This article introduces a new formulation of, and method of computation for, theprojection median. Additionally, we explore its behaviour on a specific bivariate set up,providing the first theoretical result on form of the influence curve for the projectionmedian, accompanied by numerical simulations.Via new simulations we comprehensively compare our performance with anestablished method for computing the projection median, as well as other existingmultivariate medians. We focus on answering questions about accuracy andcomputational speed, whilst taking into account the underlying dimensionality. Suchconsiderations are vitally important in situations where the data set is large, or wherethe operations have to be repeated many times and some well-known techniques areextremely computationally expensive.We briefly describe our associated R package that includes our new methods andnovel functionality to produce animated multidimensional projection quantile plots, andalso exhibit its use on some high-dimensional data examples.