Write up this chi-square test
An APA-style paragraph from the table and results above, including which cells drive the association.
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How the chi-square statistic is built
For each cell, the calculator takes (observed − expected)² ÷ expected and adds them up. Large values mean the observed table is far from what independence predicts. Under H₀ the sum follows a χ² distribution with (rows − 1)(columns − 1) degrees of freedom (k − 1 for goodness of fit).
A significant result says the variables are associated, not where. The adjusted standardised residuals (Agresti, 2013) answer that: cells beyond ±1.96 contribute notably, highlighted in the table above.
Effect size
φ (2 × 2) and Cramér's V range from 0 to 1. Cohen's rough guides for df* = 1 are .10 small, .30 medium, .50 large; for larger tables the thresholds shrink. Use Cohen's w with the power analysis calculator to plan a chi-square study.
Frequently asked questions
When do I use a chi-square test of independence?
When both variables are categorical and you have counts of cases in each combination, for example treatment group × recovered (yes/no). It tests whether the distribution of one variable differs across levels of the other.
What are the expected counts?
The counts you would see if the variables were unrelated: row total × column total ÷ grand total. The test compares observed with expected counts.
What if some expected counts are below 5?
The chi-square approximation becomes unreliable when more than 20% of expected counts are below 5 or any is below 1 (Cochran, 1954). For a 2 × 2 table use Fisher’s exact test; for larger tables combine sparse categories.
How do I report it?
χ²(df, N = total) = value, p = value, with an effect size: φ for 2 × 2 tables or Cramér’s V otherwise. Example: χ²(1, N = 200) = 6.12, p = .013, φ = .17.