The t-test is a common, reliable way to check for differences between two samples. When dealing with multivariate data, one can simply run t-tests on each variable and see if there are differences. This could lead to scenarios where individual t-tests suggest that there is no difference, although looking at all variables jointly will show a difference. When a multivariate test is preferred, the obvious choice is the Hotelling’s \(T^2\) test.
Hotelling’s test has the same overall flexibility that the t-test does, in that it can also work on paired data, or even a single dataset, though this example will only cover the two-sample case.