Lesson 8 — ANOVA

More things might
be different now

When you have 3 or more groups to compare, the t-test isn’t enough. Enter ANOVA.

T-test vs ANOVA

🐕🐱T-testExactly 2 groups

→

🐕🐱🦮ANOVA3 or more groups (also works for 2)

ANOVA = ANalysis Of VAriance. Compares means across all groups at once.

The hypotheses

H₀ — NullAll group means are equal. Golden retrievers, pugs, and St. Bernards are all the same height.

Hₐ — AlternativeAt least one group mean is different. (But ANOVA won’t tell you which!)

The F-statistic

ANOVA uses the F-statistic: the ratio of between-group variance to within-group variance.

Intuition

High F = groups differ a lot & are tight within → small p-value

Low F = groups overlap a lot → large p-value

Like the t-test, compare F against a threshold to get a p-value and decide whether to reject H₀.

The key limitation

❓ ANOVA can tell you at least one group is different — but not which ones. A significant result from ANOVA comparing 3 dog breeds just means they’re not all the same height.

To find out which pairs differ, you need a post-hoc test.

Post-hoc analysis

Run after a significant ANOVA to compare each pair of groups.

🔎Tukey HSD — Honestly Significant Difference. Compares all group pairs simultaneously.
🔎Bonferroni — More conservative; adjusts for multiple comparisons.

Now you can say: golden retrievers are taller than pugs and St. Bernards are taller than pugs.

Real-world uses

ANOVA is everywhere

💊 Drug trials 🌿 Crop yield by fertilizer 🏠 Housing prices by region 📚 Test scores by teaching method 🥊 Plant heights by treatment

Any time you have 3+ groups and a continuous variable — ANOVA is your tool.