## 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.