Enter your chi-square statistic (χ²), sample size (n), number of rows, and number of columns to compute Cramer's V — a standardized effect size measuring the strength of association between two categorical variables. You get back the V value (0 to 1), its interpretation (weak, moderate, or strong), and the underlying degrees of freedom used in the calculation.
Cramer's V
--
Strength of Association
--
Chi-Square (χ²)
--
Min(R−1, C−1)
--
Effect Size Category
--
Cramer's V is a statistic that measures the strength of association between two categorical (nominal) variables in a contingency table. It generalizes the phi coefficient to tables larger than 2×2. The value ranges from 0 (no association) to 1 (perfect association), making it easy to interpret effect size regardless of sample size.
The chi-square statistic depends heavily on sample size — a very large sample can produce a statistically significant chi-square even for a trivially small association. Cramer's V standardizes the effect size between 0 and 1, allowing you to compare association strength across different studies or datasets regardless of their sample sizes.
The formula is V = √(χ² / (n × min(R−1, C−1))), where χ² is the chi-square test statistic, n is the total sample size, R is the number of rows, and C is the number of columns. The denominator uses the smaller of R−1 or C−1 to keep V bounded between 0 and 1.
Interpretation depends on the degrees of freedom (df* = min(R−1, C−1)). For df*=1: small ≈ 0.10, medium ≈ 0.30, large ≈ 0.50. For df*=2: small ≈ 0.07, medium ≈ 0.21, large ≈ 0.35. For df*=3: small ≈ 0.06, medium ≈ 0.17, large ≈ 0.29. Always interpret V in the context of your specific field and research question.
Yes, Cramer's V is a symmetric measure. It does not matter which variable you place in the rows and which in the columns — the resulting V value will be the same. This makes it especially convenient for describing mutual association without implying directionality.
Cramer's V can be upwardly biased with small sample sizes, causing it to overestimate the true association. A bias-corrected version (Cramer's Ṽ) exists to address this. Additionally, V only measures the strength of association, not its direction or cause — it should be used alongside other descriptive statistics and domain knowledge.
Cramer's V was designed for nominal (unordered categorical) variables. While it can be computed for ordinal data, it does not take advantage of the ordering information. For ordered categories, measures like Kendall's tau or Somers' D may be more appropriate.
For a 2×2 contingency table, Cramer's V equals the absolute value of the phi coefficient. Cramer's V is a direct generalization of phi to larger tables, which is why phi is sometimes described as a special case of Cramer's V when both variables have exactly two categories.