StatReason

Cramer's V Calculator

Summarize how strongly two categorical variables are associated, on a scale from 0 (no association) to 1 (complete association), for any size contingency table.

One row per line, comma-separated counts. Works for any number of rows and columns, not just 2x2.

What this answers

Cramer's V answers "how strong is the association between these two categorical variables, regardless of table size?" Unlike a chi-square test, which tells you whether an association is statistically significant, Cramer's V tells you how large that association actually is, on a bounded 0 to 1 scale that stays comparable across tables of different sizes and sample sizes.

How it is calculated

Cramer's V is derived from the chi-square statistic, divided by the sample size times the smaller of (rows minus 1) or (columns minus 1), then square-rooted. That denominator normalizes the raw chi-square value so V always falls between 0 and 1 regardless of how many rows or columns your table has, unlike chi-square itself, which grows with sample size and table dimensions in ways that make it hard to compare across different studies.

Worked example

For a table where 10 observations fall in one diagonal cell and 10 in the other, with the off-diagonal cells at zero: every row perfectly predicts its column, a textbook case of complete association, giving Cramer's V exactly equal to 1, the maximum possible value.

Assumption audit

Calculated from your data: the chi-square statistic underlying V, and the table's row and column totals used to compute it.
Evidence to review: V summarizes magnitude only. Pair it with the Chi-Square Test of Independence if you also need to know whether this association is statistically significant, not only how large it is.
You must verify: that each observation contributes to exactly one cell, and that association, however strong, does not by itself mean one variable causes changes in the other.

What this result does not mean

Association in a contingency table, however large V measures it to be, does not establish that one categorical variable causes the other. A shared underlying cause, how categories were defined, or the specific sample collected can all produce a large V without any direct causal link.

Limitations

Cramer's V reports magnitude only, with no direction (it cannot be negative, since direction is not meaningful for unordered categories) and no significance test attached. With a small sample, V can be unstable and should be interpreted alongside a formal significance test.