In two papers, Drew Skau and I recently showed that our idea of how we read pie charts is wrong, that donut charts are no worse than pie charts, and a few more things. Here is a detailed walk-through of the results of the three studies we conducted for this purpose. Let’s go on a little journey through some real data and do a little science together!

For my talk at Information+, I redid the figures we had used in the EuroVis pie chart papers, both for the papers themselves and for the presentations. The result is much clearer, I think. I figured I’d share them here since they give a nice walk-through of the study results using the real data, but without too much detail. While the violin plots provide useful information during analysis, they’re just too detailed for presentation – I feel like I should have seen that coming.

How the Charts Work

What I’m about to show are the results of three studies, each of which had about 80–100 participants who each answered about 60 questions (for details see the papers).

The charts are all based on the difference between what people thought they were seeing and what we were showing them – called error. If we showed them a value of, say, 27% and they answered 29%, that means they were 2% off. The 2% would be the same if they had said 25%, at least in absolute terms. I’m also going to use signed error below, which would be +2% if they answered 29% in my example, and -2% if they answered 25%. In the papers, we used the logarithm of the error, which made things more complicated (but followed what others had done before).

I’m showing all the results using confidence intervals, or CIs. They have the advantage that they give us an idea of the data without being too overwhelming (I hope!). I use 95% CIs here, which means that we’re 95% confident that the real value of what we’re measuring lies inside those intervals.

For every measure, I will show two images: the raw error and absolute error, which will give us a sense the differences, and then the difference in error between each particular case and the baseline (which is always the pie chart). The latter allows us to make judgments about which of them are really different. This all might sound confusing, but it’ll become clearer below.

Study 1: Arcs, Angles, or Area

To figure out how people read pie charts, we decided to deconstruct them so that we could test their visual properties separately. A pie slice has three visual cues that all change linearly with the percentage it represents: its central angle, its area, and its arc length (the length of the circle arc on the outside).

The designs we came up with are shown below. In the top left, there’s the pie chart itself, which acts as a baseline, and next to it, the donut chart. In the second row, we have a very thin donut, which can only be read using arc length, and a chart that is round but uses only area to show the value. Finally, in the right-most column are the two arc-only charts: one that connects in the center and is based on the pie, and one that doesn’t and is based on the donut.

We had people go through a number of these – with different percentages being shown and rotated randomly –, and gauge the percentage they were showing. When we had the results, we looked at how well people had done in their guesses relative to what they were actually seeing. Here are the results.

Each of these bars shows us the signed error. That means we can get a sense of the deviation from the real value (the lengths of the bars), but especially the bias: were people systematically over- or underestimating?

However, to do this properly, we need to look at the difference of each of the cases from the pie chart. That accounts for people who always over- or underestimate. It also lets us compare the intervals for the cases other than the pie chart to a single value (that is important for statistical reasons).

All of the intervals intersect the zero line, so we can’t say that any of them lead to over-or underestimates. There is some variation, of course, but that is to be expected from a study like that. None of this clearly says that there is a difference, though.

More interesting than signed error (where we take the average of all the errors, both above and below) is absolute error, where we count all deviations in the same direction. That means errors don’t even out, so we can see how far off people are, no matter if above or below the correct value.

Now that is much more dramatic! The two final cases where people could only go by angle seem to be doing much worse. Let’s look at these relative to the pie chart.

The last two intervals clearly do not intersect the zero line. We’re clearly seeing much higher error here, not just statistical deviation. Arc length and area are no different from the pie chart, and neither is the donut chart!

Several things are surprising here. First, the donut chart is doing better than most people would have expected. I’m also still surprised how well the area-only one is doing, I had figured that would be much more difficult. And then of course the real stars here, the angle-only charts. They’re not just doing a little worse, but much worse than the others.

Study 2: Donut Radii

Given the lack of difference between the pie and donut charts, we wondered: does it matter what size the hole in the center of the donut is? Perhaps a very thin donut is harder to read than a thicker one? After all, if we read pie and donut charts by angle, the larger the hole, the less chance you have to see that properly.

So we tested six different donut hole diameters. The baseline is a pie chart, or a donut with a 0% hole in the center. The actual donuts have holes with a radius of 20%, 40%, 60%, 80%, or 97% of the donut radius.

Just like in the first study, we asked people to guesstimate the percentage they were seeing.

Looking at signed error first, there appears to be some bias in the middle two donut hole diameters. So let’s look at the error relative to the pie to see if these are in fact statistically significant.

And it turns out they’re not: all the CIs cross the zero line. The second-to-last one looks a bit different, but this might just be a coincidence. It certainly doesn’t mean that there’s a bias in that particular donut configuration.

Looking at absolute error next, we see some variation, with the second-to-last again looking different. Anything different from the baseline? Looking at the difference from the pie chart again…

Interestingly, the thinnest donut shows up here as being just about a meaningful difference. The confidence interval is so close to including the zero though that it’s hard to say. This does show up as a significant difference in the ANOVA we ran for the paper, but looking at it here it’s a lot more doubtful. It’s possible that a few more study participants would have pushed the mean down or widened the CI to include zero.

To be safe, you should probably stay away from very thin donuts. But none of the others differ from the pie chart, corroborating what we saw in the first study. This also again suggests that central angle is not important, since the absolute error would otherwise be much higher for the donuts.

Study 3: Pie Chart Variations

People do all sorts of things to pie charts. Are those okay, or are they problematic? To test that, we designed a few pie chart variations that mimicked those.

These also give us a chance to make and test some predictions. For example, we no expect the second chart to be overestimated, because the larger slice increases area and arc length of the blue segment relative to the rest of the chart, while not changing the angle. We also of course expected the irregularly-shaped charts at the end to do worse due to their distortions.

Looking at error again first, things are looking interesting.

Look at that larger-slice chart! This is the first chart we’re seeing here that has a very visible bias. Let’s do the math to see what this looks like compared to the regular pie chart.

If the decomposed pie charts in the first study didn’t convince you, this definitely should: the larger slice gets overestimated systematically. This is exactly what you’d expect if pie charts were read by arc length or area (since those are larger in comparison due to the larger radius), but not if you’re in the angle camp. This is the smoking gun, right there!

It’s also interesting to note that the square pie chart has a slight but significant-looking negative bias. This one is harder to interpret though because real irregular pie charts aren’t usually exact squares, and the ellipse doesn’t show the same bias (also, area and arc for these charts behave in very strange ways).

The latter two should be interesting when looking at the absolute error, though.

It appears that the exploded pie chart and the two irregular ones have higher error. Again, switching to absolute error relative to the pie chart…

Well, look at that! The irregular pie charts have significantly higher error than the basic pie. That is expected, but why? If angle is how we read them, how would the shape cause that error? We haven’t messed with that. Clearly, area and/or arc length must be what we read.

It’s also interesting to note that the exploded pie chart has higher error. We haven’t changed anything about the slice: not the angle, not the area, not the arc. It’s possible that there are some strange effects here because of the way the gap interacts with the rest of the chart. More research is needed here, but just looking at this, I’d say: avoid exploded pie charts.

Conclusions

These three studies clearly show that we do not read pie charts by angle. Whether it’s arc length or area is not clear from this work, but my money is on either arc length alone or arc length and area in some sort of combination.

To be clear, these studies say nothing about the suitability of pie charts. We used judgment error to gauge how well people can read different variations of pie charts as a way to find out what visual cue they were using.

What we did find, however, is that the donut chart is no worse than the pie chart. That is new, and it’s a direct consequence of the fact that we’re not reading pie or donut charts by angle. Donut charts are popular and are useable wherever a pie chart can be used.

What is more, we questioned and debunked the prevailing idea about how pie charts work that people have believed for 90 years – a paper by Walter Crosby Eells in 1926 appears to have been the basis for many assumptions about these charts. Nobody seems to have bothered to question them since. It’s time somebody did.

I have created a github repo with the code and data to recreate these images, as well as versions of them in three different formats: PDF, PNG, and SVG.