Effect size converter: d, r, η², f, odds ratio and AUC

Name: Effect size converter
Author: Richard Injante

Convert between Cohen’s d, r, eta squared, f, odds ratio and AUC to compare studies that report the effect in different ways.

Content reviewed by Richard Injante Published by SolucionesAhora.com Updated on 2026-10-08 How we verify

Calculator inputs

Enter your figures, or press “See example” to watch it work.
If your case is unusual, open “Advanced options”.
Read the result and, below it, what it means and how it was worked out.

I have the effect as

Pick the measure you have. It is first converted to Cohen’s d (the difference between two groups in standard deviations) and from there to the others. η² and f come from ANOVA and two groups are assumed here.

Effect value

The value of the chosen measure. Valid ranges: r between −1 and 1 (exclusive), η² between 0 and 1, f from 0, OR above 0 and AUC between 0 and 1.

For example 0.5 for a medium d

Advanced options

Group 1 size (optional) (optional)

The conversion between d and r depends on whether the groups are the same size. If yours are not, enter both sizes; if left blank they are assumed equal.

Only if the groups are unequal

Group 2 size (optional) (optional)

The size of the second group. Give both sizes or neither.

Only if the groups are unequal

See the result

Results

Cohen’s d0.5000

Correlation r0.2425

Eta squared (η²)0.0588

Cohen’s f0.2500

Odds ratio (OR)2.4766

AUC0.6382

Probability of superiority (%)63.8

Magnitude (Cohen’s benchmarks)medium

Equivalents and magnitude on each scale
Index	Value	Magnitude
d de Cohen	0.5	medium
r	0.2425	small
η²	0.0588	small
f	0.25	medium
Odds ratio	2.4766	medium
AUC	0.6382	medium

See the working step by step

d = 0.5000; r = 0.2425; η² = 0.0588
f = 0.2500; OR = 2.4766; AUC = 0.6382

In short

What it calculates: Convert between Cohen’s d, r, eta squared, f, odds ratio and AUC to compare studies that report the effect in different ways.
Formula used: a = (n₁+n₂)² / (n₁·n₂) (a = 4 with equal groups)
Example: Example 1. From d to the rest.

Effect size converter

One study reports a correlation, another a Cohen’s d, another an odds ratio and another an eta squared, and all of them talk about “how big the effect is”. To compare them, pool them in a meta-analysis or plan your sample with the measure your software asks for, you have to put them on the same scale.

Pick the measure you have, type its value and you get the others, with the magnitude (small, medium, large) on each scale. If your groups are different sizes, enter them: the relationship between d and r changes. The calculation runs in your browser.

How it works

Almost all these measures are versions of the same idea (how far apart two groups are), so they can be converted into one another with closed formulas. The calculator first converts what you have to Cohen’s d and gets the rest from there.

The conversions for r, η² and f are exact for two groups; those for the odds ratio and AUC assume the underlying data are roughly normal (for the odds ratio, a latent logistic variable). They are useful approximations, not perfect mathematical equivalences.

Formula

a = (n₁+n₂)² / (n₁·n₂) (a = 4 with equal groups)
r = d / √(d² + a) d = r·√a / √(1 − r²)
η² = r² f = |d| / √a (f = d/2 with equal groups)
OR = exp(d·π/√3) d = ln(OR)·√3/π
AUC = Φ(d/√2) d = √2·Φ⁻¹(AUC)
Small · medium · large thresholds:
d 0.2 · 0.5 · 0.8 r 0.1 · 0.3 · 0.5 η² 0.01 · 0.06 · 0.14 f 0.1 · 0.25 · 0.4 OR 1.44 · 2.48 · 4.27 AUC 0.56 · 0.64 · 0.71

Worked example

Example 1. From d to the rest.
With d = 0.5 and equal groups: r = 0.243, η² = 0.059, f = 0.250, odds ratio = 2.48 and AUC = 0.638. In other words, Cohen’s medium effect equals an odds ratio of almost 2.5 and a 63.8% chance that someone from the group with the higher mean scores higher than someone from the other.

Example 2. From an odds ratio to d.
A medical paper reports an odds ratio of 3. That is d = 0.606, r = 0.290 and AUC = 0.666: a medium effect (d between 0.5 and 0.8). With this conversion, an odds ratio of 2.48 equals d = 0.5 and one of 4.27 equals d = 0.8.

Example 3. Unequal groups.
With d = 0.5, 20 people in one group and 80 in the other, r = 0.196 instead of 0.243. The same difference in means gives a smaller correlation when groups are unbalanced, which is why you should enter the sizes.

Explanation

What converting effect sizes is for

**Meta-analysis**: to combine studies you have to express them all in one measure.
**Planning sample size**: each program asks for the effect on its own scale (d for a t test, f for ANOVA, r for a correlation).
**Comparing results** from papers that report different measures.
**Explaining an effect** with a more intuitive measure, such as the probability of superiority.

What each scale measures

**Cohen’s d**: the difference between two means in standard deviations.
**r**: the correlation between group membership and the outcome (point-biserial).
**η² and f**: the proportion of variance explained and its root relative to unexplained variance, typical of ANOVA.
**Odds ratio**: how much the relative odds of an outcome are multiplied between groups.
**AUC**: the probability that a case from the higher group beats one from the lower.

Common mistakes

Applying d’s thresholds to an odds ratio without converting it: an OR of 2 is a small effect, not “twice the effect”.
Converting d to r with very unequal groups as if they were equal.
Using the η² or f conversion in an ANOVA with three or more groups as if there were two.
Losing the sign: η² and f do not have one, so they cannot tell which group scores higher.
Forgetting these are approximations when the data are not roughly normal.

Limits of this calculator

The η² and f conversions assume two groups; with three or more, f does not translate to a single d.
The odds ratio and AUC conversions assume underlying normality.
It does not convert effect sizes for proportions (Cohen’s h) or chi-square (w).
The OR and AUC thresholds are those equivalent to d = 0.2, 0.5 and 0.8 under these same formulas; other sources suggest others (Chen, Cohen and Chen, 2010, give 1.68, 3.47 and 6.71 for a 1% baseline risk in the unexposed group).

Notice

This is an estimate. In a meta-analysis, check the conversions with the package you use (for example metafor or RevMan).

Frequently asked questions

How do I go from d to r?

With equal groups, r = d / √(d² + 4). With unequal groups, replace the 4 with (n₁+n₂)² / (n₁·n₂). The calculator does it for you.

How do I convert an odds ratio to Cohen’s d?

With d = ln(OR) × √3 / π (Chinn, 2000). An odds ratio of 3 equals d = 0.61.

What odds ratio equals a medium effect?

With Chinn’s conversion, d = 0.5 equals an odds ratio of 2.48 (small, 1.44; large, 4.27). Chen, Cohen and Chen (2010) give 3.47 for a medium effect if the baseline risk is 1%: the value depends on the starting risk.

Why do η² and f have no sign?

Because they are based on variances, which are always positive. They show how much, not which group scores higher.

What is AUC in this context?

The probability that a randomly chosen person from the group with the higher mean scores higher than a randomly chosen person from the other. It is the same as the probability of superiority.

Can I convert f from an ANOVA with three groups?

Not into a single d: with three or more groups there are several possible differences. This calculator’s conversion assumes two groups.

What if my groups are different sizes?

Enter both sizes. The resulting r and f are smaller than with equal groups for the same d.

Is my data sent to a server?

No. The calculation runs in your browser.

Sources and references

The formula behind this calculator is not ours. These are the sources it comes from, in APA 7 format, in case you want to go to the original or cite them in your own work.

Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to meta-analysis. Wiley. The conversions between d, r and the odds ratio, for equal and unequal groups.
Chinn, S. (2000). A simple method for converting an odds ratio to effect size for use in meta-analysis. Statistics in Medicine, 19(22), 3127–3131. The formula d = ln(OR) × √3 / π.
Chen, H., Cohen, P., & Chen, S. (2010). How big is a big odds ratio? Interpreting the magnitudes of odds ratios in epidemiological studies. Communications in Statistics - Simulation and Computation, 39(4), 860–864. The odds-ratio thresholds by baseline risk (1.68, 3.47 and 6.71 at 1%) and the AUC ones.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2.ª ed.). Lawrence Erlbaum Associates. The small, medium and large thresholds for d, r, η² and f.

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