Clash of representations

The pie chart in the above tweet jumped out of the page when it appeared in my twitter feed on September 14. My initial shock at seeing the figure 1% attached to a region of the pie chart that was evidently almost 25% of the total area of the disk did not last long, of course, since the accompanying text made it clear what the diagram was intended to convey. The 1% label referred to the section of the population being discussed, whereas the pie-chart indicated the share of taxes paid by that group. Indeed, the image was an animated GIF; when I clicked on it, the region labeled “1%” shrank, culminating with the chart on the right in the image shown below:

But here’s the thing. Even after I had figured out what the chart was intended to convey, I still found it confusing. I wondered if a lay-reader, someone who is not a professional mathematician, would manage to parse out the intended meaning. It was not long before I found out. The image below shows one of the tweets that appeared in response less than an hour later:

As I had suspected, a common reaction was to dismiss the chart as yet another example of a bad data visualization created by an innumerate graphics designer. Indeed, that had been my initial reaction. But this particular example is more interesting. Yes, it is a bad graphic, for the simple reason that it does not convey the intended message. But not because of the illustrator’s innumeracy. In fact, numerically, it appears to be as accurate as you can get with a pie chart. The before and after charts do seem to have regions whose areas correspond to the actual data on the tax-payer population.

This example was too good to pass up as an educational tool: asking a class to discuss what the chart is intended to show, could lead to a number of good insights into how mathematics can help us understand the world, while at the same time having the potential to mislead. I was tempted to write about it in my October post, but wondered if I should delay a couple of months to avoid an example that was at the heart of a current, somewhat acrimonious party-political debate. As it turned out, the September 30 death of the game-show host Monty Hall resolved the issue for me—I had to write about that—and then November presented another “must do” story (the use of mathematics in election jerrymandering). So this month, with the background political, tax votes now a matter of historical record, I have my first real opportunity to run this story.

The two-month delay brought home to me just how problematic this particular graphic is. Even knowing in advance what the issue is, I still found I had to concentrate to “see” the chart as conveying the message intended. That “1%” label continued to clash with the relative area of the labeled region.

It’s a bit like those college psychology-class graphics that show two digits in different font sizes, and ask you to point to the digit that represents the bigger integer. If the font sizes clash with the sizes of the integers, you take measurably longer to identify the correct one, as shown below:

For me, the really big take-home lesson from the tax-proposal graphic is the power of two different mathematical representations of proportions: pie charts and numerical percentages. Each, on its own, is instant. In the case of the pie chart, the representation draws on the innate human cognitive ability to judge relative areas in simple, highly symmetrical figures like circular disks or rectangles. With percentages, there is some initial learning required—you have to understand percentages—but once you have done that, you know instantly what is meant by figures such as “5%” or “75%."

But how do you get that understanding of the meaning of numerical percentages? For most of us (I suspect all of us), it comes from being presented (as children) with area examples like pie charts and subdivided rectangles. This sets us up to be confused, bigly, by examples where those two representations are used in the same graphic but with the percentage representing something other than the area of the segment (or what that area is intended to represent).

The message then, from this particular example—or at least the message I got from it—is that powerful graphics are like any powerful tool, their power for good depends on using them wisely; if used incorrectly, they can confuse and mislead. And make no mistake about it, numbers are incredibly powerful tools. Their invention alone is by far the greatest mathematical invention in human history. That’s why in every nation in the world, math is the only mandated school subject apart from the native language.