An Improved Textbook Rule on the Mean-Median Inequality for Discrete Data

Demirtaş H., Vardar Acar C., Gao R., Aydemir İ., Yabacı A.

Türkiye Klinikleri Biyoistatistik Dergisi, vol.12, no.2, pp.1-10, 2020 (Peer-Reviewed Journal)


Objective: Many introductory statistics books cover the mean-median inequality, which states that if the skewness value is positive/negative, the mean is greater/less than the median. However, this textbook rule is often violated especially when one tail is long and the other is heavy. The purpose of this paper is to propose a refinement that solves the problem to a meaningful extent by bringing the area to the left and right of the median into the picture for discrete data, where violations are more common and severe. The improved version is simple and effective enough to replace the existing rule. Material and Methods:Three distributional settings were utilized for illustration: The Poisson, binomial, and discretized normal mixture distributions. A simulation study was devised to assess the relative performances of the current and new rules for count data under the Poisson distribution assumption. Results:The new rule adds a simple layer to the current rule: For right skew, the mean is greater/less than the median if the area to the left of the median is less/greater than the area to the right. Similarly, for left skew, the mean is less/greater than the median if the area to the left the median is greater/less than the area to the right. In other words, the new component comes in the form of a comparative area restriction. Conclusion:All three distributional examples lead to the same Conclusion: The proposed version is associated with substantially better results. Although it is not a complete solution, it is a serious improvement.

Keywords: Tail behavior; Introductory statistics; Symmetry; Mean; Median