Chi-Squared and ANOVA

The t-tests from the previous lesson compare one or two group means. But what if you need to compare three or more groups, or analyse categorical data? This lesson introduces two powerful families of tests: chi-squared tests for categorical data and ANOVA for comparing multiple group means.

---

Chi-Squared Tests

The chi-squared (χ²) test is used when data consists of counts or frequencies in categories.

1. Chi-Squared Goodness-of-Fit Test

Tests whether the observed frequencies of a categorical variable match an expected distribution.

Hypotheses:

H₀: The observed frequencies match the expected distribution
H₁: The observed frequencies do not match the expected distribution

Test Statistic:

χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ

Where:

• Oᵢ = observed frequency for category i
• Eᵢ = expected frequency for category i

Degrees of freedom: df = k − 1, where k is the number of categories

Example: A die is rolled 60 times. If fair, each face should appear 10 times.

Face	Observed	Expected
1	8	10
2	12	10
3	7	10
4	14	10
5	9	10
6	10	10

χ² = (8−10)²/10 + (12−10)²/10 + (7−10)²/10 + (14−10)²/10 + (9−10)²/10 + (10−10)²/10
   = 0.4 + 0.4 + 0.9 + 1.6 + 0.1 + 0.0
   = 3.4

With df = 5 and α = 0.05, the critical value is 11.07. Since 3.4 < 11.07, we fail to reject H₀ — no evidence the die is unfair.

2. Chi-Squared Test of Independence

Tests whether two categorical variables are associated.

Hypotheses:

H₀: The two variables are independent
H₁: The two variables are associated

Expected frequency for each cell:

Eᵢⱼ = (Row total × Column total) / Grand total

Degrees of freedom: df = (r − 1)(c − 1), where r = rows, c = columns

Example: Is there an association between gender and preference for tea or coffee?

	Tea	Coffee	Total
Male	30	45	75
Female	40	35	75
Total	70	80	150

Expected values: E₁₁ = (75 × 70)/150 = 35, etc.

The test statistic and p-value then determine whether gender and beverage preference are independent.

---

Assumptions for Chi-Squared Tests

1. Data are counts (frequencies), not percentages or proportions
2. Observations are independent
3. Each expected frequency should be at least 5 (use Fisher's exact test if not)

---

Analysis of Variance (ANOVA)