Letter-value plots: Boxplots for large data

Boxplots (Tukey, 1977) are useful displays that convey rough information about the distribution of a variable. Boxplots were designed to be drawn by hand and work best for small data sets, where detailed estimates of tail behavior beyond the quartiles may not be trustworthy. Larger data sets afford more precise estimates of tail behavior, but boxplots do not take advantage of this precision, instead presenting large numbers of extreme, though not unexpected, observations. Letter-value plots address this problem by including more detailed information about the tails using “letter values”, an order statistic defined by Tukey. Boxplots display the first two letter values (the median and quartiles); letter-value plots display further letter values so far as they are reliable estimates of their corresponding quantiles. We illustrate letter-value plots with real data that demonstrate their usefulness for large data sets. All graphics are created using the R package lvplot, and code and data are available in the supplementary materials.