Variance explained, and which denominator that was out of.
Variance explained, four ways
η² = 0.934216 against ω² = 0.927084 — a gap of 0.007. When the effect is strong the bias correction barely moves anything, which is why it gets forgotten in exactly the papers where it does not matter.
3 groups, n = 30, F(2, 27) = 191.7160
η² = 0.934216
Omega squared is 0.927084 and epsilon squared 0.929343. Under a TRUE null — no group difference at all — this design would still produce an eta squared of 0.068966 on average, because eta squared cannot go below zero and every random split explains something. That figure is (k − 1)/(N − 1), and it is the bias the other two correct for.
η²
0.934216
SS effect / SS total
Expected η² if null
0.068966
(k − 1) / (N − 1)
ω²
0.927084
bias-corrected
Cohen's f
3.76845
the power-analysis unit
Four effect sizes, their denominators and what each answers
Measure
Value
Denominator
What it answers
η²
0.934216
SS total
share of ALL the variance — sums to 1 across effects and error
partial η²
0.934216
SS effect + SS error
share of the variance this effect and error share — sums to nothing
ω²
0.927084
SS total + MS error
share in the POPULATION, with the bias subtracted
ε²
0.929343
SS total
the same correction on the original denominator
On a one-way design η² and partial η² are the same number, because there is no other factor to remove from the denominator. They diverge as soon as a second factor enters, and on a two-way design the three partial values can sum to 2.4733 — 247% of a variance that only adds to 100%.
What chance alone would have given you
Mean η² across 4,000 datasets in which every group was drawn from one distribution, so every effect is chance. The measured average lands on (k − 1)/(N − 1) every time.
Measured mean eta squared under a true null at seven designs
Design
Measured mean η²
(k − 1)/(N − 1)
Mean ω²
η² above 0.05
2 groups of 10
0.052579
0.052632
0.000199
34.4%
3 groups of 10
0.068310
0.068966
−0.000527
49.5%
4 groups of 10
0.077009
0.076923
0.000189
60.8%
4 groups of 20
0.038120
0.037975
0.000162
26.6%
6 groups of 10
0.084390
0.084746
−0.000332
72.0%
3 groups of 50
0.013112
0.013423
−0.000312
2.4%
8 groups of 6
0.147253
0.148936
−0.001791
94.7%
With eight groups of six, 94.7% of completely null datasets produce an η² above 0.05 — and the average is 0.147253. Omega squared averages −0.001791 across the same datasets, which is what an unbiased estimator looks like.
η² sums to 1; partial η² sums to nothing Both are biased upward, by (k − 1)/(N − 1) Say which one you reported
What this tool shows
With eight groups of six and no real effect at all, 94.7% of datasets produce an η² above 0.05, averaging 0.147253. That is measured across 4,000 null datasets, and it lands exactly on (k − 1)/(N − 1) — the bias that is built into η² by construction. The tool prints that expectation for your own design beside the value you computed.
η², partial η², ω² and ε² from raw group data
The denominator each one divides by, stated rather than assumed
The expected η² for your design when the null is TRUE — never zero
A measured table of mean η² under a true null at seven designs
Cohen’s f, the unit a power calculation actually takes
Why partial η² across several effects can sum far above 1
Four effect sizes Denominators shown Null expectation 4,000-dataset table
η² and partial η² are different numbers with the same name in most papers.
Updated 12 September 2026 · Works in any browser, no installation
Eta squared is the share of the total variance in the outcome that a factor accounts for. It is the ANOVA equivalent of R² and, with two groups, it is exactly the squared point-biserial correlation between group membership and the outcome. Partial eta squared answers a related but different question and is the one most journals actually print.
At a glance
Formula shown
η² = SS_effect / SS_total, so across every effect and the error it sums to exactly 1. Partial η² = SS_effect / (SS_effect + SS_error), which removes the other factors from the denominator and therefore sums to nothing in particular. ω² = (SS_effect − df_effect·MS_error) / (SS_total + MS_error) subtracts the expected chance contribution, and ε² = (SS_effect − df_effect·MS_error) / SS_total applies the same correction on η²’s own denominator.
Scenario support
Reporting the size of an ANOVA effect, comparing findings across studies, power analysis inputs, meta-analysis, and any situation where a significant F needs translating into how much of the outcome the factor actually explains.
Educational estimate
Planning support from the values you enter — not professional advice.
Partial η² does not partition anything
On a one-way design the two are identical, which is why the distinction so often goes unnoticed until a second factor appears.
η² divides by SS_total, so the values for every effect plus the error add to exactly 1. It is a genuine partition of the variance, and each value is a share of the same whole.
Partial η² divides by SS_effect + SS_error, a different denominator for every row. Each answers “of the variance this effect and error share, how much is the effect?” — a separate question each time.
On the worked two-way design the three partial values are 0.8780, 0.8333 and 0.7619, summing to 2.4733. The η² values for the same three effects sum to 0.9390, which is the share they actually account for.
Partial η² also moves when you add a factor that changes nothing. A new covariate that soaks up error variance shrinks SS_error, which raises every existing partial η² without any effect changing size — so partial values are not comparable across studies that fitted different models.
η² is never zero, even when the effect is
It is a ratio of sums of squares, both of which are non-negative, so it cannot be negative — and an estimator that cannot go below zero cannot be centred on zero.
The expectation under a true null is (k − 1)/(N − 1). For three groups of ten that is 0.068966; for eight groups of six, 0.148936.
Measured across 4,000 null datasets, the means land on those figures. 0.068310 against 0.068966; 0.147253 against 0.148936; 0.052579 against 0.052632 for two groups of ten. Over the same datasets ω² averages within 0.002 of zero.
The practical consequence is a reporting hazard. With eight groups of six, 94.7% of completely null datasets give an η² above 0.05, and 72.0% do with six groups of ten.
The bias shrinks with N and grows with k, so it is worst in exactly the designs small studies use — many conditions, few observations each. Three groups of fifty has an expected null η² of only 0.013423.
The 0.01 / 0.06 / 0.14 bands ignore the design
Cohen’s small, medium and large marks for η² are widely quoted and were offered as a fallback for researchers with no substantive basis for judging an effect.
They take no account of k or N, which is a problem when the null expectation for eight groups of six is 0.148936 — above the “large” threshold.
Applying the bands to ω² instead is closer to defensible, since ω² is centred on zero and a value of 0.06 there really does mean 6% of the population variance.
They are also not comparable across effect-size families. η² = 0.06 and Cohen’s d = 0.5 are both “medium” and describe different quantities.
The question worth asking is what size of effect would change a decision, which comes from the subject matter and not from a table.
Cohen’s f is the same information, in the unit power analysis takes
Power software asks for f, not for η², and the two carry exactly the same information in different coordinates.
f = √(η² / (1 − η²)), so the conversion is exact and reversible — nothing is estimated twice.
Cohen’s bands for f are 0.10, 0.25 and 0.40, which map onto the η² bands of 0.01, 0.06 and 0.14 by that formula.
f is unbounded above while η² is capped at 1, which is why f stretches out the region near a perfect effect where η² compresses everything together.
Powering a study on f from a small pilot is the standard trap. An η² from a pilot is biased upward by (k − 1)/(N − 1), which at pilot sizes is large — so the f it implies is too big and the study that follows is underpowered. Converting from ω² instead removes most of that.
Generalised η² for designs that mix factor types
A third variant exists specifically so that effect sizes can be compared between studies whose designs differ, and it is the one most useful in meta-analysis.
Generalised η² puts the measured variance components in the denominator rather than only the ones in the current model, so a within-subjects effect and a between-subjects effect can be placed on one scale.
Partial η² cannot do that. A repeated-measures design removes subject variance from the error term, which inflates every partial η² relative to the same effect measured between subjects.
Which makes cross-design comparisons of partial η² systematically misleading, in a direction that always favours the within-subjects study.
This tool computes the one-way case, where generalised and ordinary η² coincide. For factorial designs the two-way calculator prints η² and partial η² per row so the divergence is visible.
Reporting an eta squared
Four things, and the first is the one that most often makes a reported number ambiguous.
Say whether it is η² or partial η². They are different numbers, they are usually both written “η²”, and on a factorial design they can differ by a great deal.
Give k and N. The reader needs them to know what η² a null result would have produced, and that figure is (k − 1)/(N − 1).
Report ω² alongside it when N is small. It costs one line, it is the estimate that is centred on the truth, and the gap between the two is itself informative.
And give a confidence interval where you can. Effect sizes from small samples are unstable, and a point estimate implies a precision the design does not have.
Method. All four effect sizes are computed from one source table rather than from separate formulas, so the identity that η² sums to exactly 1 across every row holds by construction and is asserted numerically. The null-expectation column is the analytic (k − 1)/(N − 1), and the shipped table beside it is a measurement: 4,000 datasets per design, every group drawn from one distribution, so every effect in them is chance. The measured means land on the analytic value at all seven designs — 0.147253 against 0.148936 at eight groups of six, 0.052579 against 0.052632 at two groups of ten — and the mean ω² over the same datasets stays within 0.002 of zero. The two-group identity η² = r²_pb is checked against the point-biserial engine, agreeing to 1.4×10⁻⁷. Fewer than two groups, a group with one value and zero within-group variation all return no result. That engine is verified on every change against 96 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Omega SquaredThe bias-corrected effect size from a published F and its degrees of freedom, exactly zero when F is exactly 1 — because the correction subtracts precisely what chance contributes.
One-Way ANOVAThe full F table with eta and omega squared, plus every pairwise gap — because a significant F says something differs and never says which, and ten groups tested pairwise carry a 90% false-positive rate.
Two-Way ANOVAFull source table with three effect sizes per row and the interaction reported first, because a crossover gives both main effects a sum of squares of exactly zero while the interaction has F = 600.
Effect SizeCohen d, Hedges g and the overlap between groups, with a sample-size control that moves the p-value while leaving the effect size fixed — the same d gives t = 1.29 at n=30 and 23.57 at n=10,000.
Point-Biserial Correlationr with Cohen's d, t and Pearson's r on the same columns computed independently, plus the split table: hold d at 1.9799 and r falls from 0.707107 at 50/50 to 0.269630 at 2/98.
Coefficient of DeterminationR-squared across five models at once with adjusted R-squared beside it, because adding a term can never lower it and the value most tools report for a curve fit was computed in log space.
An educational tool. Eta squared cannot be negative and is therefore biased upward by (k − 1)/(N − 1) — with eight groups of six, 94.7% of datasets with no real effect at all still produce an η² above 0.05 — so a value near the conventional thresholds from a small design is not evidence of an effect.
Published an effect-size tool that names the denominator each measure divides by, since eta squared, partial eta squared, omega squared and epsilon squared are routinely all written the same way.
Measured the bias rather than describing it: across 4,000 datasets per design in which every group was drawn from one distribution, mean eta squared lands on (k-1)/(N-1) every time — 0.147253 against a predicted 0.148936 at eight groups of six, 0.052579 against 0.052632 at two groups of ten.
Recorded the practical consequence: with eight groups of six and no effect at all, 94.7% of datasets produce an eta squared above 0.05, and 72.0% do with six groups of ten.
Showed that omega squared over the same null datasets averages within 0.002 of zero at every design, which is what an unbiased estimator looks like next to one that is not.
Verified the two-group identity that eta squared equals the squared point-biserial correlation, against the site's own point-biserial engine, agreeing to 1.4e-7.
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