Analyse categorical data with the chi-square test

When both variables are categorical, means make no sense. The question becomes whether the distribution of one variable differs across the categories of the other. The chi-square (χ²) test of independence answers it by comparing the counts you observed in each cell of a crosstab with the counts you would expect if the two variables were unrelated.

RQ3: is gender associated with using counselling?

Step 1: the crosstab

Expected counts are in parentheses. They are what the cell would contain if counselling use were the same in both groups: for example, 134 × 81 / 240 = 45.2 female students would have used counselling. In fact 56 did. Converting to row percentages makes the pattern clear: 41.8% of female students (56 of 134) and 23.6% of male students (25 of 106) used counselling.

Step 2: check the assumption

The chi-square test is a large-sample approximation. It needs expected counts of at least 5 in (nearly) every cell: no more than 20% of cells below 5 and none below 1. The tool checks this for you; here the smallest expected count is 35.8, so the assumption is comfortably met. With small expected counts in a 2 × 2 table, use Fisher’s exact test instead.

Step 3: the test and the effect size

Pearson χ²(1, N = 240) = 8.77, p = .003. Gender and counselling use are associated. The degrees of freedom are (rows − 1) × (columns − 1) = 1. The Yates continuity correction (χ² = 7.98, p = .005) is a more conservative version for 2 × 2 tables; most researchers now report the uncorrected Pearson value, but the conclusion is the same.

The effect size for chi-square is Cramér’s V (equal to φ in a 2 × 2 table), which runs from 0 to 1. V = .19 is a small-to-moderate association: for a 2 × 2 table, .10 is small, .30 medium and .50 large. In practical terms, the difference of 18 percentage points between female and male students is substantial for a service planner, even though the association is statistically modest.

Step 4: where is the association?

In larger tables, the overall test does not tell you which cells drive the result. Adjusted standardised residuals do: values beyond ±1.96 mark cells with significantly more (positive) or fewer (negative) cases than expected. Here, female students who used counselling have a residual of +2.96, and male students who used counselling −2.96. In a 2 × 2 table the residuals mirror each other, but in a 4 × 3 table they pinpoint the interesting cells.

Common mistakes

• Running chi-square on percentages or means instead of counts. The test needs raw frequencies.
• Counting the same person in several cells, for example with multiple-answer questions. Each case must appear in exactly one cell.
• Interpreting the association as causal. The design is correlational: women may use counselling more because they experience more stress, because they are more willing to seek help, or for other reasons.
• Forgetting the effect size. With very large samples, tiny associations become significant.

Writing it up

A chi-square test of independence showed a significant association between gender and counselling use, χ²(1, N = 240) = 8.77, p = .003, Cramér’s V = .19. Female students were more likely to have used the counselling service (41.8%) than male students (23.6%).

Related tests

For a single categorical variable compared with expected proportions (for example, whether the faculty mix matches the university’s), use the chi-square goodness-of-fit test. For two paired categorical measurements (before and after, yes/no), use McNemar’s test. For an ordered outcome with several predictors, consider ordinal logistic regression.