von Hippel, P. T. (2005), “Mean, Median, and Skew: Correcting a Textbook Rule,” Journal of Statistics Education, 13(2). (jse.amstat.org/v13n2/vonhippel.html)
Kudos to Paul von Hippel for showing us the lesser-known pitfalls of a commonly encountered rule of thumb and for ably “enhancing the desired intuition” into when and why this happens. One of the simplest specific examples I’ve used to show that skewness and unimodality need not imply a particular ordering of measures of location is the binomial (n = 10, p = .10) distribution, as suggested by Eisenhauer (2002). It is intuitive (and straightforward to verify) that this distribution is right-skewed. Introductory students can readily verify that the mean is 1 by computing np. The simple calculations Pr(X = 0) = .910 = .3486 and Pr(X = 1) = (10)(.1)(.99) = .3874 are enough to deduce that the mode and the median are also equal to 1. This is no surprise, since von Hippel states, “Since the Poisson is the limiting distribution for the binomial and hypergeometric, it follows that those distributions can break the rule as well.” However, it is worth specifically noting this example because more courses are likely to include the binomial distribution than the Poisson distribution.
There is actually a textbook (Levine, Berenson, & Stephan, 1998, p. 149) that gave this inequality for a right-skewed unimodal distribution that involved five quantities: mode < median < midhinge < mean < midrange. It might be an enriching followup to explore when this does and does not happen. In any case, it is hopefully clear that even more important to introductory students than having detailed knowledge about when one quantity might be larger than another is the knowledge to select the most appropriate measure of location in the first place, as students are asked in the first activity in the appendix of Garfield (1993).
Beyond the specific rule of thumb for mean, median, and skew, there has been broad examination and discussion about accuracy in statistics textbooks. For example, Brewer (1985) categorizes and catalogs a variety of statistical myths and misconceptions from then-bestselling behavioral statistics textbooks. Brewer suggests that some misstatements arise out of a sincere attempt at user-friendly simplification while others may simply be out of ignorance that gets maintained as the next generation of authors draws from the previous one. He does, however, offer the constructive suggestion for editors to have “manuscripts reviewed by people with collectively more knowledge of statistical theory and applications than the author of the manuscript” (p. 264). He adds this further safeguard (p. 266): “Because reviewers often see their advice ignored by publishers who are understandably more market-conscious than statistically astute, a mechanism should be devised to let the book purchaser know what each reviewer had to say about the statistical aspects of the book.”
Lawrence M. Lesser
University of Texas at El Paso
500 W. University Ave.
El Paso, TX 79968-0514
Brewer, J. K. (1985), “Behavioral Statistics Textbooks: Source of Myths and Misconceptions?”, Journal of Educational Statistics, 10(3), 252-268.
Eisenhauer, J. G. (2002), “Symmetric or Skewed?” College Mathematics Journal, 33(1), 48-51.
Garfield, J. (1993), “Teaching Statistics Using Small-Group Cooperative Learning.” Journal of Statistics Education, 1(1) jse.amstat.org/v1n1/garfield.html.
Levine, D. M., Berenson, M. L. & Stephan, D. (1998), Statistics for Managers using Microsoft Excel. Upper Saddle River, NJ: Prentice Hall.
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