Effect Size Calculator Cohen’s d: Formula & Use

This calculator works out Cohen’s d, the standardized mean difference between two groups, from the group means, standard deviations, and sample sizes you enter. It returns the effect size as a decimal, along with the variance and confidence interval, so you can judge whether a difference is small, medium, or large beyond what a p-value alone tells you.

Formula

Cohen’s d is defined as the difference between two group means divided by the pooled standard deviation. The pooled SD is a weighted average of the two group standard deviations, weighted by their degrees of freedom (n – 1).

d = (M� – M₂) / s_pooled

where s_pooled = √[ ((n� – 1) · SD�² + (n₂ – 1) · SD₂²) / (n� + n₂ – 2) ]

• Mâ‚�, Mâ‚‚ = means of group 1 and group 2 (same unit as the original measurement)
• SDâ‚�, SDâ‚‚ = standard deviations of each group (same unit)
• nâ‚�, nâ‚‚ = sample sizes of each group (counts)
• d = Cohen’s d (dimensionless, typically reported to 2 or 3 decimal places)

The formula assumes equal variances in the two populations (the same assumption as a standard two-sample t-test). If your data have very different variances, you may prefer a different effect size measure (see Alternatives below).

Worked Example

Using the default values: Group 1 mean = 100, SD = 15, n = 30; Group 2 mean = 110, SD = 15, n = 30.

1. Compute the numerator: M� – M₂ = 100 – 110 = –10
2. Compute the pooled variance: ((30–1)·15² + (30–1)·15²) / (30+30–2) = (29·225 + 29·225) / 58 = 13050 / 58 = 225.0
3. Take the square root: √225.0 = 15.0
4. Divide: d = –10 / 15.0 = –0.667

The calculator gives d = –0.667 (rounded to 3 decimals). The negative sign simply means group 1 has the lower mean; the magnitude 0.667 is what matters for interpretation. The variance of d is (30+30)/(30·30) + (–0.667)²/(2·60) = 60/900 + 0.444/120 = 0.0667 + 0.0037 = 0.0704, so the 95% confidence interval is –0.667 ± 1.96·√0.0704 ≈ –0.667 ± 0.520, giving (–1.187, –0.147).

When to Use Cohen’s d

Cohen’s d is used when you want to report the size of a difference between two groups, not just whether the difference is statistically significant. It is most appropriate when:

• You have two independent groups (e.g., treatment vs. control) and a continuous outcome.
• The data are approximately normally distributed and the variances are similar.
• You plan to compare your result with other studies or perform a meta-analysis.
• You need to calculate statistical power or sample size for a future study.

It is also commonly used in psychology, education, medicine, and social sciences to express the practical significance of an intervention. A p-value tells you if an effect exists, but d tells you how large it is in a way that is comparable across different scales.

How to Interpret the Value

Cohen (1988) suggested rough benchmarks for interpreting d:

• d ≈ 0.2 – small effect
• d ≈ 0.5 – medium effect
• d ≈ 0.8 – large effect

These are only guidelines. In some fields (e.g., medicine), even a small d can be clinically meaningful, while in others (e.g., physics), a large d might be expected. Always interpret d in the context of your research area and the cost/benefit of the intervention.

The sign of d indicates the direction: positive means group 1 has the higher mean, negative means group 2 does. When reporting, many authors report the absolute value and specify which group was higher.

Common Mistakes and Pitfalls

• Using the wrong pooled SD: Some formulas use the total sample size (nâ‚�+nâ‚‚) instead of the degrees of freedom (nâ‚�+n₂–2). This gives a slightly different value, especially for small samples.
• Confusing d with other effect sizes: Hedges’ g corrects for small-sample bias; Glass’s Δ uses only the control group’s SD. They are not interchangeable.
• Ignoring the direction: A negative d is fine, but if you expect a positive effect, check which group you labeled as 1.
• Forgetting the confidence interval: d is an estimate, and it has uncertainty. Reporting only the point estimate without a CI is incomplete.
• Using d for paired data: If your groups are paired (e.g., pre-test/post-test), the paired d is computed differently (using the standard deviation of the differences).

Limitations of Cohen’s d

Cohen’s d assumes equal variances and normal distributions. When these are violated, the value can be misleading. Also, d is a standardized difference, which means it is unitless; it does not tell you the actual mean difference in the original units, so always report the raw means and SDs alongside d. For very small sample sizes (n < 20), d tends to overestimate the true effect size; Hedges' g is a better choice.

Finally, d is only a measure of effect magnitude. It does not imply causality, and it does not tell you whether the effect is practically important in your specific context.

Alternatives to Cohen’s d

• Hedges’ g: Same as d but with a correction factor for small samples (g = d × (1 – 3/(4(nâ‚�+nâ‚‚) – 9))).
• Glass’s Δ: Uses only the control group’s SD; useful when the treatment changes the variance.
• Pearson’s r: For correlations or when you want an effect size that ranges from –1 to 1.
• Odds ratio / risk ratio: For binary outcomes.
• Eta-squared / partial eta-squared: For ANOVA designs with more than two groups.

Frequently Asked Questions

What is a good Cohen’s d value?

There is no universal “goodâ€� value. Cohen’s benchmarks (0.2, 0.5, 0.8) are rough guides. In many fields, a d of 0.5 is considered a moderate effect that is practically meaningful. The best approach is to compare your d with other studies in your field and consider the real-world implications.

Can Cohen’s d be negative?

Yes. The sign depends on which mean you subtract from which. If group 1 has a lower mean than group 2, d will be negative. The magnitude (absolute value) is what you interpret. When reporting, you can say “d = –0.67, indicating a medium-to-large effect with group 2 scoring higher.�

How is Cohen’s d different from a t-test?

A t-test gives a p-value, which tells you the probability of observing the data if there were no true difference. Cohen’s d gives the magnitude of the difference in standard deviation units. A study can have a very small p-value (e.g., 0.001) but a tiny d (e.g., 0.1), meaning the effect is statistically significant but practically negligible.

Do I need to enter standard deviations or standard errors?

You must enter the standard deviation (SD) of each group, not the standard error of the mean (SEM). If you only have SEM, convert it to SD using SD = SEM × √n, where n is the group size. Entering SEM will give a much larger d than correct.

Can I use Cohen’s d for paired data?

Not directly. For paired data (e.g., pre-test and post-test), the appropriate effect size is often the mean difference divided by the standard deviation of the differences, sometimes called Cohen’s d_z. This calculator is for independent groups only.

What is the confidence interval for d?

The calculator provides a 95% confidence interval based on the normal approximation of the sampling distribution of d. The variance formula used is an approximation that works well for moderate sample sizes. For very small samples, exact methods or bootstrapping may be more accurate.

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in m1 | Mean of group 1 | - | 100 | 0
in m2 | Mean of group 2 | - | 110 | 0
in sd1 | SD of group 1 | - | 15 | 0
in sd2 | SD of group 2 | - | 15 | 0
in n1 | Sample size of group 1 | - | 30 | 2
in n2 | Sample size of group 2 | - | 30 | 2
out d | Cohen's d | - | (m1 - m2) / sqrt(((n1-1)*sd1^2 + (n2-1)*sd2^2) / (n1 + n2 - 2)) | 3
out var_d | Variance of d | - | (n1+n2)/(n1*n2) + d^2/(2*(n1+n2)) | 4
out ci_low | 95% CI lower | - | d - 1.96*sqrt(var_d) | 3
out ci_high | 95% CI upper | - | d + 1.96*sqrt(var_d) | 3