Chi-square test of independence
The χ² test checks whether two variables are related by comparing observed counts with those expected under independence.
Formulas
- Expected: eᵢⱼ = row total · column total / grand total
- χ² = Σ (o − e)² / e
- Degrees of freedom: (rows − 1) · (columns − 1); 1 for 2×2
- χ² > critical value → reject H₀ (independence)
Your turn
Three problems on this topic. Comma or point, fractions like 1/6 work too.
Task 1 of 3Chi-square test
Task
Are subject and passing related? Test independence with the χ² test (α = 5%, 1 degree of freedom, critical value 3.841).
| passed | failed | |
|---|---|---|
| Subject A | 23 | 27 |
| Subject B | 48 | 44 |
χ² (2 decimal places)
Decision
Comma or point; fractions like 1/6 work too.
Worked example
Are subject and passing related? Test independence with the χ² test (α = 5%, 1 degree of freedom, critical value 3.841).
Worked solution
- 1.Expected count: eᵢⱼ = row total · column total / grand total
- 2.e = 28.66; 44.34; 24.34; 37.66
- 3.χ² = Σ (o − e)² / e = 0.01
- 4.0.01 ≤ 3.841 → do not reject H₀
Typical mistake: Dividing by the observed instead of the expected counts.
Frequently asked questions
When is the χ² test valid?
Rule of thumb: all expected counts should be at least 5.