Sample size required to test means independent samples

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


  • $\mu_1$ 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

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


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

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