# Effect Size Calculator (Cohen’s d)

URL: https://speedycalculator.com/calculators/statistics/effect-size/
Last updated: 2026-10-30

> The effect size calculator turns the means and standard deviations of a control and an experimental group into Cohen's d, Hedges' g and Glass's Δ, converts d to a correlation r and r², and reports overlap measures and a 95% confidence interval for d.

## Inputs

- 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

- Pooled standard deviation
- Cohen's d
- Size of the effect
- Hedges' g (small-sample corrected)
- Glass's Δ (control SD only)
- Correlation coefficient r
- Variance explained (r²)
- Cohen's U₃ – control members below the experimental mean
- Overlap of the two distributions
- Probability a random experimental member beats a random control
- Standard error of d
- 95% confidence interval for d – lower
- 95% confidence interval for d – upper

## 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))
```

## Explanation

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

dLabelU₃ (percentile of control)Overlapr (equal n)0.2Small57.9%92.0%0.100.5Medium69.1%80.3%0.240.8Large78.8%68.9%0.371.2Very large88.5%54.9%0.512.0Huge97.7%31.7%0.71

### Inputs explained

FieldWhat to enterControl group mean (x̄₁)A numberControl group standard deviation (s₁)A numberControl group size (n₁)A numberExperimental group mean (x̄₂)A numberExperimental group standard deviation (s₂)A numberExperimental group size (n₂)A numberPooled standard deviationChoose 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]

## 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.

## FAQ

**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.


## Sources

- [Penn State STAT 500 – Applied Statistics](https://online.stat.psu.edu/stat500/)
- [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e)
- [Cohen J. – A power primer, Psychological Bulletin 1992](https://doi.org/10.1037/0033-2909.112.1.155)