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

The test statistic is:

$z={\frac {{\bar {x}}-{\bar {y}}}{\sqrt {{\frac {\sigma _{x}^{2}}{m}}+{\frac {\sigma _{y}^{2}}{n}}}}}$

where:

${\bar {x}}$ and ${\bar {x}}$ 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,

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