# Relation between Two Sample Hotelling's T-test and Mahalanobis Distance?

Mahalanobis distance is a measure of distance between a point and distribution. So if we want to check if a point belongs to a particular distribution or not, we can use Hotelling's T-test, which is squared Mahalanobis distance. But if we have two sample distribution and we want to check if they belong to the same group or not, we can use two sample Hotelling's T-test, that is,

$T^2$ = $n(X-Y)^T$$(X-Y)$ /$S$

(assuming same number of samples), where $S$=$Sx$+$Sy$ is the pooled covariance. Now my question is this, Is there any relation between Two Sample Hotelling's T-test and Mahalanobis Distance (similar to 1 sample H T-test and MD)? How are they mathematically related?

My thinking is that, there is some relation (equality) between them but I can't get my head around it. Any guidance would be greatly appreciated. Thanks

• Speaking of Mahalanobis distance we usually mean a distance between a point and a clouds's centroid, or between two points in the same cloud. So its is a pointwise distance in any case, like Euclidean one. But you want to extend its definition to a set distance, a distance between two clouds. It is surely possible, but I think it will transgress the definition of "Mahalanobis distance", so why need to do it? There are other well-known kinds of distance between sets/distributions. – ttnphns Nov 1 '16 at 12:33
• As I see a clear relation between Mahalanobis and 1 sample Hotelling's T-test, I assume that two sample Hotelling's T-test will be just an extension. What do you mean by other kinds of distance sets? – tehseen fatima Nov 2 '16 at 2:08
• Does the answer here help? – Carl Mar 2 '17 at 18:32