Mean, Median, and Skew: Correcting a Textbook Rule

Abstract

Many textbooks teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew. This rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy. Most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they also contradict the textbook interpretation of the median. We discuss ways to correct ideas about mean, median, and skew, while enhancing the desired intuition.

• Asymmetry
• Central tendency
• Extreme values
• Influence
• Mean-median-mode inequality
• Mode
• Outliers
• Robustness
• Sensitivity

Acknowledgements

This paper used MathStatica 1.5 under Mathematica 5 for calculations and graphs. I thank the reviewers as well as Jim Albert, Patti Hunter, Steve MacEahern, Doug Wolfe, and Ann Watkins for helpful feedback on earlier drafts.

Addendum

Volume 13, Number 3, of the Journal of Statistics Education contains a Letter to the Editor concerning this article.