Ch8. ANOVA — Testing Mean Differences Across Three or More Groups

Why ANOVA Is Necessary

When comparing means of three or more groups, running multiple t-tests accumulates Type I error.

Comparing 3 groups with 3 separate t-tests:
P(at least 1 error) = 1 − (0.95)³ ≈ 14.3% (actual error rate ≈ 14% even with α = 0.05)

ANOVA (Analysis of Variance): Tests all group means simultaneously in a single procedure.

One-Way ANOVA

Hypotheses:

H₀: μ₁ = μ₂ = ... = μₖ  (all group means are equal)
H₁: At least one group mean differs

Core Idea:

Total Variation = Between-Group Variation + Within-Group Variation
SST             = SSB                     + SSW

Source of Variation	Sum of Squares	df	Mean Square
Between Groups	SSB	k−1	MSB = SSB/(k−1)
Within Groups	SSW	N−k	MSW = SSW/(N−k)
Total	SST	N−1

F-Statistic:

F = MSB / MSW ~ F(k−1, N−k)

If F > F_α (critical value), reject H₀

ANOVA Assumptions

Each group follows a normal distribution
Equal variances across groups (homoscedasticity)
Observations are independent

Post-Hoc Tests

After rejecting H₀ with ANOVA, determine which specific groups differ.

Method	Characteristics
Tukey HSD	Recommended when group sizes are equal
Bonferroni	Conservative correction for multiple comparisons
Scheffé	Most conservative; applicable to all contrasts
Duncan	Less strict; pairwise comparisons

Chi-Square Test

Used for categorical data analysis.

Test of Independence

Tests whether two categorical variables are independent of each other.

H₀: The two variables are independent
H₁: The two variables are not independent

χ² = Σ[(Observed − Expected)² / Expected]

Degrees of freedom: (rows − 1) × (columns − 1)

Example: Relationship between gender and job satisfaction (contingency table analysis)

Goodness-of-Fit Test

Tests whether an observed distribution matches a theoretical distribution.

Nonparametric Tests

Alternatives when normality assumptions are not met.

Situation	Parametric Test	Nonparametric Alternative
Comparing 2 groups	t-test	Mann-Whitney U
Comparing 3+ groups	ANOVA	Kruskal-Wallis
Paired samples	Paired t-test	Wilcoxon Signed-Rank

ANOVA F-Statistic ★★★★★ : F = Between-group variation / Within-group variation. A large F means the differences between groups exceed within-group noise → reject H₀. Memory tip: F = MSB/MSW; larger F = more significant group differences

Chi-Square Test of Independence ★★★★★ : Tests independence of two categorical variables. χ² = Σ(Observed − Expected)²/Expected. Memory tip: chi-square = independence test for categorical data

Post-Hoc Tests ★★★★☆ : After ANOVA, identify which specific pairs of groups differ. Choose Tukey, Bonferroni, or Scheffé depending on the situation. Memory tip: ANOVA → post-hoc → identify specific group pairs

Practice Questions

Q. You want to compare exam scores for three teaching methods A, B, and C. Which test should you use?

One-way ANOVA. H₀: μA = μB = μC. Use the F-statistic to test overall differences. If significant, apply a post-hoc test (e.g., Tukey HSD) to identify which specific pairs differ.

Q. What test should you use to examine the relationship between gender (male/female) and purchase behavior (yes/no)?

Chi-square test of independence. Create a 2×2 contingency table and compute the χ² statistic with degrees of freedom = (2−1) × (2−1) = 1.