How to use the effect size converter
- Choose the effect size you have: Cohen's d, Hedges' g, Pearson r, eta squared or an odds ratio.
- Type its value and the two group sizes. The sizes matter for the small sample correction and for unequal groups.
- Read the table of every other effect size, each with the formula that produced it, or copy the numbers as text or CSV.
- Check the note under the results before comparing with a paper, because several conversion formulas exist for the odds ratio and r.
Formula
a = (n1 + n2)^2 / (n1 n2)
r = d / sqrt(d^2 + a) and d = r sqrt(a) / sqrt(1 - r^2)
ln(odds ratio) = d x pi / sqrt(3)
eta squared = r^2
Hedges g = d x J, J = Gamma(m/2) / (sqrt(m/2) Gamma((m - 1)/2)), m = n1 + n2 - 2
Common language effect size = Phi(d / sqrt(2)), U3 = Phi(d)
Phi is the standard normal distribution function. With equal groups a = 4, which gives the familiar r = d / sqrt(d^2 + 4). The logistic conversion of d to an odds ratio assumes both groups follow a bell-shaped distribution with the same spread.
A large effect with 30 people per group
Start from d = 0.8 with n1 = n2 = 30. Then a = 60 x 60 / 900 = 4, so r = 0.8 / sqrt(0.64 + 4) = 0.3714 and eta squared = r squared = 0.1379.
The log odds ratio is 0.8 x pi / sqrt(3) = 1.4510, so the odds ratio is 4.2675. With 58 degrees of freedom the small sample correction is J = 0.9870, which makes Hedges' g = 0.7896. The common language effect size is 71.42% and U3 is 78.81%.
Why one effect can have several names
Researchers report the size of a difference in different ways: a standardized mean difference such as Cohen's d, a correlation, a share of variance or an odds ratio. They describe the same relationship, so each can be converted into the others when the groups are roughly bell shaped. This page does the arithmetic so you can put results from different papers on one scale.
The conversion between d and r depends on how the groups are split. Equal groups give the textbook formula. When one group is much larger, the same d corresponds to a smaller r, which is why the group sizes are asked for. Hedges' g shrinks d slightly for small samples, and the shrinkage disappears as the groups grow.
Limits of the conversions
Converting an odds ratio into d with the logistic method treats a yes or no outcome as a hidden bell-shaped measure. That is an approximation, and other methods give somewhat different numbers. If you are working directly with a 2 by 2 table, the odds ratio and relative risk calculator uses the counts themselves and needs no such assumption.
Eta squared from a two-group comparison equals r squared, but with three or more groups it is a different quantity, so use this page for two groups only. These numbers describe a sample. They do not say whether a difference is large enough to matter in practice.
Assumptions and limits
- The comparison is between two independent groups with outcomes that are roughly bell shaped and have similar spread.
- The odds ratio uses the logistic conversion d x pi / sqrt(3). Other conversions exist and will give different values.
- Eta squared is entered as a positive effect, because it carries no sign.
- Nothing here is a test of significance; use a test or a confidence interval for that.
Frequently asked questions
Why does the same d give a different r for unequal groups?
Pearson r treats group membership as a two-valued variable, and its variance is smaller when the groups are lopsided. The factor (n1 + n2) squared over n1 times n2 captures that, and it equals 4 only when the groups are the same size.
What is the difference between d and Hedges' g?
Hedges' g multiplies d by a correction slightly below 1 that removes a small upward bias in small samples. With 30 people per group the factor is about 0.987, and with several hundred people it is close to 1.
What does the common language effect size mean?
It is the chance that a randomly picked person from group 1 scores higher than a randomly picked person from group 2, assuming bell-shaped scores. A value of 50% means no difference, and it moves toward 100% as d grows.
Can I enter a negative value?
Yes for d, g and r. A negative sign means group 1 has the lower average, and the odds ratio then falls below 1. Eta squared and the odds ratio cannot be negative.