Cramer's V Calculator

Measure the strength of association in a contingency table from chi-square, sample size, and table shape.

What it does

A chi-square test tells you whether two categorical variables are associated. It does not tell you how strong the association is. With a large enough sample, even a tiny departure from independence can produce a small p value. Cramer's V turns the same chi-square into a number on a 0 to 1 scale, so the strength can be compared across tables of different sizes.

The formula divides chi-square by the sample size and by the smaller of the two table dimensions minus one, then takes the square root. That adjustment keeps V inside 0 to 1 and makes it comparable between a 2x2 table and a 3x5 table. For a 2x2 table, Cramer's V is the same as the phi coefficient.

The denominator uses min(r minus 1, c minus 1), not the product of the two. That is the part people get wrong. The chi-square test itself uses (r minus 1) times (c minus 1) degrees of freedom for the p value, but V uses only the smaller dimension. A 3x2 table therefore has two degrees of freedom for the test and one for V.

Cohen's rules of thumb are often quoted as 0.10, 0.30, and 0.50, but those values apply when the smaller dimension minus one is 1. For a table with two degrees of freedom in the denominator, the medium threshold drops to about 0.21. For three, it drops to about 0.17. The calculator uses the correct set for the table you enter.

Cramer's V describes strength, not pattern. Two tables can have the same V with very different arrangements of the counts. It also says nothing about causation, and it can be inflated by small expected counts or by combining categories after looking at the data. Report it with the chi-square test, the sample size, and the table itself.

How to use

1. Run the chi-square test on your contingency table and note the statistic.
2. Enter the total number of observations in the table.
3. Enter the number of rows and columns, counting every category.
4. Read Cramer's V and the effect size band for that table shape.
5. Check the p value to see whether the association is distinguishable from zero.
6. Report V with the table dimensions, sample size, and p value so readers can judge it.

Formula

V

Cramer's V, from 0 to 1.

chi-square

Pearson chi-square statistic from the contingency table.

n

Total number of observations in the table.

r

Number of rows.

c

Number of columns.

The p value comes from the chi-square distribution with (r - 1)(c - 1) degrees of freedom. The V denominator uses min(r - 1, c - 1), which is why the two degrees of freedom values can differ. For a 2x2 table, V equals phi. Cohen's thresholds are 0.10, 0.30, and 0.50 when the denominator is 1; 0.07, 0.21, and 0.35 when it is 2; and 0.06, 0.17, and 0.29 when it is 3. The bands in the result use those values.

Worked example

Reference table

Limits

• Only applies to categorical data arranged in a contingency table.
• The chi-square approximation is weak when many expected counts are below 5.
• Cohen's thresholds are rules of thumb, not universal standards.
• V does not show which cells drive the association or whether the pattern is ordered.

Common mistakes

• Using (r - 1)(c - 1) in the denominator instead of min(r - 1, c - 1).
• Reporting V without the sample size or table dimensions.
• Calling V a measure of causation.
• Applying the 0.10, 0.30, 0.50 thresholds to every table shape.
• Combining sparse categories after seeing the result and not reporting it.

Try it yourself

Quick checks. Answers sit tucked under each question.

Questions

Sources

• University of Cambridge MRC CBU: Cramer's V
• University of Washington: Assessing the strength of chi-square statistics

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