Statistics
Cohen's d effect size calculator
Enter the mean, standard deviation and size of each group to get Cohen's d, the standardised difference between them. The calculator also reports Hedges' g, Glass's Δ and an approximate 95% interval, and shows the working.
Show the working, step by step
Pool the two standard deviations, weighting each group's variance by its degrees of freedom.
sp = √(((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2)) sp = √((24 × 9.2² + 26 × 10.1²) ÷ 50) = 9.67845
Divide the difference between the means by it.
d = (x̄₁ − x̄₂) ÷ sp = (78.4 − 72.1) ÷ 9.67845 = 0.6509
d is slightly biased upwards in small samples. Hedges' g multiplies it by a correction factor.
J = 1 − 3 ÷ (4 × 50 − 1) = 0.98492 g = 0.6509 × 0.98492 = 0.6411
An approximate 95% interval uses the large-sample standard error of d.
SE(d) = √((n₁ + n₂) ÷ (n₁n₂) + d² ÷ (2(n₁ + n₂))) = 0.2848 d ± 1.96 × SE = 0.09273 to 1.209
Cohen's benchmarks — 0.2 small, 0.5 medium, 0.8 large — were offered as a last resort when nothing better is known. An effect's practical importance depends on what was measured. The sign only says which group had the higher mean.
What Cohen's d measures
A p-value tells you whether a difference between two groups is likely to be real. It does not tell you whether the difference is big. With a large enough sample, a trivial gap becomes significant. Cohen's d answers the other question: how large the difference is compared with the ordinary spread of the individuals.
A d of 0.65 means the group means sit 0.65 standard deviations apart. Because the units cancel, the same scale works for exam marks, reaction times, blood pressure or conversion rates. That is why effect sizes are what meta-analyses combine and what the APA asks authors to report next to their tests.
The formula
d = (x̄₁ − x̄₂) / sp sp = √( ((n₁ − 1)s₁² + (n₂ − 1)s₂²) / (n₁ + n₂ − 2) )
The denominator is the pooled standard deviation: each group's variance weighted by its degrees of freedom, then square-rooted. Averaging the two SDs directly gives a slightly different, and wrong, number.
Two related measures are reported as well:
- Hedges' g = d × (1 − 3 / (4·df − 1)). It corrects d's small upward bias in small samples (Hedges & Olkin, 1985).
- Glass's Δ = (x̄₁ − x̄₂) / s₂. It uses only the second group's SD. That group is usually the control, and this is the better choice when a treatment changes the spread as well as the mean.
A worked example
A class taught with a new method (group 1) scores a mean of 78.4 with SD 9.2 across 25 students. A comparison class (group 2) scores 72.1 with SD 10.1 across 27 students. These are the numbers loaded in the calculator above.
- Pooled SD: sp = √((24 × 9.2² + 26 × 10.1²) ÷ 50) = 9.67845.
- Mean difference: 78.4 − 72.1 = 6.3.
- d = 6.3 ÷ 9.67845 = 0.6509, a medium effect by Cohen's benchmarks.
- Correction: J = 1 − 3 ÷ 199 = 0.98492, so Hedges' g = 0.6411.
- SE(d) = 0.2848, which gives an approximate 95% interval of 0.093 to 1.209.
The interval is the part that is easy to overlook. With about 25 students per group, the data are consistent with an effect anywhere from negligible to large. The point estimate of 0.65 is the best single guess, but it is far from pinned down.
Reading the size of d
| |d| | Cohen's label | What it looks like |
|---|---|---|
| 0.2 | Small | Real but hard to see by eye; the two distributions overlap almost entirely. |
| 0.5 | Medium | Visible to a careful observer. |
| 0.8 | Large | Obvious; the average member of one group beats most of the other. |
Treat these as a starting vocabulary rather than a verdict. Where your field has typical effects on record, compare against those instead.
Getting the inputs from raw data
If you have each group's individual scores rather than their summaries, paste each group into the mean and standard deviation calculator using sample mode. That gives the x̄, s and n you need here. The t-test calculator takes both groups' raw values at once and reports d with the test.
When d is the wrong measure
- Paired or repeated measures. Before-and-after scores on the same people need a paired effect size, based on the SD of the differences or corrected for the correlation. This formula treats the groups as independent.
- Very unequal spreads. If one group's SD is several times the other's, pooling hides that. Report Glass's Δ, or both SDs.
- Heavily skewed data or outliers. Means and SDs are both pulled by extreme values. Check the data's shape first.
- Proportions. For yes/no outcomes, a risk difference or odds ratio is usually clearer.
Related calculators
-
T-test calculator
Test whether the difference is significant, with d reported alongside.
-
Pooled standard deviation
The denominator of d, for two or more groups.
-
Mean and standard deviation
Get each group’s mean and SD from its raw values first.
-
Confidence interval
An interval for a single group’s mean.
Common questions
What is Cohen's d?
Cohen's d is the difference between two group means divided by their pooled standard
deviation: d = (x̄₁ − x̄₂) / sp. It says how far apart the groups are in units of
standard deviation, so it can be compared across studies that measured different things on
different scales.
What counts as a small, medium or large Cohen's d?
Cohen (1988) suggested 0.2 as small, 0.5 as medium and 0.8 as large. He offered them as a fallback for when a field has no better reference point, not as fixed thresholds. A d of 0.2 can matter a great deal for a cheap intervention given to millions of people, and a d of 0.8 can be unremarkable in a lab task where effects are routinely larger.
What is the difference between Cohen's d and Hedges' g?
Hedges' g is Cohen's d multiplied by a small-sample correction,
J = 1 − 3 / (4·df − 1), where df = n₁ + n₂ − 2. Cohen's d slightly overstates the
true effect in small samples, and g removes most of that bias. With 20 or more per group
the two differ by only a few percent; meta-analyses usually report g.
Can Cohen's d be negative?
Yes. The sign only records which group had the higher mean. If group 2 scored higher, d is negative. Swap the groups and the sign flips while the size stays the same, so report the direction in words alongside the number.
How do I get Cohen's d from a t statistic?
For two independent groups, d = t × √(1/n₁ + 1/n₂). This gives the same
answer as the formula above when the t-test used the pooled variance (Student's t). If you
have the raw data, the t-test calculator reports d
alongside the test.