Binomial Test Calculator

Enter your number of trials (n), observed successes (x), expected probability of success (p), and significance level (α) to run an exact binomial test. Choose your alternative hypothesis (two-sided, less, or greater) and get back the p-value, test decision, and 95% confidence interval for the true proportion. Also try the calculate Kolmogorov-Smirnov Test KS Test Statistic (D).

Results

P-Value

Observed Proportion (x/n)

95% CI — Lower Bound

95% CI — Upper Bound

Test Decision

What is a binomial test?

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. See also our Critical Value Calculator.

What is a binomial experiment?

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.

What does the p-value mean in a binomial test?

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.

What is the difference between one-sided and two-sided tests?

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.

What is the 95% confidence interval in this calculator?

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.

What is the significance level (α) and how do I choose it?

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.

When should I use a binomial test instead of a z-test for proportions?

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.

What happens if my successes value is greater than my trials?

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.