drawn from two independent multivariate normal distributions with the same mean and covariance, and we define as the sample means, and as the respective sample covariance matrices. Then is the unbiased pooled covariance matrix estimate (an extension of pooled variance). Finally, the Hotelling's two-sample t-squared statistic is Related concepts It can be related to the F-distribution by The non-null distribution of this statistic is the noncentral F-distribution (the ratio of a non-central Chi-squared random variable and an independent central Chi-squared random variable) with where is the difference vector between the population means. In the two-variable case, the formula simplifies nicely allowing appreciation of how the correlation, , between the variables affects . If we define and then Thus, if the differences in the two rows of the vector are of the same sign, in general, becomes smaller as becomes more positive. If the differences are of opposite sign becomes larger as becomes more positive. A univariate special case can be found in Welch's t-test. More robust and powerful tests than Hotelling's two-sample test have been proposed in the literature, see for example the interpoint distance based tests which can be applied also when the number of variables is comparable with, or even larger than, the number of subjects. See also Student's t-test in univariate statistics Student's t-distribution in univariate probability theory Multivariate Student distribution F-distribution (commonly tabulated or available in software libraries, and hence used for testing the T-squared statistic using the relationship given above) Wilks's lambda distribution (in multivariate statistics, Wilks's Λ is to Hotelling's T2 as Snedecor's F is to Student's t in univariate statistics) References External links Continuous distributions