When and why do I use ANOVA instead of multiple t-tests?

ANOVA (Analysis of Variance) tests whether the means of three or more groups are all equal. While you could run pairwise t-tests, there's a critical statistical reason not to.

The Problem with Multiple t-Tests:

Every t-test has a probability α of falsely rejecting (Type I error). With many tests, the errors compound:

• Comparing 4 groups requires C(4,2) = 6 pairwise t-tests
• At α = 0.05, the chance of at least one false rejection = 1 - (0.95)^6 = 26.5%

That's unacceptably high. ANOVA solves this by testing all groups simultaneously with a single test.

How ANOVA Works:

ANOVA compares two sources of variation:

1. Between-group variance (MSB): How much group means differ from the overall mean
2. Within-group variance (MSW): How much individual observations vary within each group

F-statistic = MSB / MSW

Interpretation:

• Large F (MSB >> MSW): Groups differ significantly — the between-group variation isn't just noise
• Small F (MSB ≈ MSW): Group differences are within the range of random variation

Finance Example:

A researcher at Orion Capital tests whether average returns differ across 4 sectors (Tech, Healthcare, Energy, Finance) over 60 months.

• H₀: μ_Tech = μ_Health = μ_Energy = μ_Finance
• H₁: At least one mean differs

ANOVA Table:

F = 140 / 30 = 4.67

F-critical at α = 0.05 with (3, 236) df ≈ 2.64

Since 4.67 > 2.64, reject H₀. At least one sector has a statistically different mean return.

Key assumptions of ANOVA:

1. Observations are independent
2. Each group is normally distributed
3. Equal variances across groups (homoscedasticity)

Exam tip: ANOVA tells you that at least one group differs — but NOT which one. If the exam asks which specific groups differ, you'd need post-hoc tests (not directly tested on CFA Level I, but understand the limitation).

Practice ANOVA problems in our CFA Level I question bank.