A p-value tells you whether a difference between two groups is likely to be real; an effect size tells you how big it is. Cohen’s d expresses the difference between two means in units of standard deviation, so it can be compared across studies that use different scales – test points, kilograms, seconds or survey scores.
Jacob Cohen suggested rough benchmarks: d = 0.2 is small, 0.5 medium and 0.8 large. A d of 0.5 means the average person in the experimental group scores half a standard deviation above the average control person, which puts them at about the 69th percentile of the control group (Cohen’s U₃).
Hedges’ g corrects d’s slight upward bias in small samples, and Glass’s Δ uses only the control group’s standard deviation – useful when the treatment changes the spread. The calculator also converts d to a point-biserial correlation r, and to r², the share of variance explained by group membership, and gives the probability that a random member of the experimental group beats a random control (the common-language effect size).
How to use the Effect Size Calculator
- Control group mean (x̄₁)
- Control group standard deviation (s₁)
- Control group size (n₁)
- Experimental group mean (x̄₂)
- Experimental group standard deviation (s₂)
- Experimental group size (n₂)
- Pooled standard deviation
Results update as you type, and everything is calculated in your browser.
How it works
The pooled standard deviation combines both groups’ variances (weighted by their degrees of freedom, or a simple average when sample sizes are equal); the difference in means is divided by it. Overlap measures assume normal distributions with equal spread, and the confidence interval uses the large-sample standard error of d.
Worked example
A teacher compares a control class (mean 100, SD 10, n = 30) with a class taught with a new method (mean 105, SD 12, n = 30). The pooled SD is √[(29 × 100 + 29 × 144) ÷ 58] = √122 ≈ 11.045, so d = 5 ÷ 11.045 ≈ 0.453 – a small-to-medium effect. Hedges’ g = 0.453 × (1 − 3/231) ≈ 0.447, r = 0.453 ÷ √(0.205 + 4) ≈ 0.221, and r² ≈ 4.9% of the variance is explained by the teaching method. About 67.5% of the control class score below the new-method average (U₃), the distributions overlap by 82%, and the 95% confidence interval for d runs from −0.06 to 0.97.
Interpreting Cohen’s d
| d | Label | U₃ (percentile of control) | Overlap | r (equal n) |
|---|---|---|---|---|
| 0.2 | Small | 57.9% | 92.0% | 0.10 |
| 0.5 | Medium | 69.1% | 80.3% | 0.24 |
| 0.8 | Large | 78.8% | 68.9% | 0.37 |
| 1.2 | Very large | 88.5% | 54.9% | 0.51 |
| 2.0 | Huge | 97.7% | 31.7% | 0.71 |
Inputs explained
| Field | What to enter |
|---|---|
| Control group mean (x̄₁) | A number |
| Control group standard deviation (s₁) | A number |
| Control group size (n₁) | A number |
| Experimental group mean (x̄₂) | A number |
| Experimental group standard deviation (s₂) | A number |
| Experimental group size (n₂) | A number |
| Pooled standard deviation | Choose one: Weighted by sample size (standard), Simple average of variances √((s₁² + s₂²)/2) |
What each result means
- Cohen’s d = ([Experimental group mean (x̄₂)] – [Control group mean (x̄₁)])÷[Pooled standard deviation]
- Hedges’ g (small-sample corrected) = [Cohen’s d]×(1 – 3÷(4×([Control group size (n₁)] + [Experimental group size (n₂)]) – 9))
- Glass’s Δ (control SD only) = ([Experimental group mean (x̄₂)] – [Control group mean (x̄₁)])÷[Control group standard deviation (s₁)]
- Correlation coefficient r = [Cohen’s d]÷sqrt([Cohen’s d]^2 + ([Control group size (n₁)] + [Experimental group size (n₂)])^2÷([Control group size (n₁)]×[Experimental group size (n₂)]))
- Variance explained (r²) = ([Correlation coefficient r]^2)×100
- Cohen’s U₃ – control members below the experimental mean = (ncdf([Cohen’s d]))×100
- Overlap of the two distributions = (2×ncdf(-abs([Cohen’s d])÷2))×100
- Probability a random experimental member beats a random control = (ncdf([Cohen’s d]÷sqrt(2)))×100
- Standard error of d = sqrt(([Control group size (n₁)] + [Experimental group size (n₂)])÷([Control group size (n₁)]×[Experimental group size (n₂)]) + [Cohen’s d]^2÷(2×([Control group size (n₁)] + [Experimental group size (n₂)])))
- 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]
Formula
d = (x̄₂ − x̄₁) ÷ s_pooled · s_pooled = √[((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2)] · r = d ÷ √(d² + (n₁ + n₂)²/(n₁n₂)) · g = d × (1 − 3/(4(n₁ + n₂) − 9))
Worked example
Inputs — Control group mean (x̄₁): 100; Control group standard deviation (s₁): 10; Control group size (n₁): 30; Experimental group mean (x̄₂): 105; Experimental group standard deviation (s₂): 12; Experimental group size (n₂): 30; Pooled standard deviation: Weighted by sample size (standard).
Results — Pooled standard deviation = 11.0454; Cohen's d = 0.4527; Size of the effect = Small (0.2 ≤ |d| < 0.5); Hedges' g (small-sample corrected) = 0.4468; Glass's Δ (control SD only) = 0.5; Correlation coefficient r = 0.2208; Variance explained (r²) = 4.87 %; Cohen's U₃ – control members below the experimental mean = 67.46 %; Overlap of the two distributions = 82.09 %; Probability a random experimental member beats a random control = 62.56 %; Standard error of d = 0.2615; 95% confidence interval for d – lower = -0.0598; 95% confidence interval for d – upper = 0.9652.
Key terms
- Effect size
- A measure of the magnitude of a difference or relationship, independent of sample size.
- Pooled standard deviation
- A combined estimate of the common spread of two groups.
- Cohen's d
- Mean difference divided by the pooled standard deviation.
- Hedges' g
- Cohen's d with a small-sample bias correction.
- Point-biserial correlation (r)
- The correlation between group membership (0/1) and the outcome.
Frequently asked questions
What is a good effect size?
By Cohen's rule of thumb d = 0.2 is small, 0.5 medium and 0.8 large, but what matters depends on the field – in education a d of 0.4 is already a strong intervention.
How do I calculate Cohen's d?
Subtract the control mean from the experimental mean and divide by the pooled standard deviation. Means 100 and 105 with SDs 10 and 12 give s_pooled = √122 ≈ 11.05 and d ≈ 0.45.
Can Cohen's d be negative?
Yes. The sign only shows direction: here a negative d means the experimental group scored lower than the control group. The size is |d|.
What is the difference between Cohen's d and Hedges' g?
g multiplies d by 1 − 3/(4(n₁ + n₂) − 9) to remove a small upward bias; with 30 per group g ≈ 0.987 × d. Prefer g when groups are small.
When should I use Glass's Δ?
When the treatment is likely to change the variability, divide by the control group's standard deviation only.
How do I convert d to r?
r = d ÷ √(d² + 4) for equal group sizes. d = 0.5 corresponds to r ≈ 0.24, and d = 0.8 to r ≈ 0.37.
What does an overlap of 82% mean?
The two normal curves share 82% of their area – most people in the two groups are indistinguishable even though the means differ.
Why report a confidence interval for d?
With small groups d is imprecise. An interval that runs from about −0.06 to 0.97, as in the default example, is consistent with no effect or a large one.