The bias-corrected effect size, from a table you already have.
From a published F and its degrees of freedom
Four groups of ten, F = 4.0 and p = 0.0146. η² is 0.250000 and ω² is 0.183673 — the paper would have reported 25% of the variance explained where 18.4% is the population estimate.
4 groups, N = 40, F(3, 36) = 4.0000
ω² = 0.183673
η² for the same result is 0.250000, so the correction removes 0.066327. That is not an arbitrary shrinkage: it is df_effect × MS_error divided through, which is exactly the amount this design would produce from chance alone. The expected η² here with no real effect at all is 0.076923.
ω²
0.183673
population estimate
η²
0.250000
sample description
Correction
0.066327
η² minus ω²
ε²
0.187500
same correction, η² denominator
How much the correction takes off, across F
Your degrees of freedom held fixed at (3, 36); only F changes.
Eta squared, omega squared and the correction at eight values of F
F
η²
ω²
Correction
ω² as a share of η²
0.5
0.040000
-0.038961
0.078961
-97.4%
1
0.076923
0.000000
0.076923
0.0%
1.5
0.111111
0.036145
0.074967
32.5%
2
0.142857
0.069767
0.073090
48.8%
3
0.200000
0.130435
0.069565
65.2%
5
0.294118
0.230769
0.063348
78.5%
10
0.454545
0.402985
0.051560
88.7%
25
0.675676
0.642857
0.032819
95.1%
The correction barely moves in absolute terms while η² moves enormously. At F = 1 it removes exactly all of η² and ω² is precisely 0; across F from 0.5 to 25 the correction only falls from 0.078961 to 0.032819 while η² climbs from 0.040000 to 0.675676. Which is why the gap between the two matters overwhelmingly for weak results and barely at all for strong ones.
ω² is exactly 0 when F is exactly 1 Negative values are reported, not clamped F and its two df are a complete input
What this tool shows
Omega squared is exactly 0 when F is exactly 1. Not approximately — the correction it applies is df_effect × MS_error, which is precisely what a null effect contributes on average, and F = 1 is precisely the case where the between-group variation equals the within-group variation. Below that it goes negative, which is the honest report rather than an error. An F and its two degrees of freedom are a complete input.
Omega squared from a published F and its two degrees of freedom
Eta squared and epsilon squared for the same result, side by side
The exact size of the correction, and what it is made of
How the correction behaves as F changes, with the degrees of freedom held fixed
Negative omega squared reported as computed rather than clamped to zero
The expected eta squared for the design when the effect is truly zero
Bias-corrected F and df is enough Exactly 0 at F = 1 Negative values kept
Updated 12 September 2026 · Works in any browser, no installation
Omega squared estimates the share of variance the factor explains in the POPULATION, where eta squared describes the share it explained in your sample. The difference is a bias: eta squared cannot be negative, so it cannot be centred on zero, and it averages (k − 1)/(N − 1) even when there is no effect at all. Omega squared subtracts exactly that.
At a glance
Formula shown
ω² = (SS_effect − df_effect·MS_error) / (SS_total + MS_error). The subtracted term is the expected contribution of chance: under a true null, E[SS_effect] = df_effect·MS_error exactly. Every ratio here is scale-free, so F and its two degrees of freedom determine all of them — set MS_error = 1, then SS_error = df_error and SS_effect = F·df_effect, and nothing else is needed.
Scenario support
Recomputing a bias-corrected effect size from a published ANOVA table, meta-analysis, power planning from a small pilot, reporting the size of a marginal result honestly, and any situation where a paper printed eta squared but the population estimate is what the question needs.
Educational estimate
Planning support from the values you enter — not professional advice.
The correction is exactly what chance contributes
“Bias-corrected” sounds like a shrinkage applied for safety. It is an exact subtraction of a known quantity.
Under a true null, the expected between-group sum of squares is df_effect × MS_error. Not roughly — that is the definition of the error mean square as an unbiased variance estimate.
Omega squared subtracts precisely that term from the numerator. What is left is the part of the between-group variation that chance does not account for.
Which is why ω² is exactly 0 when F is exactly 1. F = 1 means SS_effect = df_effect × MS_error, so the numerator vanishes identically. The tool shows 0.000000, not a rounded zero.
And it is why ω² goes negative when F is below 1. The groups differed by less than chance usually manages. The population estimate is then negative, and clamping it to zero would hide that a result came in below the noise floor.
The correction is nearly a constant amount, not a constant fraction
This is what decides when the distinction matters, and it is visible directly in the sweep table.
With four groups of ten, the correction runs from 0.078961 at F = 0.5 to 0.032819 at F = 25. It falls by a factor of 2.4 over that range while η² climbs from 0.040000 to 0.675676 — a factor of seventeen.
So the gap is decisive for weak results and negligible for strong ones. At F = 1.2, η² = 0.090909 and ω² = 0.014778: the correction removes 84% of the reported effect. At F = 25 it removes 5%.
Which is exactly backwards from where it gets reported. Papers with strong results can afford to print either; papers with marginal results — where the correction changes the story — are the ones that usually print η² alone.
The correction also shrinks as N grows. More error degrees of freedom means a smaller MS_error relative to SS_total, so a large study needs the correction less — and a small one needs it most.
Measured: η² averages 0.147253 when nothing is happening
The bias is not a theoretical concern. Simulated across 4,000 datasets per design, with every group drawn from one distribution so every effect is chance:
Eight groups of six: mean η² = 0.147253, mean ω² = −0.001791. The analytic prediction for η² is 0.148936, and 94.7% of those null datasets produce an η² above 0.05.
Six groups of ten: 0.084390 against a prediction of 0.084746, with ω² averaging −0.000332 and 72.0% of null datasets above 0.05.
Three groups of fifty: 0.013112 against 0.013423, ω² at −0.000312, and only 2.4% above 0.05. The bias all but disappears once N is large relative to k.
Across every design tested, ω² stays within 0.002 of zero. That is what an unbiased estimator looks like, and it is the whole case for preferring it.
ε² applies the same correction to a different denominator
Epsilon squared is the third member of this family and is genuinely close to omega squared, which is why it is rarely worth arguing about.
ε² = (SS_effect − df_effect × MS_error) ÷ SS_total. Same numerator as ω²; η²’s denominator rather than SS_total + MS_error.
It is therefore slightly larger than ω² — by a factor of (SS_total + MS_error)/SS_total, which is close to 1 for any reasonable design.
Except when both are negative, where the ordering flips. They share a numerator and differ only in denominator, so a bigger denominator pulls a negative value CLOSER to zero: ω² is above ε² below F = 1. The rule that actually holds at every F is that they always share a sign and ω² is always the one nearer zero — which this site’s own verification suite established by first asserting the naive ordering and watching it fail on 11 of 300 random tables.
ε² is also the exact analogue of adjusted R² from regression, and it is sometimes called the Kelley correction.
Either is defensible; reporting neither is the actual problem. The tool prints both so the choice is visible rather than hidden in a software default.
Where the corrected value changes a decision
Three places, and in all three using η² propagates its bias into something else.
Power analysis from a pilot. A pilot’s η² is inflated by (k − 1)/(N − 1), which at pilot sizes is large. Powering the main study on it gives a sample size that is too small, and the study that follows is underpowered for the effect that actually exists. Converting to Cohen’s f from ω² removes most of that.
Meta-analysis. Pooling η² values across studies pools their biases, and the bias is largest in the smallest studies — which are also the ones most subject to publication selection. The two effects compound.
Comparing designs with different k. A four-group study and a two-group study have different null expectations, so their η² values are not on the same scale. ω² is.
For describing the sample in front of you, η² is fine — it is a correct description of how the variance divided up. The problem only arises when it is read as an estimate of something beyond the sample.
Reporting omega squared
Three things, and the third is the one people quietly skip.
Give F with both degrees of freedom alongside it. They are what let a reader recompute ω², η² and ε² for themselves.
Name the statistic. ω², ε² and η² are three different numbers and only the label distinguishes them in a results table.
Report a negative value as negative. Rounding −0.031 up to 0 removes the information that the result came in below the noise floor, which is exactly what a reader assessing a null finding needs to know.
And give a confidence interval if you can. Effect sizes from small designs are unstable, and an interval on ω² often includes zero even when the F test was significant.
Sources and methodology
References for omega squared and the bias it corrects.
Method. Every ratio here is scale-free, so the engine reconstructs the sums of squares from F and the two degrees of freedom by setting MS_error to 1 — SS_error = df_error and SS_effect = F × df_effect — which is why a published “F(3, 36) = 4.02” is a complete input and the raw data is not needed. The exactness of the F = 1 case is a property of that construction rather than a rounding: the numerator is SS_effect − df_effect × MS_error, which is identically zero there, and the suite asserts it across a grid of degrees of freedom. Negative values are returned as computed. The measured bias figures quoted on this page come from 4,000 datasets per design in which every group was drawn from one distribution. Degrees of freedom below 1 and a negative F 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:
Eta SquaredFour effect sizes with the denominator each one uses, and a measured table: with eight groups of six and no effect at all, 94.7% of datasets still give an eta squared above 0.05.
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.
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.
Statistical PowerPower and sample size from the non-central t rather than a normal approximation, with the gap shown — plus a live demonstration that post-hoc power is a function of the p-value alone, and 0.500044 at p = 0.05 for every study ever run.
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.
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. Omega squared is a population estimate and can legitimately come out negative when F is below 1; that is the correct output rather than an error. It corrects a bias but does not fix a small sample — an interval around ω² from a small design is wide, and frequently includes zero even when the F test cleared the conventional threshold.
Published a bias-corrected effect size that takes a published F and its two degrees of freedom as a complete input, because every ratio involved is scale-free and the raw sums of squares are not needed.
Established the exact property that makes the correction principled: omega squared is EXACTLY zero when F is EXACTLY 1, at all 25 combinations of degrees of freedom tested, because the term it subtracts is precisely df_effect x MS_error — what a null effect contributes on average.
Reported negative values as computed rather than clamping them, since a negative omega squared means the groups differed by less than chance usually manages, and that is the information a reader assessing a null result needs.
Measured how the correction behaves: across F from 0.5 to 25 at four groups of ten it only falls from 0.078961 to 0.032819 while eta squared climbs from 0.040000 to 0.675676 — so it is decisive for weak results and negligible for strong ones, which is exactly backwards from where it gets reported.
Corrected an error this site had made in its own reasoning: omega squared is NOT always below epsilon squared. They share a numerator and differ only in denominator, so below F = 1 the ordering flips. The assertion suite caught it on 11 of 300 random tables, and the rule that actually holds — same sign, omega always nearer zero — is now what is asserted.
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