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).
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
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.
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
- Siegel, S. & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences (2nd ed.). McGraw-Hill. ISBN: 978-0070573574
How to cite this page
ScholarGate. (2026, June 1). Exact Binomial Test. ScholarGate. https://scholargate.app/en/statistics/binomial-test
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Proportion TestStatistics↔ compare
- Sign TestStatistics↔ compare