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.
Resultado
—
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).
One-Proportion z-Test
Computes the one-proportion z-test statistic, used to check whether the proportion observed in a sample differs from a hypothesized value. For example: out of 100 voters, 45 approve a measure — is that compatible with 50% in the population? The statistic divides the difference between the observed and hypothesized proportions by the standard error computed under the null hypothesis. It applies when the sample size is large enough for the normal approximation. Enter the successes, the total and the hypothesized proportion.
Sign Test (Paired)
Computes the sign test, the simplest and most robust paired test of all. It completely ignores the magnitude of the differences between pairs and looks only at the sign: how many times the value went up and how many it went down. Under the hypothesis of no effect, ups and downs should balance out, like heads and tails, and the p-value comes straight from a binomial distribution. Because it uses so little information, it's less powerful than Wilcoxon, but it makes almost no assumptions. Enter the two paired series.
Two-Proportion z-Test
Computes the two-proportion z-test statistic, which compares the success rates of two independent groups — the backbone of any A/B test. For example: variant A converted 45 of 100 and B converted 30 of 100; is the difference real or noise? The statistic uses the pooled proportion of the two groups to estimate the standard error under the hypothesis that the proportions are equal. Enter the successes and the total of each of the two groups.
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.