Median as a Weighted Arithmetic Mean of All Sample Observations

Abstract:     
Median, a well known measure of central tendency, of a sample data x =(x1, x2, ... ,xn) is obtained through the traditional method as (xn-m + xm+1)/2, where m= int(n/2) and values of x are arranged in ascending (descending) order. Since, for odd n, n-m=m+1, the formula uses only one value from the sample data. For even n, only two middlemost values are used. This property of the formula invites the popular criticism of median being not based on all sample observations.

It is possible to use an alternative formula to compute median as a weighted arithmetic mean of all sample observations, where weights are non-trivial and iteratively obtained. This alternative formula yields median identical to that obtained by the conventional formula if n is odd. If n is even, the results differ, though both of them yield the same minimum absolute norm. This is due to indeterminacy of median for even n in which every z between xn-m and xm+1 minimizes the absolute norm.

The paper further makes a comparative study of other desirable properties of the conventional and the alternative methods by Monte Carlo experiments.

Number of Pages in PDF File: 6

Keywords: Median, alternative method, weighted arithmetic mean, Monte Carlo experiments, Efficiency, consistency

JEL Classification: C10, C13, C15

working papers series