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Converting Between Effect Sizes

After reading this you can turn a Cohen's d into a correlation r, an r into an odds ratio, or a proportion of variance explained into any of the others, and know how far to trust each conversion.

Effect sizes come in different currencies. A psychologist reports Cohen's d = 0.5. An epidemiologist reports an odds ratio of 2. A meta-analyst wants everything on one scale so the studies can be pooled. These numbers describe the same kind of thing, the strength of a difference or association, but in different units. Converting between them lets you compare results that were reported in incompatible ways.

The conversions here assume two groups of equal size and roughly normal data. They are approximations. Under those assumptions d = 0.5 corresponds to r \approx 0.243, an odds ratio of about 2.48, and \eta^2 \approx 0.059. The rest of this article shows where those numbers come from and when they break.

What each measure means

Four measures appear on this page. They answer slightly different questions.

Cohen's d
The difference between two group means, divided by the pooled standard deviation. A d of 0.5 means the groups differ by half a standard deviation. It has no upper bound.
Correlation r
The point-biserial correlation between group membership and the outcome, ranging from -1 to 1. It is the same r you know from regression, applied to a two-group contrast.
Eta squared (\eta^2)
The proportion of variance in the outcome explained by group membership. For a two-group comparison it equals r^2. Ranges from 0 to 1.
Odds ratio
The odds of an outcome in one group divided by the odds in the other. An OR of 1 means no difference. It ranges from 0 to \infty, and it is naturally read on a log scale.

The d, r and \eta^2 family assume a continuous outcome. The odds ratio assumes a binary outcome. Moving between the two families requires the extra assumption that a normal continuous trait sits underneath the yes/no split.

When conversion is fair, and when it is not

Use these conversions when you genuinely need one scale, most often in a meta-analysis or when comparing your result to a published benchmark. They are also handy for sanity checks: if a paper reports both an r and a d, the two should be consistent.

The equal-group-sizes assumption matters more than people expect. The d-to-r formula in wide use includes a correction factor a that depends on the ratio of group sizes. With balanced groups a = 4. With a 90/10 split a rises above 11, and the same d maps to a much smaller r. This calculator uses the balanced value, so treat its output as a best case for balanced designs.

Do not convert when the direction of causation or the shape of the data contradicts the assumptions. An odds ratio from a rare-disease case-control study does not cleanly become a Cohen's d, because there is no underlying continuous mean to compare. And never convert a number and then claim more precision than the original had. If the source d was rounded to one decimal, the derived odds ratio is no more exact.

The formulas

Start with the link between d and r for equal groups:

r = \frac{d}{\sqrt{d^2 + 4}}

Here d is Cohen's d and r is the resulting correlation. The 4 is the correction factor for two equal groups. Solving the same relationship for d gives the reverse direction:

d = \frac{2r}{\sqrt{1 - r^2}}

The proportion of variance follows directly, since for a two-group contrast \eta^2 = r^2.

The bridge to the odds ratio uses the logistic approximation. A Cohen's d becomes a log odds ratio by scaling with \pi / \sqrt{3} \approx 1.814:

\ln(\text{OR}) = \frac{d \, \pi}{\sqrt{3}}

To go the other way, take the natural log of the odds ratio and divide:

d = \frac{\ln(\text{OR}) \, \sqrt{3}}{\pi}

The intuition: a logistic distribution has a standard deviation of \pi / \sqrt{3}, so multiplying a standardized mean difference by that factor puts it on the log-odds scale. This is the Hasselblad and Hedges conversion, and it is accurate to within a few percent for moderate effects.

Worked example: starting from d = 0.5

Reproducing the demo

Set Mode to From Cohen's d and leave d = 0.5, then press Calculate. Here is every step.

  1. Compute r: r = 0.5 / \sqrt{0.5^2 + 4} = 0.5 / \sqrt{4.25} = 0.5 / 2.0616 \approx 0.2425.
  2. Compute \eta^2 = r^2 = 0.2425^2 \approx 0.0588. About 5.9% of the variance is explained.
  3. Compute the log odds ratio: \ln(\text{OR}) = 0.5 \times 1.8138 \approx 0.9069.
  4. Exponentiate: \text{OR} = e^{0.9069} \approx 2.476.

So a half-standard-deviation difference is a correlation near 0.24, explains roughly one twentieth of the variance, and corresponds to odds about 2.5 times higher in one group. These four numbers describe one effect.

To check the reverse conversions, feed each output back in. Entering r = 0.2425 in From correlation r mode should return d \approx 0.5, and entering 2.476 in odds-ratio mode should also return d \approx 0.5. Small rounding differences are normal.

How the measures track each other

The relationship between d and r is not linear. Near zero the two rise together almost one to one, but as d grows, r flattens toward its ceiling of 1. The chart below shows the curve.

r climbs steeply for small d, then bends toward 1. The marker sits at d = 0.5, where r is about 0.24.

Some landmark values are worth memorizing. They line up with the conventional small, medium and large labels for d.

Equivalent effect sizes for balanced two-group designs
Cohen's drη²Odds ratioLabel
0.20.09950.00991.44Small
0.50.24250.05882.48Medium
0.80.37140.13794.27Large
1.20.51450.26478.79Very large

Cohen's d of 0.2, 0.5 and 0.8 convert to r values of about 0.10, 0.24 and 0.37, to η² of about 0.010, 0.059 and 0.138, and to odds ratios of about 1.44, 2.48 and 4.27. As d rises, r approaches 1 while the odds ratio grows without limit.

Reading the results without over-claiming

An \eta^2 of 0.0588 sounds tiny, and a d of 0.5 sounds respectable, yet they are the same effect. That gap is a feature, not a contradiction. Variance-explained measures square the correlation, so they shrink fast: halving r quarters \eta^2. Report the measure your audience expects, but understand that a "small" percentage of variance can still be a practically meaningful mean difference.

The odds ratio deserves special care because it is multiplicative. An OR of 2.48 and an OR of 0.40 are equal and opposite: 0.40 \approx 1/2.48. When you compare odds ratios, compare their logs. On the log scale 2.48 and 0.40 sit the same distance from 1, at \pm 0.91.

Common mistakes

The errors below account for most wrong conversions.

  • Assuming balanced groups when they are not. A 70/30 split inflates the correction factor and lowers the r you get from a given d. The output here is only exact for a 50/50 design.
  • Confusing \eta^2 with r. They differ by a square. An \eta^2 of 0.09 is an r of 0.30, not 0.09.
  • Averaging odds ratios directly. The mean of 2 and 0.5 is not 1.25 in any useful sense; on the log scale they cancel to 0, so the geometric mean is 1. Pool odds ratios as log odds ratios.
  • Treating the converted value as new evidence. Converting does not add information. A shaky d yields an equally shaky odds ratio.

Related calculators

If you have raw group means and standard deviations rather than a finished effect size, compute d first with the Cohen's d Effect Size Calculator, then bring it here. To plan a study around a target effect, the Power & Sample Size for t-Tests calculator turns a d into a required sample size, and the Sample Size Calculator handles estimation with a fixed margin of error. Once you have an estimate, put bounds on it with the Confidence Interval Calculator.

Frequently asked questions

Is r the same as r-squared here?

No. The correlation r for a two-group contrast is the point-biserial correlation, and \eta^2 = r^2 is the variance-explained version. For r = 0.2425, \eta^2 = 0.0588.

Why does the odds ratio conversion use π over the square root of 3?

That factor, about 1.814, is the standard deviation of a logistic distribution. Multiplying a standardized mean difference by it converts the difference from standard-deviation units to log-odds units. It is the standard Hasselblad and Hedges bridge between the normal and logistic scales.

Can I convert an odds ratio below 1?

Yes. An OR of 0.5 gives \ln(0.5) = -0.693, so d = -0.693 \times 0.5513 \approx -0.382. The negative sign just means the effect points the other way.

How accurate are these conversions?

For balanced groups, normal outcomes and effects in the small-to-large range, the d-r-\eta^2 conversions are exact and the odds-ratio bridge is accurate to within a few percent. Accuracy falls off with unequal group sizes and skewed data.

Which measure should I report?

Match your field. Report d for experimental group differences, r for associations, \eta^2 in ANOVA contexts, and odds ratios for binary outcomes. Convert only to compare across studies.

Try it yourself in the tool

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