Chi-Square Test of Independence

The chi-square (χ²) test of independence determines whether two categorical variables are associated. It is the standard test for contingency table data — for example, whether defect type depends on the shift that produced it.

The chi-square formula

χ² = Σ (O − E)² / E Where: O = observed frequency (from data) E = expected frequency if variables are independent E[i,j] = (row total[i] × column total[j]) / grand total Degrees of freedom = (rows − 1) × (columns − 1)

Worked example

Contingency table — Defect type vs Shift: Shift A Shift B Total Dimensional 18 12 30 Surface 12 18 30 Total 30 30 60 E[Dim, A] = (30 × 30) / 60 = 15 E[Dim, B] = (30 × 30) / 60 = 15 E[Sur, A] = (30 × 30) / 60 = 15 E[Sur, B] = (30 × 30) / 60 = 15 χ² = (18−15)²/15 + (12−15)²/15 + (12−15)²/15 + (18−15)²/15 = 9/15 + 9/15 + 9/15 + 9/15 = 2.40 df = (2−1)(2−1) = 1 Critical value at α = 0.05, df = 1: χ²_crit = 3.841 2.40 < 3.841 → fail to reject H₀ → no significant association between defect type and shift

CQE exam tips

Chi-square requires expected cell counts ≥ 5 — if not, combine categories or use Fisher's exact test.
df = (r−1)(c−1): a 2×2 table has 1 df; a 3×4 table has 6 df.
Chi-square tests association, not causation.

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