Test independence between two categorical variables when sample sizes are small or expected cell counts are below 5.
Fisher's Exact Test is a non-approximation test of independence for 2×2 contingency tables. Unlike the Chi-Square test, which uses a large-sample approximation, Fisher's Exact Test calculates the exact probability of observing a table at least as extreme as the one obtained, given fixed marginal totals. It is preferred when sample sizes are small (typically when any expected cell count falls below 5) or when the total sample size is less than 20.
The test is widely used in medicine, genetics, and epidemiology when testing associations in rare events. For example, comparing the proportion of adverse events between a treatment and control group when events are rare, or testing gene-disease associations in small family studies.
StatRyx constructs the 2×2 contingency table, computes the exact two-tailed p-value using the hypergeometric distribution, and reports the odds ratio — all formatted in APA 7 style.
Read the result through the odds ratio. An odds ratio of 1 means no association; above 1 means the outcome is more likely in the first row's group; below 1, less likely. The p-value is exact — it comes from the hypergeometric distribution rather than a large-sample approximation — so it stays valid however small the cells are. In the example table, 12 of 15 treated patients recovered against 4 of 15 controls, an odds ratio of 11.0.
| Recovered | Not Recovered | Total | |
|---|---|---|---|
| Treatment | 12 | 3 | 15 |
| Control | 4 | 11 | 15 |
| Total | 16 | 14 | 30 |
Fisher's exact test indicated a significant association between treatment group and recovery status, p = .009, OR = 11.00. Twelve of 15 treated patients (80%) recovered, compared with 4 of 15 controls (27%).
Yes. You can run it without an account (3 free analyses), and a free account adds 10 more — a one-time allowance, no payment needed. Every result includes the APA 7 write-up. Paid plans (Student from $9/month) add more analyses and Word, PDF and Excel export.
A chi-square result in StatRyx warns you when expected cell counts fall below 5, and on a 2×2 table it also reports Fisher's exact p-value (as SPSS does). You can also ask for Fisher's exact test directly.
The standard Fisher's exact test applies to 2×2 tables; its r × c generalisation is the Fisher–Freeman–Halton test. StatRyx runs Fisher's test on 2×2 tables — for larger tables, merge categories into two groups or use the chi-square test.
Fisher's test has no test statistic, so report the exact p-value and the odds ratio, with the proportions in each group: "Fisher's exact test showed a significant association between group and recovery, p = .009, OR = 11.00; 80% of treated patients recovered compared with 27% of controls."
It is the extension of Fisher's exact test to tables larger than 2×2 (r × c). StatRyx runs Fisher's exact test on 2×2 tables; for a larger table it runs the chi-square test of independence and shows the expected counts. If too many expected counts are small, merge categories into two groups and use Fisher's test.
Run a Fisher's Exact Test on your data →