Fisher's Exact Test Calculator (2×2)

When to use Fisher's exact test

Fisher's exact test calculates the exact probability of the observed table, given its row and column totals, instead of relying on the chi-square approximation.

It is the usual choice when expected counts are small, typically when any expected count is below 5 in a 2×2 table.

The chi-square test relies on an approximation that is only trustworthy when no more than 20% of cells have expected counts below 5 and no cell is below 1. In a 2×2 table, a single small cell already breaks that rule. Fisher's exact test has no such requirement: it is valid at any sample size.

Worked example: in a small pilot, 1 of 10 treated participants and 7 of 10 controls had the outcome. The expected count is 4 in both outcome cells, so chi-square is unreliable. Fisher's exact test gives a two-sided p = 0.020 and a sample odds ratio of 0.05. A chi-square test on the same table gives p = 0.006, overstating the evidence because the approximation breaks down with counts this small.

When you report the result, give the counts as well as the p-value, for example: the outcome occurred in 1/10 treated and 7/10 control participants (Fisher's exact test, two-sided p = 0.020). Add an effect estimate such as the odds ratio or risk difference with its confidence interval, because a p-value alone does not show how large the difference is.

How it is calculated

• Probabilities from the hypergeometric distribution with fixed margins
• Two-sided p = sum of the probabilities of all tables no more likely than the observed one

Sources

• Kim HY. Statistical notes for clinical researchers: Chi-squared test and Fisher's exact test. Restor Dent Endod 2017;42(2):152-155
• NIST/SEMATECH e-Handbook of Statistical Methods: Chi-square test