2×2 Chi-Square Calculator

What is a 2×2 Contingency Table?

A 2×2 contingency table (also called a fourfold table) is the simplest form of cross-tabulation, with two rows and two columns. It is the most common layout in medical research, epidemiology, and A/B testing. CrossTabs.com automatically detects 2×2 tables and computes additional measures like odds ratio, relative risk, and Fisher's exact test.

Special Tests for 2×2 Tables

For 2×2 tables, CrossTabs.com provides several additional statistics beyond the standard chi-square test:

• Yates' continuity correction — adjusts chi-square for small samples by reducing each |O − E| by 0.5
• Fisher's exact test — computes the exact p-value using the hypergeometric distribution, recommended when any expected cell count is below 5
• Odds ratio (OR) — quantifies how strongly the presence of one variable is associated with the presence of another, with 95% Woolf CI
• Relative risk (RR) — measures the ratio of event probability in the exposed vs. unexposed group
• Number needed to treat (NNT) — the number of patients who need to be treated to prevent one additional adverse outcome

Worked Example

Consider a clinical trial comparing a new drug to placebo:

Results: χ²(1) = 7.50, p = 0.006. Odds Ratio = 3.00 [95% CI: 1.37, 6.58]. Relative Risk = 1.50 [95% CI: 1.12, 2.01]. The drug group had significantly higher improvement rates.

When to Use Yates' Correction

Yates' continuity correction is recommended when: (1) the table is exactly 2×2, (2) any expected cell count is between 5 and 10, and (3) you want a more conservative test. However, some statisticians argue it is overly conservative. CrossTabs.com calculates both corrected and uncorrected values so you can compare. For more details, see the chi-square test documentation.

Frequently Asked Questions

When should I use Fisher's exact test instead of chi-square for a 2×2 table?

Use Fisher's exact test when any expected cell count is below 5, or when the total sample size is less than 20. CrossTabs.com automatically recommends Fisher's exact test when appropriate.

How do I interpret the odds ratio?

An odds ratio of 1 means no association. OR > 1 means the outcome is more likely in the exposure group. OR < 1 means it is less likely. The 95% confidence interval tells you the precision; if it includes 1, the association is not statistically significant.

What is the difference between odds ratio and relative risk?

Relative risk (RR) compares probabilities directly and is easier to interpret. Odds ratio (OR) compares odds and is used in case-control studies where RR cannot be calculated. For rare outcomes (< 10%), OR approximates RR.