A novel definition of the multivariate coefficient of variation

The coefficient of variation CV (%) is widely used to measure the relative variation of a random variable to its mean or to assess and compare the performance of analytical techniques/equipments. A review is made of the existing multivariate extensions of the univariate CV where, instead of a random variable, a random vector is considered, and a novel definition is proposed. The multivariate CV obtained only requires the calculation of the mean vector, the covariance matrix and simple quadratic forms. No matrix inversion is needed which makes the new approach equally attractive in high dimensional as in very small sample size problems. As an illustration, the method is applied to electrophoresis data from external quality assessment in laboratory medicine, to phenotypic characteristics of pocket gophers and to a microarray data set.