Binomial Test Calculator

A binomial test (also called an exact binomial test) is a statistical hypothesis test used to determine whether the observed proportion of successes in a fixed number of trials differs significantly from a hypothesized probability. It is based on the exact binomial distribution, making it reliable even for small sample sizes where normal approximations may break down.

A binomial experiment consists of a fixed number of independent trials, each with only two possible outcomes (success or failure), where the probability of success remains constant across all trials. Common examples include flipping a coin, testing whether a drug cures patients, or checking whether items pass quality control.

The p-value represents the probability of observing a result at least as extreme as the one you obtained, assuming the null hypothesis is true. A small p-value (typically below your significance level α) is evidence against the null hypothesis. For example, p = 0.03 with α = 0.05 means you would reject the null hypothesis.

A two-sided test checks whether the true probability differs from p₀ in either direction (higher or lower). A one-sided test checks only one direction — either that the true probability is less than p₀ (left-sided) or greater than p₀ (right-sided). Use a one-sided test only when you have a strong directional hypothesis before collecting data.

The 95% confidence interval is computed using the Clopper-Pearson exact method, which is based on the binomial distribution. It gives you a range within which the true population probability of success is likely to fall, with 95% confidence. Unlike normal approximation methods, the Clopper-Pearson interval is valid for all sample sizes.

The significance level α is the probability threshold below which you reject the null hypothesis. The most common choice is α = 0.05 (5%), meaning you accept a 5% risk of incorrectly rejecting a true null hypothesis. In fields requiring stricter standards (e.g. medicine or physics), α = 0.01 or α = 0.001 may be used.

Use the exact binomial test when your sample size is small (typically n < 30) or when the expected number of successes or failures is fewer than 5, because the normal approximation becomes unreliable in those situations. For large samples, the z-test for proportions gives nearly identical results and is computationally simpler.

The number of successes (x) cannot exceed the number of trials (n), as this is mathematically impossible in a binomial setting. If you enter a successes value greater than trials, the calculator will flag an error. Make sure x ≤ n before running the test.