Fisher's exact test
Also known as: Fisher-Irwin test, exact test of independence, Fisher'ın Kesin Testi
Fisher's exact test is a nonparametric exact-probability test of independence for small-sample contingency tables, introduced by R. A. Fisher in 1922. Rather than relying on a large-sample approximation, it computes the exact probability of the observed table directly from the hypergeometric distribution.
Key highlights
- Gives an exact p-value with no reliance on a large-sample approximation.
- Valid for small samples and sparse tables where the chi-square test fails.
- Naturally suited to 2×2 tables and extendable to r×c tables.
- Computationally simple and available in every analysis package.
Intuition
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How it works
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When to use it
Use it to test whether two categorical variables are independent in a 2×2 or general r×c contingency table, especially when the sample is small. It is the preferred choice over the chi-square test when expected cell frequencies fall below 5, because the chi-square approximation becomes unreliable there. The observations must be independent and the data must consist of two categorical (including binary) variables.
Strengths & limitations
- Gives an exact p-value with no reliance on a large-sample approximation.
- Valid for small samples and sparse tables where the chi-square test fails.
- Naturally suited to 2×2 tables and extendable to r×c tables.
- Computationally simple and available in every analysis package.
- Computation becomes heavy for large tables or large total counts.
- Conditioning on fixed margins can make the test conservative.
- Limited to categorical data; it tests association, not the size or direction of an effect on its own.
- For large samples it offers no advantage over the chi-square test.
Common pitfalls
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Applications
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Frequently asked
When should I prefer it over the chi-square test?
Use Fisher's exact test when the sample is small or any expected cell frequency falls below 5, where the chi-square approximation is unreliable. For large samples with comfortably large expected counts, the chi-square test is the conventional and computationally lighter choice.
Does it work only for 2×2 tables?
No. The 2×2 case is the classic and simplest form, but the test generalizes to r×c contingency tables. Larger tables simply require summing probabilities over many more possible arrangements.
One-sided or two-sided?
For a 2×2 table both are available. Use the two-sided p-value by default to test association in either direction, and reserve the one-sided version for a directional hypothesis stated before seeing the data.
What effect size should I report?
For a 2×2 table the odds ratio is the natural companion to the p-value, summarizing the strength and direction of the association. The p-value alone tells you only whether an association is detectable, not how strong it is.
Sources
- 1.Fisher, R. A. (1922). On the interpretation of chi-squared from contingency tables, and the calculation of P. Journal of the Royal Statistical Society, 85(1), 87–94.
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Cite this page
ScholarGate. (2026, June 1). Fisher's exact test. ScholarGate. https://scholargate.app/statistics/fishers-exact-test