Proof by picture

One picture is worth a thousand lies

Whenever I teach statistics, I make a point of recommending to my students that they read Darrell Huff's How to Lie with Statistics, a splendid little book on the use and misuse of data. Huff warns his readers about many statistical sleights of hand, including the misleading graph.

I was reminded about Huff's cautionary tales while browsing Tim Lambert's Deltoid. His recent posts include a lovely example of how a climate-change skeptic uses a doctored graph to argue that recent global warming fits beautifully into a long-term cyclic pattern. All you have to do is screw with the time scale and suddenly any suggestion of human-induced temperature seems to vanish. It's all natural! (Lambert points us toward Stefan Rahmstorf's dissection of the bogus graph over at RealClimate. Stefan reads German so you don't have to.)

There's an old math joke whose punch-line relies on our fondness for different kinds of graph paper. Almost everyone is familiar with the good old Cartesian system: a rectangular grid of equally spaced horizontal and vertical lines. Cartesian graphs are relatively simple and highly functional, but get a little unwieldy when we try to graph things that change very rapidly or very slowly. In the graphs below, I have depicted two ways of illustrating the behavior of the curve y = 2^x. Exponential functions have extremely high growth rates, requiring us to severely compress the scale on the y axis. That's what you see in the left-hand graph. If, however, we use a logarithmic scale on the y axis (which we call a semi-log graph), the nature of our exponential graph is transformed. We get a nice straight line, On semi-log paper, the graphs of exponential functions become much neater.


There are many other forms of graph paper, too, including log-log paper, where both axes are in logarithmic scale. A good choice of scales can make a big difference in the clarity with which your functions or data are illustrated.

And that old math joke I mentioned? It's in the form of a riddle, sort of:

Q: How do you graph a linear function?
A: As a straight line—on Cartesian graph paper.
Q: How you you graph an exponential function?
A: As a straight line—on semi-log graph paper.
Q: How do you graph an arbitrary monotonic function f(x)?
A: As a straight line—on f paper!

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Hilarious, right? I'll give you a moment to recover from your fits of helpless laughter....

I imagine that most people have never seen a real-life example of f paper, but today's climate-change denialists may be in the forefront of exciting new developments in lying with statistics. In their honor, I now present my own modest contribution to the practical application of f paper. You will observe that the left-hand graph immediately below depicts a function that is both increasing and oscillatory. With the proper use of f paper, as shown in the right-hand graph, we can damp out any vestige of the oscillation, preserving only the monotonic increase. Unless one takes a hard look at the scale on the y axis (and now you know why one should always examine the axes carefully!), the embedded periodic motion of the function is completely suppressed. The right-hand graph is a breakthrough in information hiding. I fear it will not be the last example you see.