Comparing the Mean and the Median as Measures of Centrality

This paper introduces two coefficients, one for the sample mean and one for the sample median, that measure the relative advantage of one statistic over another in minimizing the variability around it. Variability is measured as the sum of squares of deviations for the mean coefficient, and by the sum of the absolute values for the median coefficient. These coefficients always range between 0 and 0.5, with larger values indicating larger advantage. The values of these coefficients are computed for international data on 40 different variables, and the results are interpreted in relationship to the skewness and kurtosis of the distributions. In order to provide a means of interpreting the magnitude of the coefficients, the values of their population analogs are computed for many members of the gamma distribution family and the lognormal distribution family. The skewness and kurtosis coefficients are also calculated for these distributions, and compared with the corresponding values of the coefficients.