Can eta squared equal zero?

Yes, when the relevant between-group sum of squares is zero and total variation is positive.

Can it be calculated when every outcome is identical?

The usual ratio has zero total variation and is undefined.

Is partial eta squared always the same?

No; equality holds in particular settings, including ordinary one-way ANOVA, rather than universally.

AI Glossary / Statistics and probability / Intermediate

Eta squared ( h 2 )

Definition

Eta squared, conventionally written eta squared or η², is an effect-size measure based on the proportion of total observed variation attributed to a specified effect in an analysis of variance. In a standard one-way ANOVA, it equals the between-group sum of squares divided by the total sum of squares. It summarizes association in the analyzed sample and does not by itself establish causation or practical importance.

What it means

The supplied term's "h 2" is a plain-text rendering of the Greek eta symbol rather than a separate quantity. Eta squared asks how much of the sample's squared variation is associated with differences among the modeled group means.
A value of 0.20 is commonly described as twenty percent of the observed variation accounted for by the effect under that analysis. It does not mean that twenty percent of people benefited, that outcomes increased by twenty percent, or that the effect has a twenty-percent probability of being real.

How it works

For a one-way model, calculate the grand mean, each group mean, and the observations' deviations. Total sum of squares measures deviations from the grand mean. Between-group sum of squares is the sum across groups of group size times the squared difference between the group mean and grand mean. Within-group sum of squares measures deviations from each group's own mean.
In this standard decomposition, total equals between plus within. Eta squared is SS_between/SS_total when total variation is positive. With a categorical group predictor fitted by ordinary least squares and an intercept, it equals the model's sample R-squared.
Distinguish eta squared from partial eta squared. The partial version uses SS_effect/(SS_effect + SS_error), excluding other modeled effects from its denominator. They coincide in the simple one-way setting, but can differ in multifactor or repeated-measures analyses. Generalized eta squared uses another design-aware denominator and should also be labeled explicitly.
In unbalanced designs, the sums-of-squares convention and hypothesis tested matter. Model terms, interactions, covariates, and term ordering can affect reported values. Do not compare software outputs without checking which effect-size version and sums of squares they use.
Sample eta squared tends to overestimate the corresponding population effect in common settings. Alternatives such as omega squared aim to reduce bias, but they are not interchangeable labels for the same number. Confidence intervals help communicate uncertainty, particularly with small samples.
Interpret effect size with group means, outcome units, study design, and consequences. Fixed small, medium, and large thresholds cannot replace substantive context. An omnibus effect does not identify which specific groups differ or whether a particular contrast is meaningful.

Three practical examples

An experiment reports eta squared alongside group means to summarize how strongly a training condition is associated with score differences.
An analyst compares the within-group and between-group components of variation in a simple service-process study.
A research synthesis checks whether reported ANOVA effect sizes are ordinary, partial, or generalized eta squared before attempting comparisons.

Think of it this way

Think of sorting a collection of measurements into groups. Some variation comes from the group centers being different, and some remains among measurements within each group. Eta squared describes the share attributed to the group-center differences in the specified decomposition.

Worked example

Use an original one-way example with group A values 2, 4, and 6 and group B values 6, 8, and 10. Their means are four and eight, and the grand mean is six.
The between-group sum of squares is 3(4 - 6) squared + 3(8 - 6) squared = 12 + 12 = 24. Within each group, squared deviations from its mean sum to eight, giving a total within-group sum of squares of sixteen.
Total sum of squares is therefore 24 + 16 = 40. Eta squared is 24/40 = 0.60. Group membership accounts for sixty percent of the observed squared variation in this particular dataset under the one-way model.
The group mean difference is four outcome units; that is a separate, directly interpretable summary. With only six observations, the effect-size estimate is uncertain and should not be presented as a precise population fraction.
For illustration, the conventional one-way omega-squared estimate is [24 - (2 - 1)(16/4)]/[40 + 16/4] = 20/44, approximately 0.455. This different value illustrates a bias-adjustment approach, not an arithmetic correction to the correctly calculated sample eta squared.

Illustrative case study

Fictional case: Linden Learning compares training formats and reports a large eta-squared value from a small pilot. Managers initially interpret it as the percentage of staff whose performance improved because of the program.
The analyst corrects that interpretation, shows group means and uncertainty, and explains the pilot's assignment process. The team plans a stronger evaluation and considers practical score improvements alongside the variance-based summary. It avoids treating the pilot effect size as a guaranteed input for every future deployment setting.

Common misconceptions and mistakes

Mistake: interpreting eta squared as a percentage change in the outcome or a percentage of participants affected.
Mistake: mixing ordinary and partial eta squared without identifying their denominators.
Mistake: assuming an observational group association is causal.
Mistake: treating a small-sample estimate as an exact population effect.
Mistake: comparing values across different designs, outcome ranges, or sums-of-squares conventions without context.
Mistake: concluding which groups differ from the omnibus effect-size number alone.

Business use

Businesses can use eta squared to complement significance tests when assessing differences among experimental conditions or operating groups. It summarizes a model-based share of observed variation, while raw group differences explain the scale of the outcome. Reports should include the effect-size version, analysis design, uncertainty, and practical interpretation. Decisions should consider whether improvements matter operationally, not merely whether the variance proportion receives a conventional size label.

Check your understanding

What is the denominator of ordinary eta squared in the example?

Answer: The total sum of squares, forty.

Does 0.60 mean sixty percent of participants improved?

Answer: No. It is a proportion of squared variation attributed to group differences in this sample.

Related terms

Effect size
Analysis of variance
Sums of squares
Omega squared ( w 2 )
R-squared
sum of squares
partial eta squared
generalized eta squared
omega squared
interaction
and practical significance

Learn next

compute ANOVA sums of squares by hand, compare ordinary and partial denominators in a multifactor model, and report raw-unit differences alongside standardized effect sizes.

Sums of squares

Sources

Sources: Howell, Statistical Methods for Psychology glossary: https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html . Lakens, Calculating and reporting effect sizes to facilitate cumulative science: https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2013.00863/full . Official effectsize package documentation: https://easystats.github.io/effectsize/reference/eta_squared.html . The two-group calculation is original.

Previous: Estimation of distribution algorithm Next: Ethics by design