Effect Size Conversion Calculator

StatisticsLast updated: August 22, 2026

Effect sizes come from different families: Cohen d from the mean difference family, correlation r from the association family, the odds ratio from categorical data, and Cohen f and eta squared from the variance family. They describe the same phenomenon on different scales and can be converted into one another. In meta-analysis, combining studies run with different designs on a single scale also runs through these conversions.

This calculator asks for whichever measure you already have and produces all the others: d ↔ r, d ↔ odds ratio, d ↔ Cohen f, and f ↔ η². Next to each measure it shows the label from that measure's own benchmarks (negligible / small / medium / large), plus intuitive equivalents such as the superiority index U₃, the common language effect size, and the overlap of the two distributions. A conversion table of reference values shows where your own value stands.

The value you enter is converted into every other measure, using Cohen d as the common unit.
The numeric value of the measure you selected. For r, between −0.999 and 0.999; for the odds ratio, a positive number; for η², between 0 and 0.999.

Effect Size Conversion Formulas

d → r:        r = d / √(d² + 4)
r → d:        d = 2r / √(1 − r²)
d → OR:       ln(OR) = d · π / √3   (π/√3 ≈ 1.8138)
OR → d:       d = ln(OR) · √3 / π
d → f:        f = d / 2
f → d:        d = 2f
f → η²:       η² = f² / (1 + f²)
η² → f:       f = √(η² / (1 − η²))
r and η²:     η² = r² = d² / (d² + 4)
Intuitive:    U₃ = Φ(d),  CLES = Φ(d/√2),  OVL = 2·Φ(−|d|/2)

The conversions run through Cohen d as the common unit. The coefficient 4 in the d ↔ r formula assumes equal group sizes, and the d ↔ OR conversion assumes a logistic distribution; both are approximations.

How to Calculate

  1. Select whichever effect size you have from the list: Cohen d, correlation r, odds ratio, Cohen f, or eta squared.
  2. Enter the value; both a decimal point and a decimal comma work. The odds ratio must be positive, and r must lie between −1 and 1.
  3. Check the label next to each measure in the results: the same effect appears as different numbers on different scales but in the same size class.
  4. For power analysis use the Cohen f value; for meta-analysis use ln(OR) or the Hedges g scale.
  5. Look at the reference values in the conversion table (d = 0.20 · 0.50 · 0.80) to see where your own value falls.
  6. If the group sizes are clearly unbalanced, keep the assumption behind the d ↔ r conversion in mind and apply the correction given in the notes.

Worked Examples

Converting a medium Cohen d

Cohen's medium threshold, d = 0.50, corresponds to r = 0.2425 on the correlation scale, 2.4766 on the odds ratio scale, and 0.2500 on the Cohen f scale. The explained variability is η² = 0.0588, only 5.9%: it shows how small a share of the total variability even a "medium" effect accounts for. The superiority index is U₃ = 69.1% and the distribution overlap is 80.3%.

Cohen d: 0.5000 · Correlation coefficient (r): 0.2425 · Odds ratio (OR): 2.4766

From a correlation to Cohen d

A moderate correlation such as r = 0.30 becomes d = 2·0.30 / √(1 − 0.09) = 0.6290 on the two-group comparison scale, a medium effect by Cohen's benchmarks. The same effect is expressed as an odds ratio of 3.1294, a Cohen f of 0.3145, and η² = 0.0900 (that is, 9.0% explained variability); the superiority index is U₃ = 73.5%.

Cohen d: 0.6290 · Correlation coefficient (r): 0.3000 · Odds ratio (OR): 3.1294

Effect size from an odds ratio

An OR = 2.5 from an epidemiological study converts through ln(2.5) = 0.9163 into d = 0.5052, an effect on the small-to-medium boundary by Chen's benchmarks. The matching correlation is r = 0.2449, Cohen f = 0.2526, and η² = 0.0600. The common language effect size is 64.0% and the distribution overlap is 80.1%.

Cohen d: 0.5052 · Correlation coefficient (r): 0.2449 · Odds ratio (OR): 2.5000

Frequently Asked Questions

Is it safe to convert effect sizes into one another?
The conversions are not exact identities but approximations that hold under specific assumptions: d ↔ r assumes equal group sizes, and d ↔ OR assumes a logistic distribution. For medium-sized effects the error is small, and the conversions are used routinely in meta-analysis. At extreme values (|d| > 3, OR > 20) the deviation becomes noticeable; there, report the converted value as a rough correspondence rather than an exact number.
Why does d = 0.50 correspond to a value as small as r = 0.24?
The two scales answer different questions: d measures the difference in standard deviation units and has no upper bound, while r squeezes the association between −1 and 1 and looks at the shared variability of two variables. As d grows, r rises more slowly because it approaches 1 asymptotically. That is why the "medium" thresholds differ as well: 0.50 for d, 0.30 for r.
How are Cohen f and Cohen d related?
For two groups of equal size, f = d / 2 holds, which is why the thresholds d = 0.20/0.50/0.80 become 0.10/0.25/0.40 on the f scale. In ANOVA designs with more than two groups this simple equality no longer applies; f is defined as the ratio of the standard deviation of the group means to the error standard deviation and is computed through η². G*Power asks for f in ANOVA power analysis.
Which formula is used to convert an odds ratio to Cohen d?
The widely accepted Hasselblad and Hedges (1995) conversion is used: d = ln(OR) · √3 / π. The factor π/√3 ≈ 1.8138 is the standard deviation of the standard logistic distribution and follows from assuming a continuous, logistically distributed propensity variable underlying the binary outcome. The same formula does not apply to a risk ratio (RR); that has to be converted to an OR first.
Are η² and r² really the same?
Numerically, yes: both are the proportion of explained variability in the dependent variable, and in a two-group design the identity η² = r² = d² / (d² + 4) holds exactly. The conceptual difference is that r² measures only the linear relationship, while η² can also capture the non-linear effect of a categorical factor. In multi-group designs η² equals the R² of a regression built with dummy variables.
How should I report a converted value in a paper?
Give both the original measure and the converted value, along with the conversion formula: "OR = 2.50 (corresponds to d = 0.51; Hasselblad & Hedges conversion)". Reporting only the converted value prevents readers from reproducing the original analysis. In meta-analyses, after putting all studies on a common scale, the method section must state clearly which conversion was used.
Can an effect size be negative?
d, r, and ln(OR) are signed and carry the direction of the effect; η², f, and the odds ratio are unsigned (for the OR, direction is set by whether it is above or below 1). So when you convert from η² or f back to d, you supply the direction, not the calculator: if you do not state which group it favors, the information is lost. Keeping the direction of comparison fixed throughout a study is the safest approach.