Cohen’s d Calculator (Effect Size)
This Cohen’s d calculator computes standardized effect sizes for the difference between means. It supports independent samples, one-sample, and paired / repeated-measures designs, with pooled standard deviation, Hedges’ g bias correction, and interpretation (small, medium, large).
Calculate Cohen’s d
All calculations assume numeric summary statistics (means, standard deviations, sample sizes).
Group 1
Group 2
You can use either dot or comma as decimal separator. All sizes must be positive.
Cohen’s d effect size – definition and formula
Cohen’s d is a standardized measure of the difference between two means. It tells you how many standard deviations apart the groups are, which makes the effect size comparable across different scales, tests and outcomes.
Two independent groups (classic Cohen’s d)
For two independent groups with means \( M_1 \) and \( M_2 \), standard deviations \( s_1 \) and \( s_2 \), and sample sizes \( n_1 \) and \( n_2 \), the most common definition is:
\( d = \dfrac{M_1 - M_2}{s_{\text{pooled}}} \)
\( s_{\text{pooled}} = \sqrt{\dfrac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \)
The pooled standard deviation \( s_{\text{pooled}} \) combines the variability of the two groups and assumes equal population variances. The direction of the effect depends on whether you compute \( M_1 - M_2 \) or \( M_2 - M_1 \); the calculator lets you choose this explicitly.
One-sample and paired designs
For a one-sample design, you compare the sample mean \( M \) to a known or hypothesized mean \( \mu_0 \):
\( d = \dfrac{M - \mu_0}{s} \)
For paired / repeated-measures designs, Cohen’s d is usually defined in terms of the difference scores (for example, post − pre):
\( d = \dfrac{\overline{D}}{s_D} \)
- \( \overline{D} \) = mean of the difference scores
- \( s_D \) = standard deviation of the difference scores
Hedges’ g (bias-corrected d)
In small samples, Cohen’s d slightly overestimates the population effect size. A common correction is Hedges’ g, which multiplies d by a factor \( J \) that depends on the degrees of freedom:
\( g = J \cdot d \)
\( J = 1 - \dfrac{3}{4\text{df} - 1} \)
For large df, \( J \) is very close to 1, so \( g \approx d \). For small samples, the correction can noticeably reduce the estimate.
Interpreting Cohen’s d (small, medium, large)
Cohen suggested conventional benchmarks:
- d ≈ 0.2 – small effect
- d ≈ 0.5 – medium effect
- d ≈ 0.8 – large effect
These are rules of thumb, not universal standards. In some fields, even d = 0.2 may be practically or clinically important; in others, an effect of d = 0.5 might be considered modest. Always interpret effect sizes in the context of:
- the outcome scale and units,
- costs and benefits of the intervention,
- baseline risk or prevalence,
- typical effect sizes in your research area.
Cohen’s d vs other effect sizes
- Standardized mean difference (SMD): Cohen’s d and Hedges’ g are both SMDs. Meta-analyses often prefer Hedges’ g because of the small-sample correction.
- Correlation coefficient (r): For bivariate relationships, r (or r²) is often easier to interpret. Roughly, d ≈ 0.2 corresponds to r ≈ 0.1, d ≈ 0.5 to r ≈ 0.24, and d ≈ 0.8 to r ≈ 0.37.
- Odds ratios, risk ratios: For binary outcomes, effect sizes are usually expressed as odds ratios or risk ratios. There are approximate transformations between d and these measures, but they rely on additional assumptions.