MEDIAN,MEAN, AND OPTIMUM AS FACILITY LOCATIONS*

ABSTRACT The mean and the median are both measures of centrality. In one dimension, the median minimizes the average absolute distance from a facility to a set of customers {x_i}, and the mean minimizes the average squared distance. In two dimensions, the median minimizes the average rectangular distance, and the mean minimizes the average squared distance. This paper investigates the “location penalty” when a nonoptimal location is substituted for the optimal location. In one dimension, the average absolute distance at the mean is never more than twice the average absolute distance at the median. Surprisingly, this happens when the median and mean are close together. In two dimensions, the ratio of the average Euclidean distance at the median to the average Euclidean distance at the optimum is never more than . However, this upper bound depends on an unlikely scenario with just two customers. With three equal-sized customers, this ratio is never more than 1.12. However, if the triangle formed by the customers is rotated relative to the rectangular grid, the ratio never exceeds 1.028.