Use of the median in Physics and Astronomy

TL;DR

This paper explores the properties, advantages, and limitations of the median in physics and astronomy data analysis, comparing it with the mean and deriving their joint distribution for better estimation methods.

Contribution

It provides a detailed analysis of the median's properties, its relation to the mean, and introduces a derivation of their joint asymptotic distribution for continuous probability laws.

Findings

Median has advantages over mean in polluted data scenarios.

Sample median can be characterized similarly to the mean using quantile methods.

Derived the asymptotic joint distribution of mean and median.

Abstract

The sample median is often used in statistical analyses of physical or astronomical data wherein a central value must be found from samples polluted by elements which do not belong to the population of interest or when the underlying probability law is such that the sample mean is useless for the stated purpose. Although it does not generally possesses the nice linearity properties of the mean, the median has advantages of its own, some of which are explored in this paper which elucidates analogies and differences between these two central value descriptors. Some elementary results are shown, most of which are certainly not new but not widely known either. It is observed that the moment and the quantile approaches to the description of a probability distribution are difficult to relate, but that when the quantile description is used, the sample median can be characterized very much in the same way as the sample mean; this opens the possibility of using it for estimation purposes beyond what is usually done. In order to relate the two approaches, a derivation is given of the asymptotic joint distribution of the mean and the median for a general continuous probability law.