Analyze contingency tables from 2×2 up to 5×5 with Pearson's chi-square test, Yates' continuity correction, Fisher's exact test (for 2×2 tables), and effect size measures including Cramér's V, phi coefficient, and odds ratios.
Enter observed counts for your 2×2 contingency table. Pre-filled with an example dataset.
Example (2×2): Observed values [30, 10; 15, 45] yield χ²(1) ≈ 16.67, p < 0.001, Cramér's V ≈ 0.41 (Medium), OR = 9.00.
Choose the dimensions of your contingency table, from 2×2 up to 5×5.
Fill in the observed frequencies for each cell. Optionally label rows and columns.
See chi-square, p-value, effect sizes, expected counts, and warnings about small expected values.
Copy the complete results to your clipboard for reporting in your manuscript or analysis.
The chi-square approximation breaks down when expected counts are too small. If any expected count is below 5, the p-value may be unreliable. Use Fisher’s exact test for 2×2 tables or consider collapsing categories for larger tables.
Fisher’s test computes an exact p-value based on the hypergeometric distribution and does not rely on large-sample approximations. It is the gold standard for 2×2 tables with small expected counts.
P-values depend heavily on sample size. A trivial association can be statistically significant with a large N. Cramér’s V (for any table size) or the phi coefficient and odds ratio (for 2×2 tables) quantify the strength of the association.
APA style recommends reporting χ²(df, N) = value, p = value, along with an effect size measure. Include the full contingency table in your results or supplementary material so readers can verify the analysis.
A chi-square calculator online automates Pearson's goodness-of-fit procedure, which Karl Pearson (1900) introduced as the first formal test of statistical significance for categorical data. The test statistic χ² = Σ(O − E)² / E compares observed cell frequencies against those expected under the null hypothesis of independence, and its sampling distribution converges to the chi-square distribution as sample size grows. The likelihood ratio test (G-test) provides an asymptotically equivalent alternative that is preferred in some fields because of its additive decomposition property, and the Cochran-Armitage trend test extends the chi-square framework to ordinal exposures by testing for a linear trend across ordered categories. Agresti (2007), in An Introduction to Categorical Data Analysis, emphasises that this large-sample approximation requires all expected counts to be at least five—a condition that frequently fails in clinical subgroup analyses with sparse cells. In systematic reviews of diagnostic accuracy, the same 2×2 table that feeds a chi-square test also yields sensitivity, specificity, and likelihood ratios; our diagnostic accuracy calculator extracts all of these measures from a single contingency matrix, making the two tools natural companions in any diagnostic test accuracy review.
The Fisher's exact test calculator embedded in this tool addresses the limitation that Pearson's asymptotic approximation becomes unreliable when expected counts are small. Fisher (1922) developed the exact test by enumerating every possible table configuration that preserves the observed marginals, computing the probability of each under the hypergeometric distribution. The Cochrane Handbook (Higgins et al., 2023) recommends Fisher's exact test over the chi-square approximation whenever any expected cell count falls below five or the total sample is less than roughly 40 participants. Because the exact test is computationally intensive for tables larger than 2×2, this calculator restricts Fisher's test to 2×2 layouts while offering Yates' continuity correction as a conservative alternative for larger tables. When zero cells occur in 2×2 tables, continuity corrections such as the Haldane (adding 0.5 to all cells) or Sweeting correction must be applied before computing odds ratios or log-transforming for meta-analysis, as zero counts make the standard estimators undefined. The Mantel-Haenszel stratified analysis extends the basic chi-square test by pooling odds ratios across strata, controlling for confounders while maintaining the simplicity of contingency-table analysis. When the research question concerns agreement between two raters rather than association between two variables, Cohen's kappa provides a chance-corrected measure of concordance; our Cohen's kappa calculator handles both 2×2 and multi-category agreement matrices with prevalence-adjusted and bias-adjusted statistics.
A contingency table calculator must also quantify the strength of the association it detects, because p-values alone conflate effect magnitude with sample size. Cramér (1946) proposed V = √(χ² / (N × (k − 1))) as a normalised measure ranging from 0 to 1 that generalises the phi coefficient to tables larger than 2×2. This tool reports Cramér's V alongside the odds ratio and its 95 % confidence interval for 2×2 tables, following the reporting standards that Agresti (2007) outlines. Researchers should be alert to Simpson's paradox, where an association that appears in several subgroups reverses direction when the data are combined, underscoring the importance of examining stratified tables before interpreting aggregate chi-square results. Translating Cramér's V into a standardised mean difference or computing the number needed to treat requires additional conversion steps; our NNT calculator bridges that gap for dichotomous outcomes, while our effect size calculator converts between d, r, and odds ratios when a unified metric is needed for meta-analytic pooling.
Planning the sample size for a study that will rely on the chi-square test is a prerequisite for adequate statistical power. Agresti (2007) notes that underpowered studies routinely produce non-significant chi-square results that are mistakenly interpreted as evidence of no association—a Type II error that propagates through any subsequent meta-analysis that includes the study. Our power analysis calculator estimates the minimum N required to detect a given Cramér's V at conventional α and β thresholds, closing the loop between design and analysis. The Cochrane Handbook (Higgins et al., 2023) further advises that systematic reviewers should evaluate whether individual studies were adequately powered before interpreting subgroup chi-square tests, because heterogeneity statistics such as Cochran's Q share the same distributional assumptions. By combining exact and asymptotic tests with normalised effect sizes, this tool supports the complete contingency-table workflow from raw counts to publication-ready results.
The Pearson chi-square test assumes: (1) observations are independent, (2) expected counts are sufficiently large — the common rule is all expected counts ≥ 5, though some sources accept 80% of cells ≥ 5 with none below 1. When expected counts are small, Fisher’s exact test (for 2×2 tables) or exact permutation tests are preferred.
Use Fisher’s exact test when any expected cell count falls below 5, when the total sample size is small (roughly N < 40), or when you want an exact p-value rather than an asymptotic approximation. Fisher’s test is computationally intensive for large tables, so it is typically applied only to 2×2 tables.
Yates’ correction subtracts 0.5 from the absolute difference |O – E| before squaring, adjusting the chi-square statistic downward to better approximate the discrete distribution. It is applied only to 2×2 tables. Some statisticians consider it overly conservative and prefer Fisher’s exact test instead.
Cramér’s V measures the strength of association in contingency tables and ranges from 0 (no association) to 1 (perfect association). Common benchmarks: V < 0.1 = Negligible, 0.1–0.3 = Small, 0.3–0.5 = Medium, > 0.5 = Large. For 2×2 tables, Cramér’s V equals the absolute value of the phi coefficient.
Report the chi-square statistic, degrees of freedom, p-value, and an effect size measure. For example: “χ²(1, N = 100) = 12.34, p < 0.001, Cramér’s V = 0.35.” For 2×2 tables, also report the odds ratio with its 95% CI. If any expected counts were below 5, report Fisher’s exact p-value instead.
Use Fisher’s exact test when any expected cell count is less than 5, or when the total sample size is less than about 40. Fisher’s test calculates the exact probability under the hypergeometric distribution and is valid for all sample sizes, while the chi-square test relies on a large-sample approximation that becomes unreliable with sparse data.
Cramér’s V is an effect size measure for chi-square tests, ranging from 0 (no association) to 1 (perfect association). It equals the phi coefficient for 2×2 tables and extends to larger tables. Cohen (1988) suggested V = 0.1 (small), 0.3 (medium), 0.5 (large) for df* = 1, with different benchmarks for higher degrees of freedom.
The chi-square test assumes: (1) observations are independent, (2) data are categorical counts (not percentages or proportions), (3) all expected cell frequencies are at least 5, and (4) the sample is randomly drawn from the population. Violating the expected frequency assumption is the most common issue; Fisher’s exact test is the appropriate alternative when this assumption fails.
Our biostatisticians can perform chi-square analyses, logistic regression, and any other statistical tests required for your systematic review or primary research study.