Chi-square Test: Are Two Categorical Variables Related?

Not every relationship is between numbers. Plan tier by gender, churn by region, response by campaign — both sides are categories, so there's no average to compare. The chi-square test of independence is built for exactly this: deciding whether two categorical variables are related or just coincidentally lined up.

Note

The idea is simple: compare the counts you actually observed to the counts you'd expect if the two variables had nothing to do with each other. A big gap is evidence of a relationship.

The Contingency Table

The data goes into a cross-tab: rows for one variable, columns for the other, counts in the cells.

Observed	Plan A	Plan B
New users	120	80
Returning users	60	140

At a glance, new users lean to Plan A and returning users to Plan B. The question is whether that pattern is real or the kind of split you'd see by chance.

Observed vs Expected

The test computes the counts you'd expect in each cell if the two variables were independent — based only on the row and column totals — and compares them to what you observed, as the bars in the figure above contrast.

Cell	Observed	Expected if independent
New · Plan A	120	90
New · Plan B	80	110
Returning · Plan A	60	90
Returning · Plan B	140	110

The chi-square statistic sums the squared gaps between observed and expected, scaled by expected. Large gaps push the statistic up and the p-value down — evidence the variables move together.

Reading the Result

A small p-value means the variables are related — the observed counts differ from independence more than chance would explain
The chi-square statistic itself grows with the size of the observed-expected gaps
Effect size (Cramér's V) tells you how strong the relationship is, not just whether it exists

Tip

As with any test, significance is not strength. With a large sample, a trivial association can be significant — report an effect size like Cramér's V alongside the p-value.

Assumptions and Limits

Independent observations — each person or record counted once
Adequate expected counts — a common rule is expected ≥ 5 in most cells
Counts, not percentages — chi-square works on raw frequencies
It shows association, not direction or cause

Watch out

When expected counts are too small, chi-square becomes unreliable — use Fisher's exact test instead. And remember: a significant result says the variables are linked, not that one causes the other.

Chi-square in the SKARI Statistical Lab

SKARI's Statistical Lab runs the crosstab and chi-square together: pick two categorical variables and it builds the table, computes observed vs expected, and reports the test with an effect size.

Crosstab (contingency table) construction from two categorical columns
Chi-square statistic, degrees of freedom, and p-value
Expected-count check with a Fisher's exact fallback for small cells
Cramér's V effect size and a plain-language read of the relationship

Takeaway

You get the table, the test, and the strength of the link in one place — and a warning when expected counts are too small to trust the standard test.

Frequently Asked Questions

What if my cells have tiny counts?

Chi-square gets unreliable below expected counts of about 5. Use Fisher's exact test, which SKARI offers as a fallback.

Does chi-square tell me the direction?

No — it flags that a relationship exists. Read the table cells (or standardized residuals) to see which combinations drive it.

Chi-square or correlation?

Chi-square for two categorical variables; correlation for two numeric ones. They answer the same 'are these related?' question for different data types.

Key Takeaways

For