Eta-Squared & Omega-Squared Calculator

Use eta-squared and omega-squared when you need to report effect size for ANOVA. While p-values tell you whether an effect exists, these measures tell you how large that effect is—how much variance in the outcome is explained by group membership.

What Are Eta-Squared and Omega-Squared?

Both measure the proportion of variance in the dependent variable explained by the independent variable(s) in ANOVA.

Eta-Squared (η²)

$\eta^2 = \frac{SS_{effect}}{SS_{total}}$

Simple and intuitive: the ratio of effect variance to total variance.

Omega-Squared (ω²)

$\omega^2 = \frac{SS_{effect} - df_{effect} \times MS_{error}}{SS_{total} + MS_{error}}$

Less biased estimate, especially for small samples.

Why Two Measures?

Eta-squared is:

• Easy to calculate and interpret
• Positively biased (overestimates population effect)
• Equivalent to R² in regression
• The descriptive effect size for your sample

Omega-squared is:

• More complex to calculate
• Less biased (better population estimate)
• Preferred for inferential purposes
• Recommended by APA guidelines

Example Calculation

ANOVA output:

Eta-squared: $\eta^2 = \frac{120}{300} = 0.40$

Omega-squared: $\omega^2 = \frac{120 - (2)(4)}{300 + 4} = \frac{120 - 8}{304} = \frac{112}{304} = 0.37$

Interpretation: About 37-40% of the variance in the outcome is attributable to group differences.

Interpreting Effect Sizes

These benchmarks come from Cohen (1988). Context matters—in some fields, "small" effects are still meaningful.

Partial Effect Sizes

For factorial ANOVA with multiple factors, use partial effect sizes:

Partial Eta-Squared

$\eta_p^2 = \frac{SS_{effect}}{SS_{effect} + SS_{error}}$

This removes variance from other factors, focusing on the specific effect.

Partial Omega-Squared

$\omega_p^2 = \frac{SS_{effect} - df_{effect} \times MS_{error}}{SS_{effect} + (N - df_{effect}) \times MS_{error}}$

When to Use Each

Calculating from F-Statistic

If you only have F and degrees of freedom:

Eta-squared: $\eta^2 = \frac{df_{effect} \times F}{df_{effect} \times F + df_{error}}$

Omega-squared: $\omega^2 = \frac{df_{effect} \times (F - 1)}{df_{effect} \times F + df_{error} + 1}$

Example: F(2, 45) = 15.0 $\eta^2 = \frac{2 \times 15}{2 \times 15 + 45} = \frac{30}{75} = 0.40$ $\omega^2 = \frac{2 \times (15-1)}{2 \times 15 + 45 + 1} = \frac{28}{76} = 0.37$

Confidence Intervals

For eta-squared, use the noncentral F distribution:

1. Find the noncentrality parameter λ = F × df₁
2. Use noncentral F to get confidence limits for λ
3. Convert back to η²

Software packages like R (effectsize package) or G*Power can compute these.

Relationship to Other Effect Sizes

Converting η² to Cohen's f: $f = \sqrt{\frac{\eta^2}{1 - \eta^2}}$

Epsilon-Squared (ε²)

A third alternative, also less biased:

$\varepsilon^2 = \frac{SS_{effect} - df_{effect} \times MS_{error}}{SS_{total}}$

Similar to omega-squared but uses SS_total instead of SS_total + MS_error in the denominator.

For Repeated Measures ANOVA

Effect sizes for repeated measures are more complex:

$\eta_G^2 = \frac{SS_{effect}}{SS_{effect} + SS_{subjects} + SS_{error}}$

Generalized eta-squared (η²_G) accounts for the correlation structure and is recommended for repeated measures designs.

Reporting Guidelines

APA format: "There was a significant main effect of treatment, F(2, 45) = 15.00, p < .001, η² = .40, 90% CI [.22, .52]."

Or with omega-squared: "...ω² = .37, indicating a large effect."

Include:

• F statistic with degrees of freedom
• p-value
• Effect size (η² or ω²)
• Confidence interval if available

Common Mistakes

Using η² for Generalization

Eta-squared describes your sample. For population estimates, prefer omega-squared.

Partial vs. Regular

Partial η² is always larger than η² (unless there's only one factor). Don't compare partial effects across studies with different designs.

Ignoring Negative Values

Omega-squared can be negative when F < 1. This indicates essentially zero effect—report as ω² ≈ 0.

Over-interpreting Small Effects

A statistically significant effect can be tiny. Always report effect size alongside p-values.

Applications

Psychology

Effect of therapy type on depression scores.

Education

Effect of teaching method on test performance.

Medicine

Effect of drug dosage on blood pressure.

Business

Effect of marketing strategy on sales.

Historical Context

Paul Hays introduced eta-squared in the 1960s. Omega-squared was developed by Hays and others as a less biased alternative.

The distinction between sample and population effect sizes became important as researchers recognized that η² systematically overestimates population effects.

Related Calculations

• One-Way ANOVA — Calculate F and SS values
• Two-Way ANOVA — Factorial designs
• Cohen's d Calculator — Effect size for t-tests
• Power Calculator — Sample size planning with effect size
• Hedges' g Calculator — Alternative effect sizes

Summary

Eta-squared and omega-squared measure how much variance an ANOVA factor explains. Eta-squared (η² = SS_effect/SS_total) is simple but overestimates population effects. Omega-squared adjusts for this bias and is preferred for inference. For factorial designs, use partial versions. Interpret using Cohen's benchmarks (0.01 = small, 0.06 = medium, 0.14 = large) but consider context. Always report effect sizes alongside significance tests to convey practical importance.