Chi-Square Tests for Categorical Data

The chi-square test is one of the most widely used statistical procedures for analyzing categorical data. It evaluates whether the observed frequencies in a contingency table differ significantly from the frequencies we would expect if the row and column variables were independent.

Two Interpretations

• •Test of independence — a single sample is cross-classified on two categorical variables. Is there an association between them?
• •Test of homogeneity — independent samples from two or more populations are compared on one categorical variable. Do the populations share the same distribution?

Despite the different study designs, the arithmetic is identical: both compute the same Pearson chi-square statistic from a contingency table.

When to Use Chi-Square Tests

The simplest contingency table is the 2x2 layout, comparing two groups on a binary outcome. Each cell contains the count of observations falling in that row-column combination.

Expected Counts

Under the null hypothesis of independence, the expected count for each cell is:

Degrees of Freedom

For a 2x2 table the degrees of freedom are:

Yates' Continuity Correction

Because the chi-square distribution is continuous but the test statistic is computed from discrete counts, Yates (1934) proposed subtracting 0.5 from each absolute deviation before squaring. For the 2x2 case:

The Pearson chi-square statistic measures the overall discrepancy between observed and expected counts across all cells in an $R \times C$ contingency table:

Degrees of Freedom

The general formula for degrees of freedom is:

For example, a 3x4 table has $(3-1)(4-1) = 6$ degrees of freedom.

P-value and Decision Rule

The p-value is the probability of observing a chi-square statistic as large or larger than the computed value under the null hypothesis:

Reject $H_0$ (independence) when $p < \alpha$. The chi-square test is always one-tailed (right tail) because larger values indicate greater deviation from independence.

Statistical significance depends on sample size: a trivially small association will reach $p < 0.05$ with a large enough N. Effect size measures quantify how strong the association is, independent of sample size.

Phi Coefficient (2x2 tables)

For 2x2 tables, the phi coefficient is the correlation between two binary variables:

Cramér's V (RxC tables)

Cramér's V generalizes phi to tables larger than 2x2 by normalizing by the smaller table dimension:

Cohen's Benchmarks

Odds Ratio (2x2 tables)

The odds ratio quantifies the multiplicative change in odds of the outcome between the two groups:

Relative Risk (2x2 tables)

The relative risk (risk ratio) compares proportions directly:

The Pearson chi-square relies on a large-sample approximation: the statistic is approximately chi-square distributed. When expected cell counts are small (commonly < 5), this approximation breaks down. Fisher's exact test avoids the approximation entirely.

Hypergeometric Distribution

Given fixed marginal totals, the probability of observing cell count $a$ follows the hypergeometric distribution:

The exact p-value sums the probabilities of all tables as extreme as or more extreme than the observed table, conditional on the margins.

When to Prefer Fisher's Over Chi-Square

• •Any expected cell count is less than 5
• •Total sample size is less than about 20–30
• •The table is very unbalanced (one margin much larger than the other)
• •You want an exact p-value rather than an asymptotic approximation

Power is the probability of correctly rejecting $H_0$ when the alternative is true. For chi-square tests, power depends on the sample size, the significance level, the degrees of freedom, and the effect size.

Cohen's Effect Size w

Cohen defined effect size $w$ for chi-square tests as:

where $p_{0i}$ are the cell probabilities under $H_0$ and $p_{1i}$ are the cell probabilities under $H_1$. For a 2x2 table, $w = |\varphi|$.

Non-centrality Parameter

Under the alternative hypothesis, the test statistic follows a non-central chi-square distribution with non-centrality parameter:

Power Calculation

Power equals the probability that a non-central chi-square variate exceeds the critical value from the central distribution:

Sample Size Determination

To find the minimum $N$ for a desired power $1 - \beta$, invert the power equation. A useful approximation:

McNemar's test is designed for paired binary data — before/after measurements on the same subjects, matched case-control studies, or any design where each observation in one condition is paired with an observation in the other.

The Discordant Pairs

Consider paired binary outcomes arranged in a 2x2 table:

Only the discordant pairs ($b$ and $c$) carry information about change. The test statistic is:

This follows a chi-square distribution with 1 degree of freedom under $H_0: p_b = p_c$ (i.e., the probability of change in each direction is equal).

When to Use McNemar vs Standard Chi-Square

• •McNemar: paired or matched data — the same subjects measured at two time points, or case-control pairs
• •Standard chi-square: independent groups — different subjects in each cell of the contingency table

Key Assumptions

• •Independence: each observation is independent of every other. Clustered or repeated-measures data violate this assumption.
• •Expected cell counts: the chi-square approximation requires all expected counts to be reasonably large. The common rule of thumb is $E_{ij} \geq 5$ for all cells.
• •Fixed margins: the total sample size (and sometimes row/column totals) is fixed by design.
• •Mutually exclusive categories: each observation falls into exactly one cell.

Alternatives When Assumptions Fail

The Chi-Square Calculator runs entirely in the browser — there is no backend API. State is captured in URL parameters so results can be shared via links. Below are the four modes and their parameters.

Mode 1: Test (2x2 and RxC)

Outputs: chi-square statistic, df, p-value, phi (2x2), Cramér's V, odds ratio (2x2), relative risk (2x2), Fisher's exact p-value (2x2), expected counts.

Example URL: /calculators/chi-square?mode=test&rows=2&cols=2&data=50,10,30,20&alpha=0.05

Mode 2: Power Analysis

Outputs: required total sample size N, non-centrality parameter, critical value.

Example URL: /calculators/chi-square/power?w=0.3&alpha=0.05&power=0.8&df=1

Mode 3: Sample Size

Outputs: required total sample size, per-cell N, derived Cohen's w, chi-square critical value.

Example URL: /calculators/chi-square/sample-size?ssrows=2&sscols=2&expected=0.3,0.2,0.2,0.3&alpha=0.05&power=0.8

Mode 4: McNemar's Test

Outputs: McNemar chi-square statistic, p-value, discordant pair count, exact binomial p-value.

Example URL: /calculators/chi-square?mode=mcnemar&a=40&b=15&c=5&d=30

1. Pearson K. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine. 1900;50(302):157–175.
2. Fisher RA. On the interpretation of chi-square from contingency tables, and the calculation of P. Journal of the Royal Statistical Society. 1922;85(1):87–94.
3. Yates F. Contingency tables involving small numbers and the chi-square test. Supplement to the Journal of the Royal Statistical Society. 1934;1(2):217–235.
4. Cramér H. Mathematical Methods of Statistics. Princeton University Press; 1946.
5. McNemar Q. Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika. 1947;12(2):153–157.
6. Cochran WG. Some methods for strengthening the common chi-square tests. Biometrics. 1954;10(4):417–451.
7. Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Lawrence Erlbaum Associates; 1988.

Last updated: April 2026 | Pearson (1900), Cohen (1988)