Cohen's d Effect Size, Explained

After reading this you can compute Cohen's d and Hedges' g from two groups (or from a t statistic), convert the result to an equivalent correlation r, and read the number without over- or under-selling it.

What Cohen's d measures

Cohen's d tells you how far apart two group means are, measured in standard deviations. A p-value tells you whether a difference is unlikely to be zero. Cohen's d tells you how big the difference is. Those are different questions, and you usually want both.

Suppose group A scores a mean of 10 and group B scores a mean of 12, with both groups having a standard deviation of 2. The raw gap is 2 points. Divide that by the standard deviation of 2 and you get d = 1.0. The groups differ by a full standard deviation. That is a large separation: the average person in group B outscores about 84% of group A.

The scale is unitless, so you can compare effects across studies that used different measurement scales. A d of 0.5 on a reading test and a d of 0.5 on a reaction-time task describe the same relative separation, even though seconds and points are not comparable in raw units.

When to use it, and when not

Use Cohen's d when you compare the means of two groups on a continuous outcome, and both groups are roughly normal with similar spread. It pairs naturally with an independent-samples t-test. If you ran that test and want to report an effect size alongside it, d is the standard choice.

Do not use d when the outcome is a proportion or a count with very skewed distribution. For proportions, an odds ratio or risk difference fits better. Do not use d to compare more than two groups at once; for that, look at \eta^2 or a set of pairwise d values. And do not read a d in isolation from its confidence interval: a d of 0.8 from 6 people per group is far shakier than a d of 0.3 from 500 per group.

Cohen's d assumes the two groups share a comparable spread. If one group's SD is 2 and the other's is 8, the pooled SD averages them and the single d becomes hard to interpret. Report the group SDs so a reader can judge this.

The formula and the intuition

The core idea is a signal-to-noise ratio. The signal is the difference between means. The noise is the typical spread within a group.

d = \frac{\bar{x}_1 - \bar{x}_2}{s_p}

Here \bar{x}_1 and \bar{x}_2 are the two group means, and s_p is the pooled standard deviation. The pooled SD combines the two group SDs, weighting by their degrees of freedom:

s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}

In this formula n_1 and n_2 are the group sizes and s_1, s_2 are the group SDs. When both groups have the same SD, s_p equals that common SD, and d is just the raw difference divided by it.

If you only have a t statistic from an independent-samples test, you can recover d without the raw means:

d = t \sqrt{\frac{n_1 + n_2}{n_1 \cdot n_2}}

For equal groups this simplifies to d = 2t / \sqrt{N} where N = n_1 + n_2. With t = 2.5 and 30 per group, that gives d = 2 \times 2.5 / \sqrt{60} = 0.645.

Hedges' g and the small-sample correction

Cohen's d slightly overestimates the true effect in small samples, because the pooled SD is a biased estimate. Hedges' g fixes this by multiplying d by a correction factor:

g = d \cdot \left(1 - \frac{3}{4(n_1 + n_2) - 9}\right)

With 30 per group the correction factor is 1 - 3/(240 - 9) = 0.987, so g is about 1.3% smaller than d. The correction barely matters at that size. With 5 per group the factor drops to about 0.90, shaving 10% off. Use g whenever a group has fewer than about 20 members; with large samples d and g agree to two decimals.

Worked example with the demo data

Two groups of 30

Use the demo values: group A has mean 10, SD 2, n 30; group B has mean 12, SD 2, n 30.

Pooled SD: with equal SDs of 2, s_p = \sqrt{(29 \cdot 4 + 29 \cdot 4)/58} = \sqrt{4} = 2.
Cohen's d: d = (10 - 12)/2 = -1.0. The sign shows group B scored higher; the magnitude is 1.0.
Hedges' g: correction factor 1 - 3/231 = 0.987, so g = -1.0 \times 0.987 = -0.987.
Equivalent r: convert with r = d / \sqrt{d^2 + 4} for equal groups, giving r = 1.0/\sqrt{5} = 0.447.

So the two groups differ by one full standard deviation, a large effect. Report it as d = -1.0, g = -0.99, r = 0.45. The sign convention is your choice; state which group you subtracted from which.

Reading the number

The common rules of thumb come from Jacob Cohen:

Cohen's benchmarks and what they mean in overlap terms
d	Label	% of group B above group A mean
0.2	small	57.9%
0.5	medium	69.1%
0.8	large	78.8%
1.0	very large	84.1%

The last column translates d into a percentage anyone can picture. At d = 0.2 the distributions overlap heavily: a randomly chosen member of group B outscores the group A average only 57.9% of the time, barely better than a coin flip. At d = 0.8 that rises to 78.8%.

Treat "small, medium, large" as rough signposts, not verdicts. In some fields a d of 0.1 changes real outcomes at scale, while in others a d of 0.5 is unremarkable. Compare your d to typical effects in your own area.

As d grows, the share of group B scoring above the group A average climbs from just over half toward near-total separation.

Common mistakes

Watch for these errors, each of which changes the number.

Dividing by the wrong SD: Using one group's SD instead of the pooled SD inflates or deflates d. With equal SDs it makes no difference, but with unequal spread it can shift d by 20% or more.
Confusing t with d: A t of 2.5 is not a d of 2.5. The t statistic grows with sample size; d does not. At 30 per group that t maps to d = 0.645, not 2.5.
Ignoring the small-sample bias: Reporting d from 5 per group without switching to g overstates the effect by about 10%.
Reading magnitude without a confidence interval: A point estimate of 0.8 can have an interval running from 0.1 to 1.5. State the interval so readers know how precise the estimate is.

Related calculators

To turn d into a target sample size for a planned study, use the Power & Sample Size for t-Tests calculator. To move between d, r, \eta^2 and odds ratios, use the Effect Size Converter. For a confidence interval around a single mean or proportion, see the Confidence Interval calculator, and to size a study for a target margin of error, the Sample Size calculator.

Frequently asked questions

Is a negative Cohen's d bad?

No. The sign only shows direction. A d of -1.0 means the second group scored higher by one SD. Report which mean you subtracted so the sign is unambiguous.

What is the difference between Cohen's d and Hedges' g?

They estimate the same thing. Hedges' g applies a small correction that removes the upward bias of d in small samples. With 30 per group g is about 1.3% smaller; with 5 per group it is about 10% smaller.

How do I convert Cohen's d to a correlation r?

For two equal groups, r = d / \sqrt{d^2 + 4}. A d of 1.0 becomes r = 0.447; a d of 0.5 becomes r = 0.243.

Can I get d from just a t statistic?

Yes, if you also know both group sizes. Use d = t\sqrt{(n_1 + n_2)/(n_1 n_2)}. Without the group sizes the t value alone cannot give d.

What counts as a large effect?

By Cohen's benchmarks, 0.2 is small, 0.5 medium, 0.8 large. These are general guides; judge your d against effects that are typical in your own field.

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