Mahalanobis distance

The Mahalanobis distance is a measure of the distance between a point ${\displaystyle P}$ and a distribution ${\displaystyle D}$, introduced by P. C. Mahalanobis in 1936. Mahalanobis's definition was prompted by the problem of identifying the similarities of skulls based on measurements in 1927.

It is a multivariate generalization of the square of the standard score ${\displaystyle z=(x-\mu )/\sigma }$: how many standard deviations away ${\displaystyle P}$ is from the mean of ${\displaystyle D}$. This distance is zero for ${\displaystyle P}$ at the mean of ${\displaystyle D}$ and grows as ${\displaystyle P}$ moves away from the mean along each principal component axis. If each of these axes is re-scaled to have unit variance, then the Mahalanobis distance corresponds to standard Euclidean distance in the transformed space. The Mahalanobis distance is thus unitless, scale-invariant, and takes into account the correlations of the data set.