Comparison
Chi-Square vs Fisher's Exact Test — Which to Use
Both the chi-square test of independence and Fisher's exact testassess whether two categorical variables are associated in a contingency table. The chi-square test is a large-sample approximation; Fisher's exact test computes an exact p-value and is preferred for 2×2 tables with small samples or low expected cell counts.
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Open the workspace →The key difference
The chi-square test compares your observed counts to the counts expected under independence and refers the resulting statistic to the chi-square distribution. That distribution is an approximation — it describes the sampling distribution of the statistic well only when the sample is reasonably large, and the approximation gets better as sample size grows.
Fisher's exact test takes a different route. Conditioning on the row and column totals, it enumerates the hypergeometric probabilities of every table at least as extreme as the one observed and sums them to get an exact p-value. There is no large-sample assumption, so it stays valid even when counts are tiny.
When chi-square is fine
The standard rule of thumb is that the chi-square approximation is trustworthy when all expected cell counts are at least 5, together with a reasonable overall sample (often n ≥ 20–30). When that holds, the chi-square distribution closely tracks the true sampling distribution of the statistic.
Chi-square also has a practical advantage: it works for tables larger than 2×2(any number of rows and columns), where the classic form of Fisher's exact test is harder to apply.
When to prefer Fisher's exact
Reach for Fisher's exact test on 2×2 tableswhenever any expected count falls below 5, or when the overall sample is small. In those situations the chi-square approximation can misstate the p-value, while Fisher's exact remains correct by construction.
On crosstabs.com, Fisher's exact test is computed for 2×2 tables. For larger tables, use the chi-square test (or the G-test) instead.
When to use it
Use it when
- Reach for chi-square when your sample is reasonably large.
- Reach for chi-square when all expected cell counts are at least 5.
- Reach for chi-square when the table is bigger than 2×2 (more than two rows or columns).
Not the right tool when
- Reach for Fisher's exact instead when you have a 2×2 table and any expected count is below 5.
- Reach for Fisher's exact instead when the overall sample is small.
- Reach for Fisher's exact instead when you want an exact p-value with no large-sample assumption — crosstabs.com computes Fisher's exact for 2×2 tables.
How to interpret it
Rule of thumb
If every expected count is at least 5, the chi-square test and Fisher's exact test usually agree, so either is fine. When expected counts are small, trust Fisher's exact test.
Frequently asked questions
- When should I use Fisher's exact test instead of chi-square?
- Use Fisher's exact test for a 2×2 table whenever any expected cell count is below 5 or the overall sample is small. It computes an exact p-value with no large-sample assumption, so it stays valid where the chi-square approximation becomes unreliable.
- What is the expected-count rule for chi-square?
- The common rule of thumb is that all expected cell counts should be at least 5 (with a reasonable total sample, often n ≥ 20–30). When that holds, the chi-square distribution approximates the test statistic's true distribution closely; when it fails, prefer Fisher's exact test.
- Can I use Fisher's exact test on tables larger than 2×2?
- In principle Fisher's exact test generalizes to larger tables, but it is computationally heavy and less commonly used there. crosstabs.com computes Fisher's exact for 2×2 tables; for larger tables, use the chi-square test or the G-test.
References & further reading
- Fisher, R. A. (1922). On the interpretation of χ² from contingency tables, and the calculation of P. Journal of the Royal Statistical Society, 85(1), 87–94.
- Cochran, W. G. (1954). Some methods for strengthening the common χ² tests. Biometrics, 10(4), 417–451.
- Fisher's exact test — Wikipedia
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