The goal of this paper is threefold. It first describes a novel way of measuring disagreement between rankings of a finite set X of n ≥ 1 elements, that can be viewed as a (mass transportation) Kantorovich metric, once the collection rankings of X is embedded in the set Kn of n×n doubly-stochastic matrices. It also shows that such an embedding makes it possible to define a natural notion of median, that can be interpreted in a probabilistic fashion. In addition, from a computational perspective, the convexification induced by this approach makes median computation more tractable, in contrast to the standard metric-based method that generally yields NP-hard optimization problems. As an illustration, this novel methodology is applied to the issue of ranking aggregation, and is shown to compete with state of the art techniques.