Pooled sample variance ($S_p^2=(S_{XX}+S_{YY})/(m+n-2)$)

For two independent normal samples of sizes $m,n$ with a common unknown variance, pooling their within-sample sums of squares gives the unbiased estimator $S_p^2=(S_{XX}+S_{YY})/(m+n-2)$. The numerator divided by the common population variance has a chi-squared distribution with $m+n-2$ degrees of freedom and is independent of both sample means. Therefore $(\bar{X}-\bar{Y}-({\mu }_X-{\mu }_Y))/(S_p\sqrt{1/m+1/n})$ has a Student t-distribution. This exact construction requires a positive residual number of degrees of freedom and the common-variance assumption.