  In data mining, it is usually to describe a set of individuals using some
summaries (means, standard deviations, histograms, confidence intervals) that
generalize individual descriptions into a typology description. In this case,
data can be described by several values. In this paper, we propose an approach
for computing basic statics for such data, and, in particular, for data
described by numerical multi-valued variables (interval, histograms, discrete
multi-valued descriptions). We propose to treat all numerical multi-valued
variables as distributional data, i.e. as individuals described by
distributions. To obtain new basic statistics for measuring the variability and
the association between such variables, we extend the classic measure of
inertia, calculated with the Euclidean distance, using the squared Wasserstein
distance defined between probability measures. The distance is a generalization
of the Wasserstein distance, that is a distance between quantile functions of
two distributions. Some properties of such a distance are shown. Among them, we
prove the Huygens theorem of decomposition of the inertia. We show the use of
the Wasserstein distance and of the basic statistics presenting a k-means like
clustering algorithm, for the clustering of a set of data described by modal
numerical variables (distributional variables), on a real data set. Keywords:
Wasserstein distance, inertia, dependence, distributional data, modal
variables.
