the same width — worth knowing at the planning stage rather than after. And the asymmetry is real: with few events the upper bound sits much further from \(\hat{p}\) than the lower one, which is correct and looks wrong to readers expecting a “±” figure.

The boundary cases are where the method earns its keep. Zero events in \(N\) trials gives a lower limit of exactly 0 and a positive upper limit — a usable statement, where the normal approximation collapses to a zero-width interval at zero. The same holds symmetrically when every trial is an event.

Choosing between the three

The handbook recommends the Wilson score interval broadly — it “does not strongly depend upon the value of \(n\) and/or \(p\)” and was recommended “for virtually all combinations of \(n\) and \(p\)” — so the practical rule is to reach for Clopper–Pearson when you specifically need the guarantee rather than as a general default.

• Use it when coverage must be guaranteed rather than approximate — a safety or compliance claim, a regulated report, a certification.
• Use it when events are very rare or the sample is tiny, where the approximations are least reliable.
• Prefer Wilson for everyday reporting, where the extra width costs precision you would rather keep.

All three assume independent trials, so the usual caution applies before the choice of formula matters at all: repeated sessions from the same user are not independent trials, and no interval method fixes a mis-specified unit of analysis. See sample size, IID, and confidence intervals.

References

• NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.4.1 “Confidence intervals”.