Circles of hell

This began as a commentary on perception of one, two, and three dimensional graphics, but that will have to wait for another time. Instead we will spend our special time together excoriating just one little graph. Here it is, from a recent issue of the New York Times.


As (almost) always, we begin with a simple question: when you first glance at the graph, what fact or idea pops out? For me, the answer is: “how long until lunch?” In other words: nada. There are a bunch of big red balls, and a whole bunch of squirrelly little lines pointing every which way, and numbers all over the place. Something is going on, but it hardly seems worth the effort to figure it out.

Upon further tedious scrutiny, we deduce that the graphic is telling us something about rates of cellphone and landline usage. OK. How fast are they changing? Which is changing faster? Did rates cross? What year? Is the rate of change of either  accelerating? Decelerating?

I venture that you were able to answer none of these questions without careful study of the graph, perhaps even with a ruler. And if you had used a ruler, you would have been wrong, but we will get to that in a minute.

Below we will enumerate the five separate things that are wrong with this graph. First, a simple conventional graph of the same data.


Simple. Easy to understand. The trends jump right out at you. No rulers required.

But as John F. Kennedy really meant to say: “How much more fun to curse the darkness, than to light a candle.” Let us turn our attention to what is wrong with the original graph.

1. The use of disks or circles to depict quantity is problematic.

As will be discussed at greater length in a subsequent post, folks are not so good at judging area. If you represent quantities by areas of disks, that may not lead to correct judgements about the relative magnitudes of quantities.

2. The area of the disks does is not proportional to the quantity depicted.

Even if people could judge the area of circles, they would get the wrong answer. Amazingly, the artist appears to have just used “artistic judgement” to decide on the size of the disks. The actual areas are plotted by the dashed lines in the next figure. Not even close.


3. The diameter of the disks is not proportional to the quantity depicted.

Perhaps the artist intended the diameter to represent the quantity? We show this by dashed lines in the next figure. Closer, but not quite. But even had it been correct, why would the artist imagine that readers would sense diameter, rather than area? This ambiguity illustrates one reason why the use of 2D or 3D markers to indicate quantity is problematic.


4. The graph distorts the time dimension.

Notice that the disks in the original graph are all spaced evenly along the time (vertical) dimension? But the years involved are not equally spaced, as can be seen in our substitute graph. The gaps range from 5 years to 1 year. Failure to correctly depict the time dimension makes it impossible to correctly judge rates of change.

5. Maximum and Minimum cannot be depicted.

The quantities depicted here are percentages, so there is a clear and inviolate minimum and maximum of 0% and 100%. But how do you indicate either of those with a disk? 0% is impossible, and 100% is ambiguous. Thus the use of disks to represent quantity makes it impossible to indicate the bounds of the data. In contrast, a simple point-and-line graph like ours easily shows these bounds, and their proximity to the data.

To conclude, big colored disks make for a fun and bold graphic, but are pretty useless when it comes to actually conveying information. Real graphing artistry consists of beautiful graphic design that also conveys a clear and accurate message.


Pump up the volume!

One of the most egregiously deceptive practices in graphology is what we might call “dimension boosting.” Like the use of a performance drug in sports, it is an effort to gain un unfair advantage by playing outside the rules. Usually this crime consists of using the  width of a two-dimensional figure, such as a circle or a square, to depict a one-dimensional quantity. But as the width increases, the area, which is what we perceive, grows as the square of the width. With this device, a small difference can be made to look much larger. If the plotted quantities differ by only a factor of two, their areas will differ by a factor of four.

That is bad enough, but sometimes the criminal decides to do all the way, and throw in not one but two extra dimensions! In other words, they depict a one dimensional quantity with a three-dimensional object. Below is an example from a recent edition of the Sunday New York Times Magazine. It illustrates the decline of drinking among American teenagers over the last three decades.

Now we will perform a little test. Quickly, without looking at the axes, look at the two images at the beginning and end of the time interval and tell me by what factor drinking declined over that period. Got your answer? OK. Lets review. Well… in 1980 it looks like a 1.5 liter jug, while in 2010 they evidently had one shot glass (3 ounces?). You can fit about 17 shot glass servings in a 1.5 liter bottle. A 17x decline! Wow! Those kids sure have cut back!

But suspecting that todays teens are not quite so abstemious, and having been burned by criminal graphologists before, we examine the plot more carefully. First, we notice that even though the little bottles and glasses vary in not one, not two, but three dimensions, the axis on the left is a simple linear scale. Presumably the top of each vessel is the relevant aspect. Also, the axis is labeled in %. On that basis we realize that  the incidence of drinking has only declined from 70% of teens to 40%, a decline of only 1.75x. An impressive decline, but not 17x.

Now that this graph has been caught red-handed, and we have it in a holding cell while it calls its lawyer, we can investigate further. Notice that the vertical axis only goes down to 40%? That is another devious trick to exaggerate the magnitude of a difference. If the axis had extended all the way to zero, the difference between 1980 and 2010 would not seem quite so impressive. (that would provide what we call a “ratio scale,” for the technically inclined). And since we are plotting a fraction of teenagers, maybe it would be fair to extend that axis all the way from 0 to 100%, further reducing the apparent magnitude of the change.

And another thing: why are the bottoms of the bottles and glasses jumping all over the place? If the top is meant to indicate the value, it would only be fair to keep the bottom stationary.

And while it feels like piling on, what is going on with the horizontal position of the containers? Their positions seem to jump around a bit, and there are different numbers in each decade. Did they forget to make the measurement is certain years? Or is the artist just exploiting their “artistic license?”

This graph is an instance of what is often called an “infographic.”  An infographic is to a graph what an infomercial is to information. A bastard form in which information takes second place to entertainment or marketing. Look! Little bottles! What fun! One could imagine a form in which entertainment was provided, but truth was retained, but regrettably that is rarely to be seen.

In the printed version of the magazine, this graph is attributed to O.o.p.s. They should be ashamed. But the Times cannot escape the blame for this many-count indictment of graphical crime.

For completeness, we show a less entertaining but more accurate plot of the same data. It shows the full range from a fractions from zero to one, and does not introduce extraneous dimensions. The change in teenage behavior is significant, but not exaggerated by multidimensional trickery.


New York Times

Well: The Kids Are More Than All Right


Published: February 2, 2012

Attack of the little people

Where did they come from, the little people? Like a horde of replicants they have streamed forth to cover the world of infographics. No trendy depiction of any statistic related to humans is complete without the little people. Consider todays freshly populated example, from our favorite whipping boy, the New York Times.

The graphic is an attempt to put “into perspective” the numbers of people in poverty in the US. It does this by rounding up a bunch of little people, and penning them in various corrals that seem to have something to do with states or demographic groups. Hard to tell, since it is an expository jumble.

Let us ask a few questions of this graphic. First, the question that we ask of every such graphic: does the point leap out at you, in a flash of effortless cognition? Uh…lets see, half the people in poverty live in New York, and half in Texas? Fail!

Some more questions. If the orange little people are women and girls, why are they all wearing men’s business suits, albeit in a saucy feminine color? And do all the impoverished women and girls live in Texas? Rick Perry, are you aware of this? The state could at least provide more appropriate apparel for those in need. If you are a woman or girl, going to a job interview in an orange men’s business suit is not advisable, especially in Texas.

There seem to be a lot of impoverished white people (31.7 million), but amazingly, none of them live in Texas or New York. And if you think that is amazing…wait for it…none of them are men, boys, women, or girls. Maybe they are little people.

Ok, but here is where it really gets crazy. There are 16.4 million aged 17 or younger in poverty. But evidently none of them are girls or boys!

What is the lesson? The little people are no substitute for clarity of expression. The artist is to be commended for attempting to make the numbers more meaningful, but the exercise is doomed from the start. First of all, there is a fundamental difficulty in trying to carve up a total population (those in poverty) into a large number of overlapping sets. To be an accurate depiction, the corrals (technically, we call these Venn diagrams) should contain the correct number of little people, but so also should the intersections between two or more corrals (e.g., Asian and male and living Texas). Easier said than done (and it wasn’t that easy to say). Second, comparisons with state populations are problematic, since most americans have only a dim sense of the population of any state, even their own.

As is so often the case, traditional methods of data representation are perfectly adequate, and much clearer than the sad corrals of little people. Below is my quick draft of a bar chart of the same data. I have used different colors to group the different sorts of comparisons (gender, age, ethnicity), and as sop to the New York Times, included horizontal lines indicating populations of a few states (source

I hope you will agree that though my chart may be conventional, it is clear, and allows the viewer to make the comparisons that the Times felt were important.

The lesson? Beware the invasion of the little people. They look cute, and you figure they are so small they can’t do any harm. But invite them into your graphic, and they can create havoc. Advanced lesson: Venn diagrams are tricky to depict when many categories are involved.


New York Times

The Impoverished States of America


Published: September 17, 2011

State populations in my chart:

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