StatReason

Fisher's Exact Test

Test whether two binary variables are associated in a 2 by 2 table, computing an exact p-value from the hypergeometric distribution rather than relying on the chi-square test's large-sample approximation.

What this answers

This test answers "are these two binary variables associated?" for a 2 by 2 table, exactly, without the chi-square test's requirement that every expected cell count be reasonably large. It is the standard fallback whenever a contingency table has small or sparse counts.

How it is calculated

Conditional on the table's observed row and column totals (its margins), every possible 2 by 2 table with those same margins is enumerable, and each has an exact probability under the hypergeometric distribution. The p-value sums the probabilities of every such table that is at least as extreme as the one you observed, in the direction specified by your alternative hypothesis. This calculator also reports the simple sample odds ratio, (a times d) divided by (b times c), as a descriptive estimate of the association's direction and rough size, distinct from the more complex conditional maximum-likelihood odds ratio some statistical packages report by default.

Worked example

For a table with 1 and 9 in the first row and 11 and 3 in the second row, the margins are quite lopsided, exactly the situation where a chi-square test's expected-count assumption would be violated. Fisher's exact test handles this table directly through the hypergeometric distribution, with no warning needed about small expected counts, since none was ever assumed in the first place.

Assumption audit

Calculated from your data: the table's row and column margins, the exact p-value from the hypergeometric distribution, and the sample odds ratio.
Evidence to review: your table's orientation; which count is "a" versus "b" versus "c" versus "d" changes which direction "greater" and "less" refer to, though the two-sided p-value is unaffected by orientation.
You must verify: that the row and column totals used to condition this test are meaningful fixed margins for your design, and that your four counts come from independent observations.

What this result does not mean

A significant Fisher's exact test means the observed table is unlikely under independence given these margins, not that one variable causes the other, and not a measure of effect size beyond the sample odds ratio reported alongside it.

Limitations

This calculator reports the simple sample odds ratio rather than the conditional maximum likelihood estimate some packages compute; for very large tables, use the Chi-Square Test of Independence instead, since Fisher's exact enumeration becomes computationally heavier as counts grow.