Pie is a continued fraction

The pie chart is a venerable and effective way of showing how some total, say the federal budget, is divided up into its constituent elements, each represented by a “slice” of appropriate angular size. Of course, often these slices will change over time, and it is tempting to portray that trend in a series of pie charts. To paraphrase Richard Nixon, “you could do that, but it would be wrong.”

It would be wrong because it fails to make trends in the data effortlessly and immediately evident to the viewer, because it fails to exploit the human visual systems automated mechanisms for perceiving trends. As we have noted with tiresome regularity, the eye is tuned to see contours, and to judge their orientation (look! the market is going up!) but not to quickly judge areas of complex shapes depicted in separate, unconnected parts of a figure.

Here is an example, taken from a New York Times article on how medical device companies bribe doctors to use their products, rather than their competitors, irrespective of the value to the patient. (This practice would be outlawed under ObamaCare, but perfectly ok under BoehnerCare). The series of pie charts attempts to show how one company (Biotronik) rapidly achieved near-complete market dominance for its pacemaker at one Nevada Hospital, after paying the hospitals cardiologists for “consulting.”

As always, I ask you to take a quick look at the chart, and see what pops out at you. I think the sad answer is: nothing. It requires careful scrutiny, with endless searching back and forth between pies, and between labels and slices of pies, until the presumed point is made: Biotronik went up, suddenly, and everyone else went down, suddenly, to almost nothing. And in fact, everyone else is primarily one company: Boston Scientific. So the point is equally well made by just showing the two companies.

That is what we have done in the following graph.

The trends in the two companies fortunes is perceived immediately and effortlessly. And because the graph shows percent market share, and Biotronik is almost at 100%, it is clear they have achieved a near monopoly. There is no need to plot “Other.”

The use of filling in this chart (coloring in the areas below each line) is a judgement call. While filling uses more ink, it can convey a contour better than a line. And since this graph is showing market share, it feels appropriate.

This graph could have been shown in color, but since there are only two categories, and since their trends are so clear, there is no need. Nonetheless we show an example here. As we note below, when many categories are involved, it is helpful to use color as a linking device.

One of the problems in attempting to show trends over a series of pie charts is that the categories within the several charts must be linked. To use the current graph as an example, we need to know which slice in each pie belongs to “Biotronik.” In the Times graphic, two strategies are used to link categories: shades of gray, and text labels.

The use of shades of gray to link the categories in the four pies is particularly weak. As any vision scientist will tell you, the human eye is very bad at remembering or identifying particular shades of gray. You have to remember, because you have to move your eye from one pie to the next. Colors, such as red or blue, suffer from no such weakness. We say they are perceived “categorically.”

Text is also a poor way of identifying the categories. It is unambiguous, to be sure, but requires reading, and moving the eyes back and forth from the label to the slice, all of which disrupts what should be an immediate, effortless “grokking” of the categories.

Consider also the additional clutter introduced by all of the labels. The words “Boston Scientific” are printed out in full three times, as are the words “Biotronik,” while the word “Other” gets repeated four times. The last is particularly ironic, since the category “Other” contributes almost nothing to the discussion.

It might be argued that the timeline, and paragraphs of text, that float above the graph provide historical context. In this case, not much context is really required. In any case, the graph that we have provided can be stretched to suit, or the text, which is really supplementary material, could be attached to the graph through arrows marking significant events in the chronology.

In the present example, there are really only two categories of interest. But often there will be more, in which case another method of plotting might be considered. This is the so-called “stacked,” or as I prefer, “accumulated” graph. In this variant, we add the values of each series on top of each other, so the data for each category is represented by the vertical extent that it occupies. Here is our current example, rendered in this way.

This chart has the advantage that, like a pie chart, it slices up the total into all of its constituents. The share of each category is immediately evident by the share of the vertical height it occupies. But in contrast to the series of pie charts, the trend is immediately and effortlessly evident, because the “slices” are connected.

There are problems with this form of presentation. One must choose the order in which to place the elements, top to bottom. There is no correct order, and this introduces an opportunity for bias or inadvertent misrepresentation. For example here, it seems natural to put the “Other” category at the bottom, but what about the other two? In the example above, placing Biotronik in the middle causes the upper edge of its share to climb precipitously, which matches its growth in share. But this has the unfortunate consequence of causing the lower edge of Boston Scientific’s share to also climb, a visual cue that is contrary to its declining share. Plotting in the reverse order, as shown below, merely reverses the problem.

So caution should be used when employing stacked graphs, and only use them when there are more than two significant categories, and the data cannot be better shown with simple line plots as in our first figure above.

The lesson here is: don’t use a series of pie charts to show a trend. It doesn’t work. A line graph is always better.

A subsidiary lesson is to use caution when using stacked graphs. The arbitrary order of stacking can convey different impressions.


New York Times

Tipping the Odds for a Maker of Heart Implants By BARRY MEIER Published: April 2, 2011


Turn the tables

Today’s note is about the very worst sort of graph: a table. Of course, a table is not a graph, but that is the point: it almost always should be. Here is a table that appeared recently to make a very simple and powerful point. To further my argument, however, I won’t tell you what that point is, and ask you to deduce it from the table. I will time you. ready? Go!

Ok. Stop! Time’s up. Did you get the point? I thought not.

Now here are the same data as a graph. Now do you get the point? The graph shows changes in the Gross Federal Deficit, as a percent of Gross Domestic Product, for the last nine presidential administrations. Above the zero line are increases in the deficit (bad). Most of the colored areas above the line (bad) are red (republican) and most of those below the line (good) are blue (democrat). So much for the notion of “tax and spend” democrats and “fiscally responsible” republicans!

But of course this column is scrupulously fair and balanced, so our point here is not the political one, but the graphical one.

The main lesson here is a very elementary one. Never use a table when a graph is possible. Graphs by their nature render data into visible patterns that jump out at the reader. Tables have a purpose – to provide access to numerical values – but they rarely make anything self evident, and they rarely show trends without scrutiny, thought, and calculation.

It may be worth mentioning some of the design decisions that went into my graph.

First, note that there is no horizontal axis. We leave out the dates because they add little to the story. The rectangles are the correct width (4 or eight years), and are in the correct order. adding dates would just be clutter.

Second, we used a rectangle graph, in which the width of each rectangle represents the length of each presidential term. And the vertical axis is deficit change per year. So the area of each rectangle represents the total change during each administration. This is appropriate, since the visual impact is proportional to area, and the total deficit from each administration is the key quantity we wish to convey.

We add the names of the Presidents, since that is of some interest, but put them in light gray, since their identities are not central to the point being made.

We add a small key, to remind readers of the meaning of the two colors. While these colors have become conventional for the two parties, we can’t assume the key will be obvious.

We leave out any extraneous lines, text, shading, or decoration. Just the facts, ma’am.

Reference:  The Atlantic, January 2, 2011.


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