When should I use Fisher's exact test?

Use Fisher's exact test when you have a 2x2 contingency table with any expected frequency below 5 or a small total sample size (typically N < 20-30). It provides an exact p-value when the chi-square approximation is unreliable.

What is the difference between Fisher's exact test and chi-square?

The chi-square test uses an approximation suitable for large samples, while Fisher's exact test computes exact probabilities using the hypergeometric distribution. For small samples or sparse cells, Fisher's test is more accurate. With large samples, both tests give nearly identical results.

How do I interpret the odds ratio?

An odds ratio of 1 means no association. OR > 1 indicates a positive association (higher proportion in the first row for the first column), and OR < 1 indicates a negative association. If the 95% confidence interval includes 1, the association is not statistically significant.

Fisher's Exact Test Calculator

Exact test for 2×2 contingency tables. Ideal when expected cell frequencies are less than 5 or sample sizes are small. Results include exact p-value, odds ratio, and confidence interval.

What is Fisher's Exact Test?

Fisher's exact test is a statistical significance test used to determine whether there is a non-random association between two categorical variables in a 2×2 contingency table. Unlike the chi-square test, which relies on a large-sample approximation, Fisher's exact test computes the exact probability of observing the data (or more extreme data) under the null hypothesis of independence. This makes it especially appropriate when sample sizes are small or when expected cell frequencies fall below 5.

When to Use Fisher's Exact Test

Use Fisher's exact test instead of the chi-square test when one or more of the following conditions apply: your total sample size is small (typically N < 20-30), any expected cell frequency is below 5, or you have a 2×2 table with highly unbalanced marginals. It is the gold standard for small-sample categorical analysis and is commonly used in clinical trials, epidemiology, and biomedical research where sample sizes may be limited.

Fisher's Exact Test vs. Chi-Square Test

Feature	Fisher's Exact	Chi-Square
Method	Exact (hypergeometric)	Approximate
Table size	2×2 only	Any size
Sample size	Any (ideal for small)	Large (N ≥ 20)
Expected freq < 5	No problem	Unreliable
Effect size	Odds ratio, Phi	Cramer's V

Worked Example: Fisher's Exact Test

A clinical trial tests whether a new treatment improves patient outcomes compared to a control. With only 20 patients, the chi-square approximation may be unreliable, so Fisher's exact test is used.

	Improved	Not Improved	Total
Treatment	8	2	10
Control	1	9	10
Total	9	11	20

Results

Fisher's exact test, p = .003, OR = 36.00, 95% CI [3.26, 397.53]

There was a statistically significant association between treatment and outcome. Patients in the treatment group were significantly more likely to improve than those in the control group (OR = 36.00).

Assumptions of Fisher's Exact Test

Fisher's exact test has fewer assumptions than the chi-square test, but the following must still be met:

1. 2×2 Contingency Table

The data must be organized in a 2×2 table with two binary categorical variables. For larger tables, consider the chi-square test or the Freeman-Halton extension of Fisher's test.

2. Independent Observations

Each observation must be independent. Each subject contributes to only one cell of the table. For paired or matched data, use McNemar's test instead.

3. Fixed Marginals

The test assumes that either the row totals, column totals, or both are fixed by the study design. This is automatically satisfied in most experimental and observational studies.

Understanding the Odds Ratio

The odds ratio (OR) quantifies the strength and direction of the association in a 2×2 table. It compares the odds of the outcome in one group to the odds in the other group:

OR Value	Interpretation
OR = 1	No association between the variables
OR > 1	Positive association (higher odds in first row)
OR < 1	Negative association (lower odds in first row)
95% CI includes 1	Association is not statistically significant

How to Report Fisher's Exact Test in APA Format

When reporting Fisher's exact test results in APA format, include the test name, p-value, odds ratio, and 95% confidence interval:

Template

Fisher's exact test indicated a [significant/non-significant] association between [Variable 1] and [Variable 2], p = .XXX, OR = X.XX, 95% CI [X.XX, X.XX].

Example Report

Fisher's exact test indicated a significant association between treatment condition and patient improvement, p = .003, OR = 36.00, 95% CI [3.26, 397.53]. Patients receiving the treatment were significantly more likely to improve than those in the control group.

Note: Report p-values to three decimal places, using p < .001 when below that threshold. Always include the odds ratio and its 95% confidence interval. If any cell contains zero, note that the odds ratio may be undefined or infinite.

Common Mistakes to Avoid

Using chi-square with small samples: When expected frequencies are below 5, the chi-square approximation is unreliable. Fisher's exact test should be used instead for 2×2 tables with small samples.
Ignoring the confidence interval: A significant p-value alone does not tell you the magnitude of the effect. Always report the odds ratio with its 95% confidence interval to convey both the direction and uncertainty of the association.
Confusing odds ratio with relative risk: The odds ratio and relative risk are different measures. When the outcome is common (> 10%), the odds ratio overestimates the relative risk. Report both when appropriate.
Applying to non-binary data: Fisher's exact test is designed for 2×2 tables. For larger tables, use the chi-square test or the Freeman-Halton extension.
Using with paired data: Fisher's exact test assumes independent observations. For paired or matched binary data, use McNemar's test.

Calculation Accuracy

StatMate's Fisher's exact test calculations have been validated against R's fisher.test() function and SAS output. The implementation uses log-factorials to avoid numerical overflow and enumerates all possible tables with fixed marginals to compute exact two-tailed p-values. All results match R output to at least 4 decimal places.

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