Fisher's Exact Test Calculator

Fisher's exact test computes the hypergeometric probability of obtaining a specific contingency table, given the marginal totals. For a 2x2 table with cells A, B, C, and D, the exact probability p of that specific arrangement is calculated using the formula: P = ((A+B)! (C+D)! (A+C)! (B+D)!) / (A! B! C! D! N!), where N is the grand total. The calculator sums these exact probabilities for all possible tables that are as extreme or more extreme than your observed data, depending on your chosen tail (one-tailed or two-tailed). It also computes the odds ratio as (A × D) / (B × C) to measure the strength of association between the variables.

Imagine a clinical study testing a new medication for reducing fever. Out of 20 patients, 10 received the drug and 10 received a placebo. In the drug group, 8 patients recovered and 2 did not. In the placebo group, 3 patients recovered and 7 did not. This gives us our 2x2 contingency table values: A = 8, B = 2, C = 3, and D = 7. The row and column sums are calculated as: A+B = 10, C+D = 10, A+C = 11, B+D = 9, and total N = 20. Plugging these values into the hypergeometric probability formula and evaluating the two-tailed extreme configurations yields an exact p-value of approximately 0.035. Because this is less than the standard significance threshold of 0.05, we reject the null hypothesis and conclude there is a statistically significant association between the medication and faster recovery.

When setting up your 2x2 matrix, ensure that your categories are mutually exclusive and that each observation belongs to only one cell. While Fisher's exact test is famous for handling small sample sizes where expected cell frequencies drop below 5, it is equally valid for larger datasets, though computational time may increase. Always select your tail type—one-tailed or two-tailed—before looking at the results to avoid confirmation bias. Finally, interpret your odds ratio alongside the p-value; a significant p-value tells you an effect exists, while the odds ratio tells you the magnitude of that effect.