Exact Binomial Test calculator

Compare a sample proportion to a theoretical probability. Compute two-sided exact binomial p-values online.

What it does

Tests whether an observed proportion differs from a hypothesised one, using the exact binomial distribution rather than a normal approximation.

When to use it

A single yes/no outcome counted over a fixed number of independent trials, compared against a known or expected rate.

Cautions

• Trials must be independent. Repeated measurements on the same subjects break that and understate the true uncertainty.
• Being exact, it is conservative at small n — the actual error rate can sit below the nominal alpha.
• Report the proportion with a confidence interval, not just the p-value.

Alternatives

• Chi-square goodness of fit — there are more than two categories
• Fisher's exact test — you are comparing proportions between two groups rather than against a fixed value

How to read the output

Observed rate and the reference p₀

The count of events out of the non-blank rows, against the rate you nominated. The gap between them is the finding. p₀ must come from outside this dataset, and it is treated as EXACT — a published rate carries its own uncertainty, and one drawn from a different population may not be the right comparator at all. Choosing p₀ after seeing your own rate guarantees the answer you were hoping for.

Exact binomial p-value

The probability of a count at least as far from p₀ as yours, if the true rate were exactly p₀. Exact rather than a normal approximation, so it stays honest at small n and at rates near 0 or 1. It is not double the smaller tail — that is a different test, and the two disagree whenever p₀ is not 0.5.

95% confidence interval (Wilson)

The range of true rates compatible with your data. This is the number to report: it says how precisely you have pinned the rate down, which the p-value never does. Wilson is the default because it keeps close to 95% coverage at every n. The Clopper-Pearson interval shown beside it is guaranteed to be at least 95% and is therefore wider — use it when you need the conservative bound. The textbook Wald interval is deliberately not offered: it can put bounds below 0, and with zero events it collapses to a single point.

Blank rows

Rows blank in the outcome column are excluded from the denominator. They are NOT counted as non-events. Doing that would shrink the rate towards zero, which on a safety measure is the flattering direction — but if the missing rows are missing BECAUSE of the outcome, no analysis here can repair that.

p-value

How surprising a difference at least this large would be IF the groups really were identical on this measure. A small p means the data would be surprising under that assumption. It is not the probability that the result is due to chance, and not the probability that the groups are the same — those read the conditional backwards. It also answers 'would this be surprising?', never 'is this big enough to matter?': with a large sample a trivial difference returns a small p, and the result looks identical to an important one.

How this calculator is validated · Which statistical test should I use?