Discussion of “Graphic Comparisons of Several Linked Aspects” by John W. Tukey

Abstract

A graphical method for data analysis has two components. The first component is information to be displayed, a statistic that is sometimes just the data themselves. The second component is a visual part, a selection of a display method that encodes the information by position, geometric magnitude, and other rendering attributes such as color. For a boxplot (Tukey 1977) the statistical information is a five-number summary of the empirical distribution together with end values; the display method is the familiar box with a line or filled circle inside, with two appendages, and with plotting symbols to show end values. This article by Tukey addresses both the selection of information and methods of display, but the emphasis is on the latter. It continues a current of thinking begun earlier by Tukey (1990), and so we will discuss both articles. Because they are so rich in ideas, and because the discussion must as a practical matter be limited, we will confine our comments to the display method of boxplots. Devoting all of the available space to the boxplot is entirely reasonable because it has become a standard of statistical graphics. And the points we make apply, at least broadly, to the other display methods discussed by Tukey. We will strike two themes: 1. The success of a display method depends on the effectiveness of the visual decoding of the encoded information. Thus to create display methods with maximal impact, as Tukey would put it, we must study visual perception. Success cannot be assessed without explicit attention to visual perception. For example, to determine whether on a boxplot it is better to encode the median by a horizontal line segment or a filled circle, we must appeal to visual perception. 2. As a practical matter, it is healthy to regard a display method as incomplete until software exists that implements it for the full spectrum of data types to which it is meant to apply. The reason is that code provides a complete specification; issues of visual perception that are easy to gloss over in a discussion with a few examples loom large when it comes time to write and test code that is meant to be general. One example is Tukey's suggestion that numbers go inside the bars of bar charts. As we will argue later, this is easy to suggest and tricky to fully specify.