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Nicholas J. Cox
Quantile-box plots as discussed here are variations on a plot originally proposed by Emanuel Parzen in 1979. The main ideas are 1) the quantiles of a batch of data for a single variable, meaning the order statistics or ordered values, are plotted vertically as point or marker symbols against fraction of the data or plotting position plotted horizontally; 2) a box showing median and quartiles is shown either in the same space or alongside. Optional choices include 3) plotting the quantiles on a transformed scale (for example, logarithmic); 4) plotting fraction of the data on a transformed scale (for example, normal quantile); 5) adding details in the tails of the box plot (for example, spikes extending to paired percentiles or quantiles); 6) adding indications of other summary measures (for example, means or geometric means shown as extra horizontal lines); 7) showing quantile traces as connected curves rather than a series of marker symbols; 8) smoothing of quantiles to reduce minor noise; 9) plotting quartile or midgap plots instead of more conventional box plots; and 10) showing confidence intervals for any suitable measure of level (location, central tendency) instead of median-quartile boxes. Quantile-box plots provide indications of the level, spread, and shape of distributions together with indications of detailed features whenever they occur, such as outliers, spikes, and gaps without values. They do not supersede other useful displays such as histograms or density estimates, but they do not entail possibly awkward choices of bin width or start or of kernel type and bandwidth, nor do they require tuning of such choices. Quantile plots show essentially the same information as plots of the cumulative distribution or its complement, which may well be preferred by researchers as already familiar or as conventional in their field. Although quantile-box plots for individual batches may be interesting or useful, their main value often lies in comparison of two or more batches of data. While it starts from Parzen’s ideas, the discussion extends earlier and later. The history of quantile plots and box plots, usually under other names, is longer and more diverse than is often appreciated. The use of median-quartile boxes by Arthur L. Bowley around 1897 and by William C. Marshall in 1921 has escaped most reviews.In Stata terms, the community-contributed command, qplot, as updated in thisissue, is a convenient workhorse for quantile and associated plots. The communitycontributed commands pctilesets, quantilesets, and cisets, discussed in the previous issue (Cox, 2026, Stata Journal 26: 291–322), may be found to be convenient for calculation of quantile and other summaries. The official command egen continues to be useful for similar calculations. A strategy of divide and © 2026 StataCorp LLC gr0104 N. J. Cox 495 conquer—dividing the problem into steps of numerical calculation and graphical display—will often be helpful.
