How to Measure Multidimensional Variation?

The coefficient of variation, which measures the dispersion of a distribution around its mean, is not uniquely defined in the multidimensional case, and so is the Gini index, which measures the dispersion of a distribution in terms of the mean differences among its observations. In this paper, we bridge these two notions of dispersion by proposing a multidimensional coefficient of variation which generalizes the Gini index. We show that our proposed multivariate coefficient of variation coincides with Voinov-Nikulin’s coefficient of variation up to a multiplicative factor. We then introduce and study a set of four properties that a multivariate coefficient of variation should have and show that our proposal possesses all of them. Our proposal is practically compared with the existing multivariate coefficients of variation on both real and simulated data.