Part 1: A Waffle Is Not A Pie
Data visualization is still a very young field. Many of its more recognizable expressions such as the bar chart and the scatter plot are quite settled in terms of best practices. The case for their usefulness has been made by their adoption as standards. However, dispute over some standards continues. One of the most hotly debated visualizations remains the pie chart.
An individual pie chart displays the proportional relationships among n parts of a whole, and each part’s relationship to that whole. Its highly simplistic structure makes it very easy to understand. Common criticisms of the pie chart include humans’ supposed lack of skill in discerning angular variation –more on that later–, the univariate information provided by the chart, and the rapid degradation of readability that occurs as n increases.
The debate may continue to echo in the crowded halls of data viz bloggers forever, but one branch of discussion that began in 2006 is of particular interest. Robert Kosara –a researcher at Tableau Software– made a post on his blog, eagereyes, called “Women in IT — Squaring the Pie?” that briefly examines the idea that pie charts can be reconfigured into a square format. In a traditional pie chart, 1% of the whole is equal to the area formed by an arc segment of a circle with a central angle of 3.6 degrees. Kosara argues that by dividing a square into a 10×10 grid instead and letting 1 section of it to equal 1% of the whole, visual interest can be created for the reader (by way of the unfamiliar encoding). More importantly, a more consistent decoding at an accuracy of 1% can be made possible.
In a follow up post from 2008 titled “Engaging Readers with Square Pie/Waffle Charts,” Kosara is more assured of the usefulness of the square pie chart, stating that one particular example found in the New York Times“strikes a good balance between being interesting and not distorting the data.” The chart makes a striking comparison between the saving and spending habits of an average American household in 2008.
Kosara is quick to note, however, that the graphic is not exactly a square pie chart. He uses the term “waffle chart” interchangeably with square pie and later refers to it as a “matrix diagram.” The semantics are unclear, but I see some very compelling use cases for this chart, specifically within the realm of persuasive writing in sustainability and policy. The distinction between the type of chart pictured and the square pie with which it occasionally shares a name has not yet been explicitly made, so I figured I would do it.
The Details, The Difference:
The Square Pie
Effective square pie charts are necessarily divided into 100 sections, each one equivalent to 1%. Scaling one section of a square pie to some other value would only serve to obscure its meaning and quash the accurate decoding that setting 1 section equal to 1% allows. For example, this chart from a 2015 feature published in The Economist features an 8×12 grid. This makes each section equal to 0.96%, which is, above all else, weird.
As their name implies, square pie charts are meant to serve as a replacement of the often ridiculed pie chart. In the introduction, I alluded to the notion that humans can not accurately discern angles. This claim forms the basis of a lot of anti-pie arguments. It arose at least as early as 1926 in a study done in by Walter Crosby Eells. In one component of the study, participants self reported which part of the pie chart they used to derive numeric values. In the end, the angle was least often used. That survey result may be the root of the idea that angles do not aid visual perception.
In a 2016 essay Robert Kosara and Drew Skau sought more robust answers to the questions of whether or not people can accurately decode pie charts and what visual components of the chart most strongly facilitate that decoding. In short, they found that of all the components of a pie chart, angles result in higher rates of decoding error. They concluded that pie charts are very likely decoded primarily on the basis of area. This conclusion was corroborated by the results of a parallel study on pie chart variations by Kosara and Skau.
In a follow up post on eagereyes, Kosara married the analytical methodology from the two studies just mentioned with the data from an older paper. Largely, he was aiming to validate the method of crowdsourced surveyingused in his work on visual perception thereafter. The analysis in the post compared square pies, stacked bars, traditional pies, and donut charts. He compared the accuracy with which survey participants could decode the value of a single percentage from them. Additionally, he recorded the confidence of the participants’ readings of the chart. In both reading accuracy and confidence, the square pie chart came out on top. Contrary to what contrary popular belief would lead one to expect, the stacked bar chart actually performed worst in terms of accuracy and reader confidence.
Error rates of participants reading the square pie in the study fell within an interval of ~1.1 to -1.9%, but it is worth noting that the chart used in the study did not utilize grids on its square pies. Adding a grid to a square pie opens the door to a level of decoding accuracy that is not reasonably doable for a circular pie. I suspect that if a grid had been used in the Kosara study, decoding accuracy would have fallen much closer to 100%. A reader can quickly and easily be informed that a single filled square equals 1% and that a filled row/column equals 10%. By squarifying and gridding pie charts, significant reading accuracy can be gained, and none of the simplicity, accessibility, or scalability of the traditional pie is lost.
There are two primary differences between waffle and square pie charts. First, waffle charts are not restricted to showing parts of a whole. Where all varieties of pie are limited to relative comparisons, the waffle allows for absolute comparisons. The second distinction –rooted in the first– is that waffle charts do not have to fit a 10×10 grid, and in fact, they usually do not. It follows that a single square can be set to equal any value, further increasing its versatility. Since waffle charts and square pies are built from the same graphical pieces, it’s a safe assumption that they are approximately as readable as one another.
Waffle charts can be used in lieu of traditional bar charts. They’re particularly effective when comparing numbers that are highly variant, which makes them easily tolerant of outliers.
The above charts show the same meaningless data, but note how much more easily the quantities of each category can be read in the waffle chart. All the guesswork of decoding the value of the pink variable is eliminated without needing to apply any extra labelling to the chart. The values of larger variables, like the green for example, can be decoded easily by simply counting the full columns (1 column = 10) of that color and then adding in the incomplete bars.
If waffle charts are arranged in multiples, they can replace stacked and clustered bar charts as well. This is quite compelling, as Kosara recently exposed stacked bar charts as The Worst in his continued inquiries on visual perception.
Stacked bar charts suffer from the unfortunate ambiguity of a baseline that shifts for each subcategory of a measure. In the first graph above, the baseline of every measure but the purple changes five times. Its clustered sibling encourages jittery readings; The reader must jump from point to point along the x axis and over the intense hills and valleys of the variant measures in order to make meaningful comparisons. Both graphs suffer from small outliers that disappear and large ones that call attention to themselves disproportionately.
Reconfiguring the same meaningless data into a series of waffle charts creates a significantly more immersive picture. Every single measure is given precise visual representation, including outliers. Each one can be explored at the finest granularity, and the macro comparison of categories against one another is as plainly logical as a traditional bar chart. While the waffle variation is not as compact as the bars, it certainly offers a fresh, pragmatic alternative.
Part 2: But, What Would Edward Tufte Do?
In the blog post that started it all on eagereyes, Hadley Wickham –creator of the popular R visualization package ggplot– has a lot to say in the comments section. He suggests that waffle charts accomplish nothing new or useful, resembling treemaps but with less depth. Indeed, the information provided by a square pie can usually be conveyed in a sentence or two. The information in a waffle chart only requires a few more. Are waffles and square pies to be relegated to fluffy chart status?
Edward Tufte, one of the most well recognized names in data visualization, has made points on both sides of the argument. On the side of Wickham and opponents of novel visualization, Tufte spends the entire seventh chapter of his first book[¹²] singing the praises of “multi-functioning” graphical elements. The charts that are most worth their salt, by his estimation, can facilitate multiple readings of the data. Waffle and square pie charts simply don’t provide the same depth as something like a bubble chart or an annotated time series. They merely provide a platform for high-resolution comparison.
Tufte chides the makers of ornamental charts for perpetuating the idea “that numbers and details are boring, dull, and tedious, requiring ornament to enliven.”[¹³] Very often, the most effective way to communicate the information offered by data is by simply showing and discussing it; a table of numbers and a paragraph. In this school of thought the role of data visualization is to express facts that are too complex and voluminous for words alone to capture. Waffles and pies have no place here.
On the other end of the spectrum, Tufte weighs in again. In Envisioning Information, he discusses a newspaper clipping from 1987. The clipping summarizes a federal conspiracy case, United States v Gotti, which was infamously upended by an information display. John Gotti, a well known Mafia leader, was acquitted after his defense lawyer, Bruce Cutler, presented the jurors with a table. Its column headers displayed the names of the prosecuting witnesses. In the rows, the variety of crimes they had committed were checked off. By providing the jury with a tool that allowed them to analyze the facts and make assessments themselves, Cutler put himself in a position to affirm the thinking of the jurors rather than merely telling themwhat to think. When he closed his arguments by calling the testimonies of the prosecution untrustworthy, members of the jury were in a position to say “Yes, indeed. I knew that because I learned it myself by examining this display of raw facts.”
That sublime instance of persuasion through facilitated discovery is the basis of my own perspective on visualization and information display: Quantitative measures are an abstraction of fact, and when they are large, complex, or numerous they quickly become murky to they who do not live their lives thinking about numbers. A simplistic visualization like a waffle chart may not seem valuable to a statistician, designer, or scientist, but their audiences may very well reap great benefits from seeing facts presented twice. Nothing is learned in one go, and by complementing a numeric abstraction with a visual one –a waffle chart and a written analysis– we have the opportunity to grace our readers with a second shot at understanding. If we can facilitate learning rather than assuming ourselves perfect teachers, why would we ever choose not to?
This blog was originally posted on Medium.