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.



Less ink, more think

Occasionally I am asked to give a lecture on how to draw good graphs. While I am always tempted to drone on interminably about abstract principles such as minimalism, balance, and consistency, I have discovered that it is much more fun to criticize bad graphs, and to show how they can be improved. But how to quire a truly bad graph? Easy! Just use the defaults in Microsoft Excel!

Here is an actual example of a graph drawn with those defaults. The data are fictitious, but the ugliness is breathtakingly real. It is sometimes said (unfairly) that engineers lack all sense of graphical design, but I think they must have hired specialists to create something so painfully wrong. vss2011workshop.010 But what specifically is wrong, and how can we make it better? Michelangelo once said “I saw the angel in the marble and carved until I set him free.”  So here too we will chip away at the obscuring excess, to reveal the beauty that Microsoft tried to hide.

First of all, what purpose is served by the heavy black rectangle that surrounds the graph? It serves two purposes: 1) to obscure useful information, and 2) to waste ink. Let’s remove it.


Better, but still bad. Next we note that the quantity being plotted is identified in three separate places: the vertical axis label, a title above the plot, and a key to the left. Is this really necessary? I think not. Lets get rid of two of them. Of course a key can be useful when several quantities are plotted together, but not when there is only one. Likewise labels above a plot have their uses, but should be avoided when they are redundant with other information, such as the axis label. We remove the key and the title. Apart from reducing clutter, this substantially increases the area available for the useful parts of the graph.


Now we ask the question: what purpose is served by the gray background? It serves two purposes: 1) to reduce the contrast and thus visibility of the data points, and 2) to waste ink. Get rid of it!


Aaah…so much more cheerful and relaxing to look at! But a few troubling questions remain. For example, what purpose is served by those shadows behind each data point? Do they indicate some exciting three dimensional aspect to the data? Of course not. But they do serve two purposes: 1) to render ambiguous the actual locations of the data points, and 2) to waste ink! Please people, can all just agree to never, never, use little shadows to suggest that our data are floating above the page? Thank you. The corrected graph is below. We have removed the shadows and also changed the diamonds to discs for the very important reasons that 1) they are simpler, and 2) I like them better.


Next we note that graphs are usually employed to show a pattern or trend. This pattern is not communicated well by a set of individual points floating out there, each an island, entire of itself. Only connect! A line drawn between the points aids enormously in conveying the visual sense of the data.


Next we correct an obvious (except to the Microsoft designers) flaw: the axis number labels running through the middle of the graph. We move them where they belong: to the axis, outside the graph.


Now we are getting somewhere. It almost looks ok. But we can do better. Gridlines can serve a purpose – for example, to let the reader easily judge approximate values – but there is never a reason for them to be dark and heavy, and to mask the useful information in the figure. Lighten up! In fact, the gridlines should generally be as light as possible, and still be visible. In this example, we make one gridline a bit darker than the others, to identify the y = 0 line.

vss2011workshop.018Now we see that the data really stand out. But we can do better still. What remains to distract the eye from the data? Well we could try removing the gridlines altogether, and then there is no need for the top and right borders of the frame.


Next we ask: what is the purpose of the bold font on the axis labels? Of course, it is to waste ink. Using a bold font for your labels is like writing your emails in all upper case. It is the digital equivalent of shouting. Don’t do it. Use your indoor voice.


And finally (yes, finally) we can reduce the line weight of the remaining axes. All we really need is enough weight to see them, and note their positions.


Thus we arrive at our final graph. It is not particularly exciting, but the data are clear, the trends are evident, and there is little to distract the eye from the essential information. Clearly, not all graphs are this simple, and there are often reasonable justifications for more elaborate presentations. But it is often a good idea to start with the simplest possible presentation, and elaborate from there. 

We conclude with the motto of this presentation, and indeed of this entire blog:

“Less ink, more think.”

Size matters, but only as a ratio

The other day I received yet another breathless email demanding my urgent attention. This one trumpeted the outcome of a recent poll for “the generic Congressional ballot.” Oh, yeah, that ballot. I hope you have mailed yours in by now.

The poll outcome was illustrated with the graphic below.

In any case, the democratic advantage looked pretty impressive until my eye drifted over to the left hand edge, where I noticed that the bars began at 30%. Why start at 30%, rather than zero? TOO MAKE THE DIFFERENCE LOOK BIGGER! Forgive me for shouting but this is such an elementary error, or transparent subterfuge, that I can’t help but be exasperated.

The principle here is that you cannot appreciate the size of the difference between the two bars without knowing the absolute size of each bar. The difference between them is meaningful only as a fraction of the total. This is why a scale extending to zero is called a “ratio scale.”

But lest your eyes glaze over in anticipation of a boring lecture, let me illustrate the idea with a few more graphs. We take the same data shown above, and plot it several times, in each case changing only the starting point of the bars.

Which is “correct?” They all show the same data. The first one reproduces the original figure, starting at 30. But why not start at 40 (second graph) or even 42 (third graph)? That appears to show a gargantuan advantage for the blue party, but only because we can’t see the total lengths of the bars. The correct depiction is the last, starting at zero, which visually presents a much more accurate, and less impressive picture.

When should you use a ratio scale? The question has some depth to it, which we will not fathom today, but it is always the case that percentages should be plotted on a ratio scale.


Email from Democratic Congressional Campaign Committee

Received: June 22, 2012

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

Friends don’t let friends use bar charts

A recent article in the New York Times drew attention to the lack of economic mobility in the United States, as compared to Canada and much of Western Europe. The article was illustrated with a graphic, which we reproduce below. It illustrates, for the US and Denmark, the percent of men raised in each fifth (quintile) of the economic range who end up in each fifth.

This graphic is not terrible: with enough scrutiny you can probably figure out the point being made. But because we enjoy picking on the Times, we will explore its various failings just for fun.

First, if I have told you once, I have told you a thousand times: no bar charts! They are almost always inferior to a comparable point and line chart. They waste ink, obscure trends, and – most relevant here – make it hard to compare two quantities. Note the extremes to which the Times artist has gone: the the Denmark data are a fat light gray bar, while the US data are a superimposed thin dark shaft. This trick to display two quantities in one location violates several canons of graphology. The first is that it is not “equitable:” the two nations are not plotted with symbols of equal visual weight. The second is that it makes it hard to see trends. And the third is that, worst of all, it actually makes it hard to compare, at a glance, the data from the two countries.

Here is a roughly similar graph that uses boring old points and lines. Color is used to distinguish the two countries.

We immediately see two things. First, only the first panel shows interesting differences between the two countries. This is the graph for men raised in the bottom fifth, and it clearly illustrates the point of the article: most men raised poor stay poor, and this trend is much more severe in the US than in Denmark.

The authors might have done us a favor by pointing out that in a true “opportunity society,” in which everyone regardless of economic origins has an equal chance at success, all of the graphs should be flat at 20%. The middle three graphs approximate this ideal, but both the leftmost and the rightmost graphs are very non-flat. This shows that the poorest and the richest are the least mobile; only the middle classes approximate the ideal of equal opportunity. This is true in both countries, but more severe in the US for those raised in the poorest quintile.

It is interesting to look for a way to depict the mobility within a single country, that does not require five separate graphs. One solution is to plot the percentages as a surface or an array.

Here is an example. Here we represented the starting and ending quintiles by rows and columns. Each cell shows the percentage of men that started in a given quintile (row), and ended up in a given quintile (column).  We scale the colors so that a percentage larger than the ideal 20% is red, less than 20% is green, and the equal opportunity ideal of 20% is a neutral color.

In the US, it is evident that the two “hot spots” are the lower left and upper right corners. These are the too many folks born poor who stay poor, or who are born rich and stay rich, respectively. The Danes suffer only from too many born rich who stay rich; those Danes born poor appear to experience nearly perfect mobility.

What are the lessons?

  1. Avoid bar charts, especially when trying to depict the covariation of two quantities.
  2. Don’t use five graphs when one (or two) will do.
  3. Something is rotten in Denmark, but two things are rotten in the US.


New York Times

“Harder for Americans to Rise From Lower Rungs”


Published: January 4, 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:

All that glitters is not Silver

“Love is blind.”

So begins a teasing article in the New York Times Sunday Magazine, by Nate Silver, the current wunderkind of popular statistics.  “Popular statistics,” now that I think about it, is almost the definition of an oxymoron, and it is to Nate’s credit that he has made it possible to utter such a phrase without puzzlement. The gist of the article is that in the dating game you are more likely to get lucky on a wednesday night than on any other night of the week. The article is accompanied by a massive “infographic” that occupies more than half of a page.

Debate has raged over the years about “decoration” of graphs, and while I am obviously  firmly in the minimalist camp, I am not a wild-eyed fundamentalist. A little furbelow here and there is harmless, provided that it does not obscure or distort the data.

Regrettably, young Nate has been kidnapped by the graphic artistes at the Times, who have never met a graph that could not be obscured or distorted. Witness below their artsy creation.

Note that there is an overall graph, for the days of the week, and within each day, a graph for hours of the evening. From a visual point of view, the most prominent effect is the trend over days. What exactly is plotted by this larger graph, for days of the week? A little scrutiny will reveal that it plots: nothing! The top of each bar is offset from the actual data, for any hour, by bizarrely random amounts. This is not decoration, it is desecration.

But suppose we extract the data, and plot them correctly. For days of the week, which is the primary focus of the article, it might be sensible to take the average “score” over the evening hours, and plot that. If we do so, we get the graph below.

Wow! No wonder they call it hump-day! Look at that massive effect! Except of course, that a glance at the scale reveals that the needle, so to speak, has barely budged. A more correct rendition of the data, showing the variation as a fraction of the total score (a ratio scale), is shown below.

Umm…never mind.  For all practical purposes, every night is the same. The main point of the article is, how shall we put it, nonsense.

And what about the numbers for the different hours of the evening? Even though they are hard to see, at least they are big effects, right? Of course not. Here is the average score for the various hours of the evening, plotted on a ratio scale.

I don’t want to be Miss Grundy, and I know even serious statistics wonks need a night out every once in a while, but even if “love is blind,” Nate really ought to reconsider the artsy types he hangs out with. Whichever night it was, he didn’t get lucky.

New York Times MAGAZINE

Wednesday Night Is All Right for Loving


Published: June 3, 2011

Approaching the singles scene statistically.

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