Eta Squared (η²): Effect Size in ANOVA#

Stage 2 · 🔗 Associations & Correlation · Lesson 16 of 56 · beginner

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How big, not just whether#

A statistical test tells you whether a categorical variable relates to a numeric one — do taxi fares differ by payment type? But “significant” does not mean “large”. Eta squared (\(\eta^2\)) answers the other half: how big is the effect? It is the effect size that partners the ANOVA test, filling the “continuous ↔ categorical” slot of the association taxonomy.

A number to a category#

The setting is ANOVA (analysis of variance): one categorical variable splits the data into groups, and you ask how much of the spread in a continuous variable is explained by which group a point falls in. ANOVA works by splitting the total variability into a between-group part (differences among the group means) and a within-group part (scatter inside each group).

The formula#

Eta squared is simply the between-group share of the total:

\[\eta^2 = \frac{SS_{\text{between}}}{SS_{\text{total}}}, \qquad SS_{\text{total}} = SS_{\text{between}} + SS_{\text{within}}.\]

It ranges from \(0\) (the grouping explains nothing) to \(1\) (the grouping explains everything). An \(\eta^2\) of 0.30 means 30% of the variance in the numeric variable is accounted for by the category.

The cousin of r²#

Eta squared is the categorical twin of a familiar quantity: it is analogous to \(r^2\), the coefficient of determination. Both report the proportion of variance explained in a continuous variable — \(r^2\) when the predictor is also continuous (as in regression, Stage 5), \(\eta^2\) when the predictor is a category. Seen this way, correlation and ANOVA are two faces of the same question: how much does one variable tell you about another? (In multi-factor models, a variant called partial eta squared isolates one factor’s contribution.)

Hint