 Difference of sample means

For independent samples of sizes $n,m$ from populations with means ${\mu }_X,{\mu }_Y$ and finite variances ${\sigma }_X^2,{\sigma }_Y^2$, the difference of their sample means has expected value ${\mu }_X-{\mu }_Y$ and variance ${\sigma }_X^2/n+{\sigma }_Y^2/m$. This uses independence between and within the samples. With dependent samples, a covariance term is required instead.