The outlying tendency of any case in a multiple regression ofppredictors may be estimated by drawing all subsets of sizepfrom the remaining cases and fitting the model. Each such subset yields anelemental residualfor the case in question, and a suitable summary statistic of them can be used as an estimate of the case's outlying tendency. We propose two such summary statistics: an unweighted median, which is of bounded influence, and a weighted median, which is more efficient but less robust. The computational load of the procedure is reduced by using random samples in place of the full set of subsets of sizep. As a byproduct the method yields useful information on the influence (or leverage) of cases and the mutual masking of high leverage points.