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.
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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.
Is the coin fair? The exact answer
When you count successes in a fixed number of trials and want to know whether the observed rate matches an expected probability, the temptation is to use the normal approximation. But with few trials that approximation misleads. The exact binomial test skips the shortcuts: it computes the true probability directly from the binomial distribution.
The procedure starts from the exact probability of the observed outcome and adds to it the probabilities of all outcomes equally or less likely, on both sides of the distribution. That sum is the two-sided p-value. Because everything is computed from the binomial, with no approximation, the test is valid even for tiny samples, where the z-test for a proportion simply doesn't apply.
Enter the successes, the total number of trials and the hypothesized probability. The tool returns the exact p-value. The classic example is the coin: eight heads in ten tosses seems like a lot, but the test shows they're still perfectly compatible with a fair coin. It's the right tool for any counting situation with little data.
Related Tools
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.
Fisher's Exact Test (2×2)
Computes Fisher's exact test for a 2×2 contingency table, the right choice when the counts are small and the chi-square approximation no longer holds. Instead of approximating, it computes the exact probability of observing the current table — and tables even more extreme — holding the margin totals fixed, via the hypergeometric distribution. The tool returns the one-sided and two-sided p-values (the latter by summing the probabilities of tables as likely or less likely than the observed one). Enter the four cells a, b, c and d.
Binomial Distribution
Compute P(X = k) and P(X ≤ k) in binomial distribution B(n, p).
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.