Occasionally, things go exactly as I’d hoped. We’re discussing scaling in my Physics of Life class, starting with things like the scaling of volume and area with size. I mentioned in passing that this issue comes up in advertising, and since students seemed interested, I brought the following to the next class — an interactive example adapted from Edward Tufte’s classic The Visual Display of Quantitative Information:
Inflation, the students hopefully know, refers to the change in purchasing power of a currency over time. Tufte shows a political ad in which the evils of Carter-era inflation are graphically depicted:
I asked the students, “Just looking at the images: A dollar in the bottom year is worth X times as much as a dollar in the top year. What’s X?”
The first three responses were 1/2, 1/3, and 1/4, so I made these the options for a clicker question for the whole class and then polled them. Here’s the outcome:
Two thirds of the class would assume, given the image, that the purchasing power of the Carter era dollar was ~1/4 that of the Eisenhower dollar — a very reasonable response. The true value:
So, I asked, were the makers of the ad being dishonest? The first few responding students guessed that the images were simply unrelated to the values, or that they were deliberately mis-scaled. I replied that there’s a way the makers of the ad could state that they were completely, perfectly honest. Then, a student cleverly suggested that the linear dimensions differ by 0.44. In other words, the length of the small dollar is 0.44 x the large one’s length, the width is 0.44 x the large one’s, and so the area is 0.44 x 0.44 = 0.19 x that of the large one! (You can measure the dollar images yourself and see that this is really the case.) So it’s a perfectly honest data visualization, but one that exploits scaling as well as the difficulty of accurately perceiving areas and lengths to manipulate the viewer. Watch out!