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Cohen's d for independent two-sample design

For studying the standardized group mean difference in an independent two-sample design, the most popular effect size measure is defined as

$$ \delta = \frac{\mu_1 - \mu_2}{\sigma} $$

where $\mu_1$ and $\mu_2$ are the population means of the two groups and $\sigma$ is the common standard deviation for the two populations.

Cohen's d

One estimator of the population effect size is Cohen's d. It is calculated as

$$ d = \frac{\bar{y}_1 - \bar{y}_2}{\sqrt{\frac{(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}} $$

where $n_1$ and $n_2$ are sample sizes, $\bar{y}_1$ and $\bar{y}_1$ are sample means, and $s_1^2$ and $s_2^2$ are sample variances under the two different conditions, respectively.

Confidence Intervals

Algina and Keselman (2003) constructed a confidence interval for the population effect size based on a non-central t-distribution. In the method, one first gets the lower and upper bounds of the non-centrality parameter as $\lambda_L$ and $\lambda_U$ by solving




where $pt$ is the cdf of the t-distribution.

Now the confidence interval is given by

$$ \left[\lambda_L \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}, \lambda_U \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}\right] $$


  • Normally distributed data
  • Equal group variances


Cohen's d is a biased estimator of the population effect size for standardized group mean difference. An alternative measure is Hedges' g.


Algina, J., & Keselman, H. J. (2003). Approximate confidence intervals for effect sizes. Educational and Psychological Measurement, 63, 537-553.

effectsize/cohen_s_d.txt · Last modified: 2021/08/08 16:17 by johnny zhang