## Sample size required to test means dependent samples

The $ES$ is defined as $$ES=\frac{|\mu_d|}{\sigma_d}$$

where

• $\mu_d$ is the mean of the difference,
• $\sigma_d$ is the standard deviation of the difference.

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

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

where

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