Cramér's V Calculator

The p-value moves with the sample size; V does not

This is the whole reason the coefficient exists, and multiplying a table through is the cleanest way to see it because nothing about the pattern changes.

Take the tool’s first preset at n = 90: χ² = 13.2554, V = 0.27136852, p = 0.0100930. Multiply every cell by 20 and you have the identical proportions in a sample of 1,800.

V is still 0.27136852 — every digit. χ² is 265.1071, exactly twenty times larger, and p has fallen to 3.6172×10⁻⁵⁶.

Chi-square is exactly proportional to the sample size for a fixed pattern. That is arithmetic, not an approximation: each cell’s contribution scales linearly when observed and expected counts scale together.

So “p < 0.001” on a large survey says almost nothing about strength. Any association at all becomes significant with enough respondents, and V is the number that stays still while significance accumulates.

The divisor uses the smaller dimension, not the degrees of freedom

The two most common errors in computing V are using the wrong k and confusing k − 1 with the degrees of freedom. They are different numbers on any non-square table.

k is min(rows, columns). On the tool’s 4×3 preset, k = 3 and the divisor is k − 1 = 2, while the degrees of freedom for the chi-square test are (4−1)(3−1) = 6.

Using the larger dimension makes V too small, and using the degrees of freedom in place of k − 1 makes it far too small on a large rectangular table — both silently, because the result is still a plausible-looking number between 0 and 1.

The cap exists because √(χ²/n) alone exceeds 1 above 2×2. That quantity is the mean-square contingency, and its maximum grows with the table’s smaller dimension. Dividing it back down is precisely what Cramér contributed.

On a 2×2 the correction vanishes — k − 1 = 1 — which is why V and |φ| coincide there and only there.

V has no natural scale, and the published thresholds depend on df

Unlike a correlation, V is not anchored to anything interpretable. 0.27 is not “27% of something”, and the conventional bands are weaker than they look.

Cohen’s benchmarks vary with the degrees of freedom. For df = 1 the small, medium and large marks are 0.10, 0.30 and 0.50; for df = 2 they are 0.07, 0.21 and 0.35; for df = 3 they are 0.06, 0.17 and 0.29. Quoting the df = 1 bands for a larger table overstates the effect.

V is also not a proportion of variance the way r² is. Squaring it gives χ²/(n(k−1)), which has no clean interpretation as shared variance.

And it is always non-negative, so it says nothing about direction. On tables larger than 2×2 direction has no single meaning anyway, but for a 2×2 the signed phi coefficient carries information V discards.

The most useful reading is comparative. V = 0.27 against V = 0.09 on tables of the same shape is a real statement; V = 0.27 read against an absolute band is a convention.

V inherits every assumption of the chi-square underneath it

It is computed from χ², so anything that makes the chi-square unreliable makes V unreliable in the same way and by the same amount.

The cells must be counts of independent observations. Percentages, rates and repeated measurements of the same people all break it — and entering percentages is the single most common way to get a wrong V, because n becomes 100 regardless of the real sample.

Small expected counts distort the chi-square, and therefore V. The usual guidance is that no expected count should fall below about 5; the goodness-of-fit calculator measures what actually happens below that floor.

For a small 2×2, use Fisher’s exact test for the p-value and keep V as the effect size. The two come apart cleanly, and there is no rule that the test and the effect size must share a distribution.

V is also biased upward in small samples, because χ² has a positive expectation even under independence. Bias-corrected variants exist and matter mainly when n is small relative to the number of cells.

A significant table does not say which cells caused it

Both the test and V are omnibus. They describe the whole table and name no part of it.

Standardised residuals are the usual next step. Each cell’s (observed − expected)/√expected shows where the table departs from independence, and values beyond about ±2 mark the cells driving the statistic.

Collapsing categories changes the answer. Merging two columns changes k, the degrees of freedom and the counts at once, so V before and after a merge are not comparable — and merging after seeing the residuals is selecting on the outcome.

Splitting the table into 2×2 sub-tables is multiple testing. Four such comparisons need a correction, the same as any other family of tests.

And an association is not a direction of causation, which matters more here than in most places because contingency tables are so often built from observational survey data where the grouping variable was not assigned.

Reporting Cramér's V

Four things, and the table dimensions are the one that makes the value interpretable.

Give the table’s dimensions and n. V = 0.25 on a 4×3 and V = 0.25 on a 2×2 sit against different benchmarks, and without the shape a reader cannot pick the right one.

Give χ² with its degrees of freedom and p. “χ²(6) = 50.09, p < 0.001, V = 0.250, n = 400” is a complete result in one line.

Say which Cohen benchmark you used, if you label the size at all. The bands move with df, and an unlabelled “medium effect” is not checkable.

And give the full table where space allows. It is a handful of integers, it lets a reader recompute everything including the residuals, and it is the only way anyone can see which cells the association lives in.

Sources and methodology

References for Cramér's V and the chi-square it rescales.

• Cramér, Mathematical Methods of Statistics, Princeton University Press, 1946 — where V is introduced
• Cohen, Statistical Power Analysis for the Behavioral Sciences, 2nd ed., 1988 — for the df-dependent benchmarks
• NIST/SEMATECH e-Handbook — the chi-square test of independence and its expected-count guidance
• Phi Coefficient Calculator — the signed 2×2 case, with its marginal ceiling
• Chi-Square Calculator — the test this rescales
• How we calculate — methodology

Method. V is computed from the chi-square statistic and the smaller of the two table dimensions, and the scaling table is generated by multiplying the user’s own table rather than by a stored example — which is what makes the invariance claim checkable on any input, not only on the preset. The preset itself is exact: V = 0.27136852 at n = 90 and at n = 1,800, with χ² = 13.2554 and 265.1071 and p = 1.0093e-2 and 3.6172e-56. Proportional rows return a chi-square of exactly zero rather than a rounded near-zero, because observed and expected coincide cell by cell. The equality with |φ| on 2×2 tables is asserted numerically across randomly generated tables. Ragged rows, negative counts and a single row or column all return no result rather than a number. That engine is verified on every change against 69 assertions. The count and the per-case breakdown are published on the formula verification page.

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