Clopper-Pearson Binomial CI (Exact)
Computes the exact Clopper-Pearson confidence interval for a binomial proportion. Unlike the Wald formula and even Wilson's, which are approximations, Clopper-Pearson is built directly on the binomial distribution, guaranteeing coverage of at least the nominal level — which is why it's regarded as the reference conservative interval. It's the choice when you need a rigorous guarantee, even with small samples. Enter the successes, the total and the confidence level.
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Clopper-Pearson Binomial CI (Exact)
Computes the exact Clopper-Pearson confidence interval for a binomial proportion. Unlike the Wald formula and even Wilson's, which are approximations, Clopper-Pearson is built directly on the binomial distribution, guaranteeing coverage of at least the nominal level — which is why it's regarded as the reference conservative interval. It's the choice when you need a rigorous guarantee, even with small samples. Enter the successes, the total and the confidence level.
The conservative guarantee for a proportion
When you estimate a proportion from successes and failures, there's a whole family of confidence intervals vying for preference. Wald's and Wilson's are approximations, fast but with coverage that sometimes runs above, sometimes below what was promised. The Clopper-Pearson interval follows another philosophy: it guarantees coverage of at least the nominal level, whatever the cost.
It achieves this by building the interval directly on the binomial distribution, without approximating by a normal. The bounds come from the relationship between the binomial and the beta distribution: the lower and upper limits are quantiles of betas with specific parameters derived from the successes and failures. The price of that guarantee is that the interval tends to be a bit wider than necessary — hence the conservative nickname.
Enter the successes, the total and the confidence level. The tool returns the observed proportion and the exact bounds. It's the choice when rigor matters more than width: regulatory approval, clinical validation, any situation where promising a confidence level and delivering less would be unacceptable.
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Exact Binomial Test
Computes the exact binomial test, which checks whether the observed proportion of successes is compatible with a hypothesized probability, without resorting to the normal approximation. For small samples, where the z-test fails, this is the correct test: it sums the exact probabilities, given by the binomial distribution, of all outcomes as likely or less likely than the observed one. For example: are 8 heads in 10 tosses compatible with a fair coin? Enter the successes, the total and the hypothesized probability.
CI for Correlation (Fisher z Transform)
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Wilson Interval (Proportion)
Computes the Wilson confidence interval for a proportion, the modern alternative to the Wald interval (the p̂ ± z·SE formula you learn first). The Wald interval fails badly with small samples or proportions near 0 or 1 — it can even produce negative bounds. Wilson's fixes this by inverting the score test, which keeps it inside [0, 1] with much better coverage. It's the recommended interval for proportions. Enter the successes, the total and the confidence level.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.