## Sample size to test mean difference $\mu_d$ in dependent samples

Use this calculator to find the minimum sample size required to test mean difference $\mu_d$.

Sample Size to test means (paired) | ||
---|---|---|

Confidence Level ($1-\alpha$) | ||

Power ($1-\beta$) | ||

Mean of difference: ($\mu_d$) | ||

Standard Deviation : ($\sigma_d$) | ||

Results | ||

Effect Size ($ES$) | ||

Z value: $Z_{1-\alpha/2}$ | ||

Z value: $Z_{1-\beta}$ | ||

Required Sample Size : ($n$) | ||

## Sample size to test means of 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.