Effect Size Converter
Effect sizes put the magnitude of a result in unit-free terms so you can compare studies and judge practical importance. Drop in a Cohen's d, correlation, odds ratio or ANOVA effect, and this tool translates it into every other common metric, plus plain-language readouts like CLES (the chance one group beats the other) and NNT, with exact confidence intervals.
r = 0.243
How this conversion is computed
Which effect size is which
Every metric below measures the same underlying signal on a different scale. The converter routes each one through Cohen's d, the common currency, and out to your target.
| Metric | What it measures | Small / medium / large |
|---|---|---|
| Cohen's d | Difference between two group means in pooled-SD units. | 0.2 / 0.5 / 0.8 |
| Hedges' g | Cohen's d with a small-sample bias correction (factor J). | 0.2 / 0.5 / 0.8 |
| Pearson's r | Linear correlation, or point-biserial r when one variable is binary. | 0.1 / 0.3 / 0.5 |
| Odds ratio | Ratio of the odds of an event between two groups (binary outcome). | 1.5 / 2.5 / 4.3 |
| η² (eta-squared) | Share of total variance explained by a factor in ANOVA. | 0.01 / 0.06 / 0.14 |
| Cohen's f | ANOVA effect for power tables: SD of group means over within-group SD. | 0.1 / 0.25 / 0.4 |
| CLES | Common-language effect size: P(a random case from group 1 beats group 2). | 0.56 / 0.64 / 0.71 |
| NNT | Number needed to treat (Kraemer-Kupfer): 1 / (2·CLES − 1). | smaller is stronger |
The conversion formulas
All conversions pass through Cohen's d. The core identities:
r = d / sqrt(d^2 + a) | a = 4 for equal groups, else (n1+n2)^2/(n1*n2) |
d = 2r / sqrt(1 - r^2) | correlation back to d |
OR = exp(d * pi / sqrt(3)) | logistic-latent bridge (Hasselblad & Hedges 1995) |
f = sqrt(eta2 / (1 - eta2)) | and eta2 = f^2 / (1 + f^2) |
CLES = pnorm(d / sqrt(2)) | probability of superiority |
g = d * J | J from lgamma; exact small-sample correction |
The d confidence interval inverts the noncentral t distribution (exact, as in MBESS::ci.smd), and the r interval uses the Fisher z transform. The d↔OR bridge assumes a logistic latent variable, so treat it as an approximation across very different designs.
Frequently asked questions
How do I convert Cohen's d to a correlation r?
Use the point-biserial relationship r = d / sqrt(d² + a), where a = 4 for equal group sizes and a = (n1+n2)²/(n1·n2) for unequal groups. For example, d = 0.5 with equal groups gives r = 0.243. Squaring r gives eta-squared.
What counts as a small, medium, or large effect size?
By Cohen's benchmarks, d of 0.2, 0.5 and 0.8 are small, medium and large; r of 0.1, 0.3 and 0.5; and eta-squared of 0.01, 0.06 and 0.14. These are rules of thumb, not hard cutoffs. In some fields a d of 0.2 is a meaningful, hard-won effect.
Why use Hedges' g instead of Cohen's d?
Cohen's d runs slightly high in small samples. Hedges' g multiplies d by an exact correction factor J (computed from log-gamma functions) that removes this bias. The tool uses the exact J, not the common 1 − 3/(4N−9) approximation, so g is preferred whenever the total sample is under roughly 40.
How is the confidence interval for Cohen's d computed?
This tool inverts the noncentral t distribution (the exact method used by MBESS::ci.smd and effectsize), not a normal approximation. It converts d to a t value, finds the noncentrality parameters whose distributions place t at the tails, and scales them back to the d metric.
Does the emitted R code reproduce these numbers?
Yes. The R block uses base R only (no packages required) and reproduces the displayed target value exactly. Every formula in this tool is verified against R's own pnorm(), qt() with a noncentrality parameter, and lgamma() across 259 checks.