Read the output in this order
| Field | What it means | What to ask |
|---|---|---|
| Observed counts | Counts actually recorded in each category or table cell | Are these counts, not percentages or duplicated observations? |
| Expected counts | Counts predicted under the null model | Are they large enough for the reported approximation? |
| χ² statistic | Sum of scaled observed-versus-expected discrepancies | Is it large relative to its reference distribution and df? |
| df | Degrees of freedom determined by the table or fitted constraints | Does the reported df match the design? |
| p-value | Tail probability under the null model and assumptions | Is the result incompatible with the null—not “the probability the null is true”? |
| Residuals | Cell-level observed-minus-expected discrepancies, often standardized | Which cells contribute most, and in which direction? |
For an independence test, the null says the two categorical variables are independent in the population represented by the sampling process. For a goodness-of-fit test, the null specifies a set of expected category proportions. The same χ² symbol can appear in both outputs, but the expected-count model differs.
A worked table
Suppose a 2×3 table reports χ² = 9.84, df = 2, p = .007. The p-value indicates that a discrepancy at least this large would be unusual under the fitted independence model. It does not tell you which category combination differs, how large the association is, or whether the sampling design supports a causal claim.
Now inspect standardized residuals. A large positive residual means more observations than expected in that cell; a large negative residual means fewer. Read the complete residual pattern rather than declaring one cell “significant” from an unlabeled rule. Multiple cell inspection creates its own multiplicity issue.
Expected counts are an approximation check
The familiar Pearson χ² p-value uses an asymptotic reference distribution. Sparse expected cells can make that approximation poor. R’s official chisq.test documentation exposes expected counts, residuals, standardized residuals, and simulation options for cases where the usual approximation is questionable.1
There is no single expected-count slogan that replaces judgment for every table. Check the software warning, the smallest expected counts, the proportion of sparse cells, table dimensions, and whether exact or simulation-based inference fits the design. Do not merge categories solely to make a warning disappear; category meaning comes first.
Independence, association, and effect size
A small p-value supports evidence against independence under the model. It does not measure association strength. Large samples can make small discrepancies detectable, while small samples can leave important patterns uncertain. Report the table, an appropriate association measure such as phi or Cramér’s V when justified, and its uncertainty.
This is the categorical-output counterpart to reading a t-test table. If the output compares means across several groups, use the ANOVA table guide instead; the tests answer different questions even when both end with a p-value.
Common reading errors
- Treating the p-value as the probability that the variables are independent.
- Reporting χ² without df, p, table dimensions, or observed counts.
- Ignoring expected-count warnings.
- Saying “the groups are different” without locating the contributing cells.
- Reading percentages as though they were the counts used by the test.
- Inferring causation from an association test.
A defensible reporting sentence
“The variables were associated in the observed table, χ²(2) = 9.84, p = .007. Expected counts were checked, and standardized residuals showed that the largest discrepancies occurred in the named cells.”