## Sample size required to test means independent samples

The $ES$ is defined as $$ES=\frac{|\mu_1-\mu_2|}{\sigma}$$

where

• $\mu_1$ is the mean of the first group,
• $\mu_2$ is the mean of the second group,
• $\sigma$ is the standard deviation.

The formula for determining the sample size required in each group to ensure that the test has a specified power is

$$n =2\bigg(\frac{Z_{1-\alpha/2}+Z_{1-\beta}}{ES}\bigg)^2$$

where

• $\alpha$ is the selected level of significance,
• $1-\beta$ is the selected power and
• $ES$ is the effect size.