Exact Binomial Test
Also known as: exact binomial test, binomial probability test, exact test for a proportion, Tam Binom Testi
The exact binomial test checks whether the observed number of successes in a fixed number of independent trials is consistent with a pre-specified success probability p₀. Because it computes exact binomial tail probabilities rather than relying on a normal approximation, it is the gold standard for testing a proportion in small samples; this two-sided formulation follows Siegel & Castellan's classic treatment (1988).
Key highlights
- Exact: computes true binomial probabilities, so it is valid even in very small samples with no normal-approximation error.
- Few assumptions — only a binary outcome, independent trials, and a fixed pre-specified p₀.
- Simple to interpret: it directly answers whether an observed success rate is compatible with a hypothesised probability.
Intuition
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How it works
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When to use it
Use the exact binomial test when you have a single binary outcome, independent trials, and a fixed reference probability p₀ that is specified in advance — for example, asking whether a success rate differs from 50%. It needs only a small sample (at least about 5 observations) and makes no normality assumption, so it is preferred over the normal-approximation proportion test precisely when the sample is small or the proportion is near 0 or 1.
Strengths & limitations
- Exact: computes true binomial probabilities, so it is valid even in very small samples with no normal-approximation error.
- Few assumptions — only a binary outcome, independent trials, and a fixed pre-specified p₀.
- Simple to interpret: it directly answers whether an observed success rate is compatible with a hypothesised probability.
- Limited to a single binary outcome compared against one fixed probability; it cannot model covariates or compare several groups.
- The success probability p₀ must be specified in advance rather than estimated from the same data.
- For large samples it offers little advantage over the simpler proportion (z) test, and its discreteness can make it conservative.
Common pitfalls
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Applications
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Frequently asked
How is the binomial test different from the proportion (z) test?
The proportion test approximates the binomial with a normal distribution and needs np₀ and n(1−p₀) to be reasonably large. The binomial test computes the exact binomial probabilities, so it stays valid in small samples or when the proportion is near 0 or 1, where the normal approximation can fail.
How small a sample can I use?
There is no normality requirement, so the test is valid even for very small samples; in practice about five or more observations is a sensible minimum to have any power.
Can I use it to compare two groups?
No. It tests one observed proportion against one fixed reference probability p₀. To compare two proportions use a proportion test or a chi-square test of association.
Why must p₀ be set in advance?
The null hypothesis is that the true success probability equals a specific value p₀. If you read p₀ off the same data you are testing, the test no longer controls the error rate and the p-value becomes meaningless.
Sources
- 1.Siegel, S. & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences (2nd ed.). McGraw-Hill.ISBN 978-0070573574
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ScholarGate. (2026, June 1). Binomial Test. ScholarGate. https://scholargate.app/statistics/binomial-test