## Monday, May 26, 2014

### Poor Comparisons: circles or spheres

Here is a post from the Wall Street Journal's Numbers Guy from earlier this month. It purports to show the number of Major League Baseball pitchers that have undergone Tommy John surgery to repair their torn elbow ligaments. I don't dispute the numbers, but the visual is wrong. We are, I suppose, to imagine baseballs whose size represents the numbers displayed. The ratio of the number of surgeries from 2005-14 to those of the previous decade 1995-2004 is 181/87 = 2.08. The graphics should agree. They don't. The ratio of the diameters of the circles for these decades is 1.8. If we interpret them only as circles, then their areas would have a ratio of 1.82 = 3.24, that is, more than three of the red-colored 1995-2004 circles can fit inside of the blue-colored 2005-2014 circle, not ratio of 2.08 given by the numbers.

But it gets worse. The graphic designer has displayed baseballs, keeping with the theme. Viewed as balls we are now would be visually comparing volume not area. The ratio of the volume of the baseballs for these two decades would be 1.83 = 5.832.

One would have thought that such well known and documented graphical mistakes would be in a catalog of standard bad examples to avoid. Such examples are famous, this one from the Washington Post in 1978.
Here the graphic artist scaled both the height and the length of each dollar bill by the ratio of the purchasing power for each year relative to 1958, resulting in an area of the dollar that visually represented the square of the ratio, shrinking the visual impression even more. Compared to a dollar in 1958, the Carter dollar, was labeled as 44 cents, indicating that the dollar's purchasing was less than half of what is was is 1958. But since both the dimensionas of the bill was scaled by 0.44, the area of the Carter dollar is now 0.1936, that is more than 5 Carter-sized dollars would fit in the Eisenhower dollar. This leaves the visual impression that the purchasing power of the Carter dollar was less than 20 cents.

Still more examples are in the chapter on The One-Dimensional Picture in the old standard How to Lie with Statistics by Darrell Huff.