On Fences and Asymmetry in Box-and-Whiskers Plots

摘要

This note provides a new explanation for Tukey's definition of "(inner) fences" for box-and-whiskers plots. The starting point is explicit bounds for the sample mean based only on the box plot. Starting from these bounds we define a dataset to contain outside values if at least one of the latter bounds is outside of the box. This leads to a new, yet simple definition of fences. They are symmetric around the box if, and only if, the median is in the middle of the box. In that case, the new definition coincides with Tukey's rule of 1.5 times the inter quartile range. To avoid instabilities for small (sub-) samples we propose to complement the box-and-whiskers plot of the original data with a box-and-whiskers plot of Walsh means.