  The concept of median/consensus has been widely investigated in order to
provide a statistical summary of ranking data, i.e. realizations of a random
permutation $\Sigma$ of a finite set, $\{1,\; \ldots,\; n\}$ with $n\geq 1$
say. As it sheds light onto only one aspect of $\Sigma$'s distribution $P$, it
may neglect other informative features. It is the purpose of this paper to
define analogs of quantiles, ranks and statistical procedures based on such
quantities for the analysis of ranking data by means of a metric-based notion
of depth function on the symmetric group. Overcoming the absence of vector
space structure on $\mathfrak{S}_n$, the latter defines a center-outward
ordering of the permutations in the support of $P$ and extends the classic
metric-based formulation of consensus ranking (medians corresponding then to
the deepest permutations). The axiomatic properties that ranking depths should
ideally possess are listed, while computational and generalization issues are
studied at length. Beyond the theoretical analysis carried out, the relevance
of the novel concepts and methods introduced for a wide variety of statistical
tasks are also supported by numerous numerical experiments.
