Independent Samples Z-Test - Comparing Two Means with Unequal Variances

The test statistic is:

$z=\frac{\overline{x}-\overline{y}}{\sqrt{\frac{{\sigma }_x^2}{m}+\frac{{\sigma }_y^2}{n}}}$

where:

$\overline{x}$ and $\overline{x}$ are the average values of variables $x$ and $y$ respectively, where each of these variables represents the data from two independent groups,

the groups have sample sizes of $m$ and $n$ respectively,

${\sigma }_x^2$ and ${\sigma }_y^2$ are the variances in the two groups,

$p=2(1-\mathrm{\Phi }(|z|))$