Cohen’s d tells you how big a difference between two groups is, in units of standard deviation. A p-value only says whether a difference is likely to be real; d says whether it matters. A d of 0.5 means the group means are half a standard deviation apart, whatever the original units were – test points, kilograms or milliseconds – so effects from different studies can be compared directly and combined in a meta-analysis.
The calculator divides the difference between the two means by the pooled standard deviation, the weighted average spread of both groups. Because d is slightly too large in small samples, it also reports Hedges’ g, which applies a small-sample correction, and Glass’s Δ, which uses only the second (control) group’s standard deviation – useful when a treatment changes the spread as well as the mean.
To make the number easier to explain, you also get the probability of superiority (the chance that a random member of group 1 scores higher than a random member of group 2), Cohen’s U3 and the percentage overlap of two normal curves that are d standard deviations apart.
How to use the Cohen’s d Calculator
- Enter
- Group 1 data (treatment)
- Group 2 data (control)
- Group 1 mean (x̄₁)
- Group 1 standard deviation (s₁)
- Group 1 size (n₁)
- Group 2 mean (x̄₂)
- Group 2 standard deviation (s₂)
- Group 2 size (n₂)
Results update as you type, and everything is calculated in your browser.
How it works
Enter both groups as lists, or switch to summary statistics and type each mean, standard deviation and sample size. The calculator pools the two variances weighted by their degrees of freedom, divides the mean difference by the pooled standard deviation and then derives the corrected and converted effect sizes. The confidence interval uses the large-sample standard error √((n₁ + n₂) ÷ (n₁n₂) + d² ÷ (2(n₁ + n₂))).
Worked example
A reading programme group scored 23, 25, 28, 30, 32 and 27 (mean 27.5, s = 3.2711) and a control group scored 20, 22, 25, 24, 26 and 21 (mean 23, s = 2.3664). The pooled standard deviation is √((5 × 10.7 + 5 × 5.6) ÷ 10) = √8.15 = 2.8548, so d = (27.5 − 23) ÷ 2.8548 = 1.576 – a very large effect. Hedges’ g = 1.576 × (1 − 3 ÷ 39) = 1.455, and the 95% confidence interval for d runs from about 0.28 to 2.87, which is wide because each group has only six people. The probability that a random programme pupil beats a random control pupil is Φ(1.576 ÷ √2) ≈ 86.7%.
How to interpret Cohen’s d
| |d| | Label | Overlap of the groups | Probability of superiority |
|---|---|---|---|
| 0.2 | Small | 92% | 56% |
| 0.5 | Medium | 80% | 64% |
| 0.8 | Large | 69% | 71% |
| 1.2 | Very large | 55% | 80% |
| 2.0 | Huge | 32% | 92% |
Inputs explained
| Field | What to enter |
|---|---|
| Enter | Choose one: Raw data for both groups, Mean, SD and size of each group |
| Group 1 data (treatment) | A list of numbers separated by commas, spaces or new lines |
| Group 2 data (control) | A list of numbers separated by commas, spaces or new lines |
| Group 1 mean (x̄₁) | A number |
| Group 1 standard deviation (s₁) | A number |
| Group 1 size (n₁) | A number |
| Group 2 mean (x̄₂) | A number |
| Group 2 standard deviation (s₂) | A number |
| Group 2 size (n₂) | A number |
What each result means
- Pooled standard deviation = sqrt((([Group 1 size] – 1)×[Group 1 standard deviation]^2 + ([Group 2 size] – 1)×[Group 2 standard deviation]^2)÷([Group 1 size] + [Group 2 size] – 2))
- Cohen’s d = ([Group 1 mean] – [Group 2 mean])÷[Pooled standard deviation]
- Hedges’ g (small-sample corrected) = [Cohen’s d]×(1 – 3÷(4×([Group 1 size] + [Group 2 size]) – 9))
- Glass’s Δ (uses group 2 SD) = ([Group 1 mean] – [Group 2 mean])÷[Group 2 standard deviation]
- d using the plain average SD √((s₁² + s₂²)/2) = ([Group 1 mean] – [Group 2 mean])÷sqrt(([Group 1 standard deviation]^2 + [Group 2 standard deviation]^2)÷2)
- Standard error of d = sqrt(([Group 1 size] + [Group 2 size])÷([Group 1 size]×[Group 2 size]) + [Cohen’s d]^2÷(2×([Group 1 size] + [Group 2 size])))
- 95% confidence interval for d – lower = [Cohen’s d] – 1.959964×[Standard error of d]
- 95% confidence interval for d – upper = [Cohen’s d] + 1.959964×[Standard error of d]
- Equivalent correlation r = [Cohen’s d]÷sqrt([Cohen’s d]^2 + ([Group 1 size] + [Group 2 size])^2÷([Group 1 size]×[Group 2 size]))
- Probability of superiority (group 1 > group 2) = (ncdf([Cohen’s d]÷sqrt(2)))×100
- Cohen’s U3 (group 1 above group 2 mean) = (ncdf([Cohen’s d]))×100
- Overlap of the two distributions = (2×ncdf(-abs([Cohen’s d])÷2))×100
Formula
d = (x̄₁ − x̄₂) ÷ s_p · s_p = √[((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2)] · Hedges g = d × (1 − 3 ÷ (4(n₁ + n₂) − 9)) · Glass Δ = (x̄₁ − x̄₂) ÷ s₂
Worked example
Inputs — Enter: Raw data for both groups; Group 1 data (treatment): 23, 25, 28, 30, 32, 27; Group 2 data (control): 20, 22, 25, 24, 26, 21.
Results — Group 1 mean = 27.5; Group 2 mean = 23; Group 1 standard deviation = 3.2711; Group 2 standard deviation = 2.3664; Group 1 size = 6; Group 2 size = 6; Pooled standard deviation = 2.8548; Cohen's d = 1.5763; Effect size = Very large (≥ 1.2); Hedges' g (small-sample corrected) = 1.455; Glass's Δ (uses group 2 SD) = 1.9016; d using the plain average SD √((s₁² + s₂²)/2) = 1.5763; Standard error of d = 0.661; 95% confidence interval for d – lower = 0.2808; 95% confidence interval for d – upper = 2.8717; Equivalent correlation r = 0.619; Probability of superiority (group 1 > group 2) = 86.7 %; Cohen's U3 (group 1 above group 2 mean) = 94.3 %; Overlap of the two distributions = 43.1 %.
Key terms
- Effect size
- A number describing how large a difference or relationship is, independent of sample size.
- Pooled standard deviation
- The combined spread of two groups, each variance weighted by its degrees of freedom.
- Hedges' g
- Cohen's d multiplied by a correction factor that removes its small-sample bias.
- Probability of superiority
- The chance that a randomly chosen member of group 1 scores higher than a randomly chosen member of group 2.
Frequently asked questions
What is a good Cohen's d?
Cohen's rough benchmarks are 0.2 small, 0.5 medium and 0.8 large. In education a d of 0.4 is already a strong intervention effect, while in lab experiments 0.8 or more is common – compare with your own field.
Can Cohen's d be negative?
Yes. The sign only shows which group has the higher mean. d = −0.6 means group 2 is 0.6 standard deviations above group 1; the size of the effect is the absolute value.
What does d = 0 mean?
The two means are identical, so there is no standardized difference at all, whatever the spread of the groups.
What is the difference between Cohen's d and Hedges' g?
g multiplies d by 1 − 3 ÷ (4(n₁ + n₂) − 9). With 6 + 6 participants the factor is 0.923; with 50 + 50 it is 0.992, so the two are almost equal in large samples.
When should I use Glass's Δ?
When the treatment may change the spread, use the control group's SD alone. If the treatment group's SD is inflated, pooling would hide part of the effect.
How do I convert Cohen's d to a correlation r?
For equal group sizes r = d ÷ √(d² + 4), so d = 0.8 corresponds to r ≈ 0.37. The reverse is d = 2r ÷ √(1 − r²), which is the same as d² = 4r² ÷ (1 − r²).
How many people do I need to detect d = 0.5?
About 64 per group for 80% power with a two-sided test at α = 0.05; for d = 0.2 it is about 394 per group.
Is Cohen's d the same for paired data?
No. For before–after measurements divide the mean difference by the SD of the differences (d_z) or by the average SD of the two time points (d_av) – they answer slightly different questions.