Cramér's V

Cramér's V scales the chi-square statistic to a [0, 1] measure of association between two categorical variables. A value near 1 indicates near-perfect association; near 0 indicates none, complementing the chi-square test of independence.

ByHoang TruongUpdated July 3, 2026

See it move

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A survey of 800 customers cross-tabulates purchasing channel by age group. The chi-square test returns p = 0.001, confirming a statistically significant association. Cramér's V, however, equals 0.12 on its 0-to-1 scale, showing the relationship is in fact very weak — detectable at this sample size but of limited practical importance for decisions.

Where it fits

SubjectData Analysis & StatisticsAdvancedTopicCommon Significance TestsAdvanced

The formula

LaTeX

V = \sqrt{\frac{\chi^2}{n \cdot \min(r - 1,\; c - 1)}}

Variables

$V$: Cramér's V association measure (dimensionless)
$χ^{2}$: chi-square test statistic from the contingency table (dimensionless)
$n$: total sample size (dimensionless)
$r$: number of rows in the contingency table (dimensionless)
$c$: number of columns in the contingency table (dimensionless)

V ∈ [0, 1]. The denominator normalises χ² by its theoretical maximum given the table dimensions, making V comparable across tables of different sizes.