Cramer's V Calculator

Cramer's V is computed from the Pearson chi-square statistic using the formula:

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

Where:

• χ² — the Pearson chi-square test statistic from the contingency table
• n — the total number of observations (sample size)
• r — the number of rows in the contingency table
• c — the number of columns in the contingency table
• min(r−1, c−1) — the minimum of the row and column degrees of freedom

The denominator normalizes the chi-square statistic so that V always falls between 0 and 1 regardless of table size. For a 2×2 table, Cramer's V equals the absolute value of the Phi coefficient. For larger tables, V provides a comparable measure that accounts for the additional degrees of freedom.

Cramer's V is symmetric — swapping rows and columns gives the same result. It is also dimensionless, making it directly comparable across studies with different sample sizes and table dimensions.

A bias-corrected version exists for small samples:

$$\tilde{V} = \sqrt{\frac{\tilde{\phi}^2}{\min(r-1, c-1)}}$$

where $$\tilde{\phi}^2 = \max\left(0, \phi^2 - \frac{(r-1)(c-1)}{n-1}\right)$$. This correction reduces upward bias in small samples.

The interpretation of Cramer's V follows Cohen's effect size conventions, though the thresholds depend on the degrees of freedom. For general guidance:

• V < 0.10: Negligible association
• 0.10–0.30: Small association
• 0.30–0.50: Medium association
• V ≥ 0.50: Large association

Always report Cramer's V alongside the chi-square test result and p-value, as a statistically significant chi-square does not necessarily imply a practically meaningful association, especially with large sample sizes.