## Sample size to test means $\mu_1-\mu_2$

Use this calculator to find the minimum sample size required to test mean $\mu_1-\mu_2$.

Sample Size to test means | ||
---|---|---|

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

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

First group mean : ($\mu_1$) | ||

Second group mean : ($\mu_2$) | ||

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

Results | ||

Effect Size ($ES$) | ||

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

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

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

## Sample size to test means $\mu_1-\mu_2$

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.