The formula everyone quotes is <em>r</em> = <em>d</em>/√(<em>d</em>²+4). That 4 is not a constant: it is (n₁+n₂)²/(n₁·n₂), and it only equals 4 when both groups are <strong>the same size</strong>. At 90 against 10 it is 11.1, and using 4 inflates the correlation.
r = d / √(d² + a), with a = (n₁+n₂)² / (n₁·n₂) · d = 2r / √(1 − r²)
The full conversion is r = d/√(d² + a) with a = (n₁+n₂)²/(n₁·n₂). When both groups are the same size that quotient is exactly 4, and the formula collapses into the one quoted everywhere.
As soon as they become unbalanced, a grows fast: 60/40 gives 4.17; 80/20 gives 6.25; 90/10 gives 11.11. And since a sits in the denominator, using 4 with unequal groups overestimates the correlation.
In meta-analysis this matters more than it sounds, because the studies with badly unbalanced groups tend to be the observational ones — which are also the ones carrying the most weight through sample size.
Yes, to pool. A meta-analysis mixing experimental studies (which give d) with correlational ones (which give r) has no alternative, and converting to a common metric is standard practice. You convert, you say you converted, and you check in a sensitivity analysis whether the result depends on the conversion.
No, to flatter. Converting a small d into r — or the reverse — because the number reads better is something a reviewer spots immediately: nobody switches metric without a reason, and the reason gets asked about.
And beware dichotomization. If your grouping variable came from splitting a continuous one at the median, the converted r does not recover the original correlation: dichotomizing already destroyed information, and no formula gives it back.
One sentence is enough: "effect sizes were converted to r using the Borenstein formula, applying the correction factor for unequal group sizes". Without it, a reader cannot reproduce your numbers.
And if you are going to pool correlations, the next step is not averaging them: you transform with Fisher’s z, average there, and transform back. Averaging raw correlations biases the result downward, and with high correlations the bias is large.
If what you need is the starting d, it is in from t to Cohen’s d; and to say what that d means in people, the translation to overlap and percentiles.
All thirteen effect size conversions, grouped by what each one is for, are in the index.
If you use it in your thesis or in a paper, cite it so that readers of your Methods section know how the calculation was done.
Ciudad-Fernández, V. (2026). Cohen's d to r converter [Web application]. Ciudad Science. Retrieved October 14, 2026, from https://analisisdedatospsicologia.com/en/convert/d-to-r
For your Methods section: Effect sizes were converted between Cohen's d and r, using the Ciudad Science cohen's d to r converter (Ciudad-Fernández, 2026).
@misc{ciudadfernandez2026convdar,
author = {Ciudad-Fern{\'a}ndez, V{\'i}ctor},
title = {{Cohen's d to r converter}},
year = {2026},
publisher = {Ciudad Science},
howpublished = {Web application},
url = {https://analisisdedatospsicologia.com/en/convert/d-to-r},
urldate = {2026-10-14},
note = {Accessed 2026-10-14},
}Whichever you give it: the conversion does not know which of your groups is the "high" one. It is computed on the absolute value and you attach the sign matching the direction you are describing.
Not with this formula. A partial r already has covariates controlled and its metric is not comparable with a raw d. Converting it and pooling it mixes two different things.
That one is exact and needs no assumptions: r = t/√(t²+df). It is the better route when you have the t, because it assumes nothing about an underlying distribution.
With balanced groups, yes, down to the last decimal. With unequal groups, not quite: the reverse uses 4 and cannot know how yours were split.