What is eta-squared and how does it differ from omega-squared?

Eta-squared measures the proportion of total variance explained by a factor in ANOVA. Omega-squared is a less biased estimator of the same quantity. Eta-squared tends to overestimate the population effect, especially with small samples. Cohen suggested benchmarks of 0.01 (small), 0.06 (medium), and 0.14 (large) for both measures.

Effect Size Calculator

Calculate and visualise effect sizes including Cohen's d, Hedges' g, Glass's delta, eta-squared, omega-squared and Cohen's f. See animated distribution overlap, step-by-step formulas, CLES probability and interpretation guidelines.

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About Effect Size Calculator

Understanding Effect Sizes in Research

Effect sizes are essential statistics that quantify the magnitude of a phenomenon, complementing the information provided by p-values. While a p-value tells you whether an effect is statistically significant, the effect size tells you how large that effect is. This distinction is critical for judging practical significance — a statistically significant result with a tiny effect size may have no real-world importance.

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Cohen's d

Standardized mean difference between two groups using pooled SD

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Hedges' g

Bias-corrected Cohen's d, preferred for small samples (n < 20)

Glass's Delta

Mean difference standardized by control group SD only

η²

Eta-Squared

Proportion of total variance explained in ANOVA

ω²

Omega-Squared

Less biased variance-explained estimate than η²

Cohen's f

Effect size for ANOVA, derived from η²

How to Calculate Cohen's d

Cohen's d measures the standardized difference between two group means:

$$d = \frac{M_1 - M_2}{SD_{pooled}}$$

where the pooled standard deviation is:

$$SD_{pooled} = \sqrt{\frac{(n_1 - 1) \cdot SD_1^2 + (n_2 - 1) \cdot SD_2^2}{n_1 + n_2 - 2}}$$

A Cohen's d of 0.5 means the two group means differ by half a standard deviation. Hedges' g applies a correction factor $J = 1 - \frac{3}{4 \cdot df - 1}$ to reduce the upward bias of d in small samples.

Interpreting Effect Size with CLES

The Common Language Effect Size (CLES) translates Cohen's d into an intuitive probability: the chance that a randomly selected individual from Group 1 will score higher than a randomly selected individual from Group 2. It is calculated as:

$$CLES = \Phi\left(\frac{d}{\sqrt{2}}\right)$$

where $\Phi$ is the standard normal CDF. For example, d = 0.5 corresponds to a CLES of about 64%, meaning there is a 64% chance a random Group 1 member outscores a random Group 2 member.

Eta-Squared vs. Omega-Squared

In ANOVA, eta-squared (η²) represents the proportion of total variance explained by the independent variable:

$$\eta^2 = \frac{SS_{between}}{SS_{total}} = \frac{F \times df_{between}}{F \times df_{between} + df_{within}}$$

However, η² tends to overestimate the population effect. Omega-squared (ω²) provides a less biased estimate:

$$\omega^2 = \frac{df_{between} \times (F - 1)}{df_{between} \times (F - 1) + N}$$

Converting Between Effect Size Measures

From	To	Formula
Cohen's d	Point-biserial r	$r = \frac{d}{\sqrt{d^2 + \frac{(n_1+n_2)^2}{n_1 \cdot n_2}}}$
Correlation r	Cohen's d	$d = \frac{2r}{\sqrt{1 - r^2}}$
t-test (independent)	Cohen's d	$d = t \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$
t-test (paired)	Cohen's d_z	$d_z = \frac{t}{\sqrt{n}}$
η²	Cohen's f	$f = \sqrt{\frac{\eta^2}{1 - \eta^2}}$

When to Use Each Effect Size

Scenario	Recommended	Why
Two equal-variance groups	Cohen's d or Hedges' g	Standard measure; g is preferred when n < 20 per group
Unequal variances	Glass's delta	Uses only the control group SD, unaffected by treatment variance
Paired / repeated measures	Cohen's d_z	Based on difference scores; accounts for within-subject correlation
One-way ANOVA	η² or ω²	η² for descriptive use; ω² for less biased population estimate
Correlation analysis	r and r²	r measures strength; r² gives proportion of shared variance
Meta-analysis	Hedges' g	Bias correction is essential when pooling across diverse sample sizes

Frequently Asked Questions

What is an effect size?

An effect size is a quantitative measure of the magnitude of a phenomenon or the strength of a relationship between variables. Unlike p-values that indicate whether an effect exists, effect sizes tell you how large the effect is. They are essential for meta-analysis, power analysis, and evaluating practical significance in research studies.

What is Cohen's d and how do I interpret it?

Cohen's d measures the standardized difference between two group means, calculated as the mean difference divided by the pooled standard deviation. Jacob Cohen suggested benchmarks of 0.2 (small), 0.5 (medium), and 0.8 (large), but these should be interpreted within the context of your specific research field. In some fields, a d of 0.2 may represent a meaningful intervention effect.

What is the difference between Cohen's d and Hedges' g?

Hedges' g is a bias-corrected version of Cohen's d. Cohen's d slightly overestimates the population effect size, especially with small samples (n < 20 per group). Hedges' g applies a correction factor J to reduce this bias. For large samples, the values are nearly identical.

What is the Common Language Effect Size (CLES)?

CLES expresses Cohen's d as a probability: the chance that a randomly selected person from one group will score higher than a randomly selected person from the other group. A CLES of 69% means there is a 69% probability that a random Group 1 member outscores a random Group 2 member. This is far more intuitive than raw d values for non-technical audiences.

When should I use eta-squared vs. omega-squared?

Eta-squared (η²) is commonly reported but tends to overestimate the population effect size, especially with small samples. Omega-squared (ω²) provides a less biased estimate and is generally recommended for inferential purposes. For descriptive purposes within your specific sample, η² is acceptable. Both share the same interpretation benchmarks: 0.01 (small), 0.06 (medium), 0.14 (large).

Can effect sizes be negative?

Yes, directional effect sizes like Cohen's d and correlation r can be negative. A negative Cohen's d means Group 2 has a higher mean than Group 1. A negative r indicates an inverse relationship. The sign indicates direction, while the absolute value indicates magnitude. Variance-based measures (η², ω², r²) are always non-negative.

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"Effect Size Calculator" at https://MiniWebtool.com/effect-size-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-16

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