For a distribution F on \(R^ p\) and a point x in \(R^ p\) the simplicial depth D(x), which is the probability that x be inside a random simplex whose vertices are \(p+1\) independent observations from F, is introduced. D(x) can be viewed as a measure of depth of the point x relative to F, and its empirical version gives rise to a natural ordering of the data points from the center outward.
This ordering provides an approach for detecting outliers in a multivariate data cloud and leads to the introduction of affine equivariant multivariate generalizations of the univariate sample median and L-statistics. This sample median is shown to be consistent for the center of any angularly symmetric distribution.