A More Sophisticated Definition of the Sample Median

Abstract

As the conventional definition of sample medians is not apt for special investigations belonging to the behaviours of the L₁ norm in case of small samples, the authors propose a more sophisticated way to determine the sample median. If the latter values are denoted by MED and Q_a is the asymptotic semi-intersextile range (calculated as 0.9674 times the asymptotic scatter), the authors have found the curious result in case of Cauchy-type parent distribution that $ Q_{⤪ MED}=Q_{a}/sqrt{n} $ holds not only asymptotically but also for very small samples, too.