On This Issue's Cover
Katherine Furgol Castellano · Educational Measurement Issues and Practice · 2015
On this issue's cover is a winning submission to the 2014 EM:IP Cover Graphic/Data Visualization by Nedim Yel, a doctoral student in Educational Psychology at Arizona State University. This figure, suitably titled A Footnote to Anscombe's Quartet, follows Anscombe's (1973) example of using four sets of data to illustrate the importance of pairing any statistical analysis not only with a graphical analysis but an appropriate graphical analysis. The purpose of this graphic is twofold: to demonstrate the importance of graphing data and choosing the right method for graphing such data. In particular, A Footnote to Anscombe's Quartet shows how the 5-point summary in the box-and-whisker plot may mask huge differences in the univariate distribution. The figure includes two ways of graphing four variables that all have the same 5-point summary. The first four box-and-whisker plots create the impression that the four variables have the exact same distribution. In contrast, by including the bee swarm with the box-and-whisker plot, as illustrated in the second set of four box-and-whisker plots, it is clear that there are differences in the distribution of data points. The general idea behind a bee swarm plot is plotting each data point in such a way that points are close to each other but do not overlap. For example, if two data points have the exact same value, these data points are plotted side by side. The size of these data points can also be controlled, for instance, by manipulating options in Eklund's (2013) “beeswarm” function in R (R Core Team, 2013). This feature is useful when plotting large number of data points. It is also possible to change the color and shape of each data point, adding more information to the plots. For example, if the sample data come from three different groups, each group can be denoted with a different color or shape. As demonstrated in this graphic, the inclusion of a bee swarm in graphing data allows for a more refined visualization and better judgment about the distribution of the data. A Footnote to Anscombe's Quartet simply, but elegantly, illustrates a fundamental statistics lesson about the importance of graphing data. Although not particularly eye-catching, this figure aligns well with the specific purpose of the EM:IP Cover Graphic competition and the broader mission of the EM:IP editorial (at least with regard to the EM:IP cover page)—to encourage the thoughtful use of graphics in educational data analyses from exploration of the data to presentation of data analysis results. The bee swarm augmented boxplots provide a much finer grain visualization of the distributions in a compact format that has several advantages over some common approaches for graphing distributions and thus provides practitioners and education researchers with a novel visual tool that they can easily use in practice. For instance, alternatively, we may use histograms or smoothed density plots to graph the distributions. Figure 1 presents the density plots overlaid on each other in the same plot for the same four variables as in the cover graphic, and Figure 2 displays the histograms for these variables side by side. Both of these plots, like the bee swarm boxplots, show clear differences among the distributions. They also illustrate some of the same key features of each distribution. For instance, the same peaks in the distributions are indicated by the longer lines in the bee swarm boxplots as the peaks in the histograms and smoothed densities. However, Figures 1 and 2 give coarser pictures of the distributions (which is partly due to my choice of bin widths and smoothing parameters), whereas the bee swarm displays markings for each data point. We could adjust the “resolution” of the histograms and densities with choices of bin widths and degrees of smoothing or particular smoothing kernel. But subjective adjustments to these features can also result in distorting, hiding, or masking key information about the shape of a variable's distribution. If the data were discrete, like test scale scores, we could plot a discrete histogram, but that is not possible in this case as the variables are continuous. The first set of box-and-whisker plots in the cover graphic shows that the four variables have the same five-number summary, but these plots illustrate that five numbers are not sufficient to define a distribution—at least not these five numbers. The means, standard deviations, skewness, and kurtosis of the four variables all differ to varying degrees, which point to differences in the distributions masked by the five-number summary. However, the moments of the distribution can be highly influenced by outliers, whereas the 25th, 50th, and 75th percentiles of the five-number summary are more robust to outlying points. The “Box-and-whisker with Beeswarm” in A Footnote to Anscombe's Quartet provides a useful alternative to other distributional graphs, such as smoothed density curves and histograms shown in Figures 1 and 2. Bee swarm plots may also be manipulated with user-specified choices about the size and shape of the added swarm, but they provide a finer grain picture of the data points in a single, clean graphic for several variables than the density or histogram graphs. What do you think? Please send us your feedback by emailing [email protected].