Chi-Square Test

Tests categorical frequencies against an expectation — for independence, goodness of fit, or a required variance.

Also known as: χ² test, chi-square test of independence, goodness-of-fit test

The chi-square test compares observed with expected frequencies. The test statistic sums the squared deviations, each normalised by the expectation:

χ^{2} = i, j \sum \frac{( n _{ij} - e _{ij} ) ^{2}}{e _{ij}}

Test of independence — operates on a contingency table. $H_{0}$ : the two characteristics are independent. Degrees of freedom $df = (r - 1) (c - 1)$
Goodness-of-fit test — compares an observed distribution with a theoretical one. Degrees of freedom $df = k - 1 - m$ with $m$ estimated parameters
Variance test — tests $H_{0} : σ^{2} = σ_{0}^{2}$ via $χ^{2} = (n - 1) s^{2} / σ_{0}^{2}$ with $df = n - 1$ ; assumes normality and is sensitive to its violation

If the p-value falls below the significance level, $H_{0}$ is rejected — for a contingency table that means the characteristics are associated. How strong the association is, the test does not say; that needs an association measure such as Cramér's V.

Used in

Sources

Agresti, A. — Categorical Data Analysis, 3rd Edition, Wiley, 2013
Cochran, W. G. — The χ² Test of Goodness of Fit, Annals of Mathematical Statistics 23 (3), 1952
Montgomery, D. C. & Runger, G. C. — Applied Statistics and Probability for Engineers, 7th Edition
Pyzdek, T. & Keller, P. — The Six Sigma Handbook, 5th Edition

Chi-Square Test

See also

Used in

Sources