Independent Group Standard Error of the Mean

The formula for calculating Standard Error of the Mean (SEM) is:

$\raisebox{1ex}{$\sigma $}\!\left/ \!\raisebox{-1ex}{$\sqrt{n}$}\right.$

This can be expressed mathematically in the following different ways, all of which are identical:

1. $\sqrt{\frac{{\sigma }^2}{n}}$

2. $\sigma {\left(\frac{1}{n}\right)}^{0.5}$

3. ${\left({\sigma }^2.\frac{1}{n}\right)}^{0.5}$

4. ${\left(\frac{{\sigma }^2}{n}\right)}^{0.5}$

In calculating the SEM of independent groups of samples, $n$ is split into two groups, $x$ and $y$. Where these groups have unequal variance, the variance $s^2$ is also split into 2 groups, $x$ and $y$.

The SEM for independent groups with unequal variance is:

${\left(\frac{S_x^2}{n_x}+\frac{S_y^2}{n_y}\right)}^{0.5}$

This corresponds to the SEM above displayed as (4)

The SEM for independent groups with equal variance uses a weighted, ‘pooled’ variance:

$S_p^2=\left(\left(n_x-1\right)S_x^2+\left(n_y-1\right)S_y^2\right)/\left(n_x+n_y-2\right)$

The square root of this is the pooled, estimated standard variance used in the calculation of the SEM:

$S_p{\left(\frac{1}{n_x}+\frac{1}{n_y}\right)}^{0.5}$

This corresponds to the SEM above displayed as (2)