Chi-square test of independence calculator

Chi-square compares the frequencies you observed with those you would expect if the two variables were independent.

It is the standard test for contingency tables in surveys and studies with categorical variables.

For each cell the expected frequency is derived from the marginal totals, and the squared difference relative to that expectation is accumulated.

Cramér's V rescales the result between 0 and 1, because chi-square grows with sample size.

What it is actually testing

The test compares what you observed with what would be expected if there were no relationship between the variables. If the discrepancy is large, independence is judged implausible. It does not measure the strength of the association, only whether chance is a reasonable explanation.

A significant result is not an important result

With very large samples, tiny differences of no practical relevance come out significant. That is why the test should be accompanied by a measure of effect size, such as Cramér's V, which does indicate how strong the association is on a scale from zero to one.

Expected frequencies rule

The approximation stops being reliable when expected frequencies are very small. The usual rule asks that they all exceed 5, or at least that the great majority do. In two-by-two tables with sparse cells, Fisher's exact test is the correct alternative.

Independence of observations

Each individual must appear in only one cell. If you measure the same people before and after, the data are paired and McNemar's test is needed instead.