Home Statistical Dictionary Chi-Squared Test

Chi-Squared Test

The broad family of hypothesis tests built around a test statistic that follows a chi-square distribution under the null hypothesis, most commonly used for categorical data.

Hypothesis TestingBeginnerFree

In Plain English

Chi-squared test is really an umbrella term covering a whole family of tests that share one thing in common: their test statistic, once you calculate it, follows a chi-square distribution if the null hypothesis is true. The two most famous members of this family test whether two categorical variables are related (the chi-square test of independence) and whether a set of categories matches some expected proportions (the chi-square goodness-of-fit test), but many other tests across statistics also produce chi-square-distributed statistics.

Definition

A chi-squared test is any hypothesis test whose test statistic is constructed so that, under the null hypothesis, it follows a chi-square distribution with a specified number of degrees of freedom. The most widely recognized applications compare observed categorical frequencies to expected frequencies, $χ^{2} = \sum_{i} \frac{( O _{i} - E _{i} ) ^{2}}{E _{i}}$ , summed over categories or contingency-table cells, but the chi-square distribution also arises as the asymptotic reference distribution for the likelihood ratio test, the score test, and the Wald test in a wide range of parametric models, making 'chi-squared test' as much a description of a test statistic's distributional behavior as a single specific procedure.

Formula

χ^{2} = i \sum \frac{( O _{i} - E _{i} ) ^{2}}{E _{i}}

Notation

O_{i}

—Observed frequency in category i

E_{i}

—Expected frequency in category i, under the null hypothesis

Properties

The chi-square distribution's origin as a sum of squared standard normal variables is what connects seemingly unrelated chi-square tests, the categorical-data chi-square statistic and the asymptotic chi-square distribution of a likelihood ratio statistic both ultimately trace back to this same underlying quadratic-form structure.
A common rule of thumb (often attributed loosely to Cochran) is that the chi-square approximation for categorical tests becomes unreliable when expected cell counts fall below about 5, in that low-count regime, an exact test (like Fisher's exact test) is generally preferred over the chi-square approximation.
'Chi-squared test' without further qualification is ambiguous in applied writing, precisely because it covers such a broad family, careful statistical writing specifies which chi-squared test is meant (of independence, of goodness-of-fit, or as the reference distribution for some other underlying test construction).

At a Glance

Core formula (categorical case)χ² = Σ (Oᵢ - Eᵢ)² / Eᵢ

Most famous memberschi-square test of independence, chi-square goodness-of-fit test

Also arises asthe asymptotic reference distribution for LRT, score, and Wald tests generally

Requiressufficiently large expected cell counts for the chi-square approximation to be reliable

Last updated August 8, 2026← Back to the Dictionary