Proofiness - Charles Seife [14]
There are endless ways to polish mathematical apples; it would be impossible to describe them all, especially since inventive fruit packers are inventing new ones all the time. But there are a few common tricks worth mentioning.
Graphs—visual depictions of data—are particularly vulnerable to apple-polishing. A fruit packer can choose to display data in an endless variety of ways, fiddling with the look so that the graph makes the data look more impressive than they actually are.
Take the case of Quaker Oats. It’s a bland and relatively unappetizing product—not easy to come up with an ad campaign for. Yet people will eat anything that they think will improve their health, so ad executives launched a blitz to make the barely digestible oat fiber appear to be a medicinal vacuum cleaner, sucking the cholesterol right out of your bloodstream. They emphasized the point with a graph:
Figure 1. A deceptive graph about Quaker Oats.
The message was clear: eat Quaker Oats, and within a few weeks your cholesterol levels will drop dramatically. However, if you look carefully, you will discover that this graph is deceptive. We normally assume that the line at the bottom of the chart represents zero cholesterol—the little oat-fiber machines have gobbled up every single dollop of cholesterol in your blood. But if you examine the vertical axis of the chart, you see that the bottom isn’t zero, but 196. This makes the data seem much more dramatic than they actually are, as you can easily see if you look at a more honest graph:
Figure 2. A less deceptive graph about Quaker Oats.
By tweaking the scale of the chart just so, Quaker Oats made it look as if oatmeal was having a huge effect when it wasn’t. (After receiving complaints, Quaker withdrew the chart.) Of the many ways to manipulate data in graphs, this is probably the most common.
Another form of apple-polishing exploits the term “average” to make numbers seem smaller or larger than they really are. Most people think that “average” means “typical”—that if, say, the average salary at a company is $100,000, then each employee earns $100,000, more or less. In fact, that’s often not the case.
The average of a set of numbers—more precisely, the mean—has a precise mathematical meaning: you add everything together and then divide by the number of data points that you added together. For example, if you had a company of ten people, each of whom earned roughly $100,000, you add those ten salaries together ($100,000 + $101,000 + $98,500 + $99,700 + $103,200 + $100,300 + $99,000 + $96,800 + $100,000 + $101,500 = $1,000,000) and then divide by the number of salaries ($1,000,000 ÷ 10 = $100,000). In this case, the average, $100,000, does in fact represent a typical salary. However, consider a company where the CEO earns $999,991 per year, and there are nine interns who each earn $1. The mean, again, is the sum of those salaries ($999,991 + $1 + $1 + $1 + $1 + $1 + $1 + $1 + $1 + $1 = $1,000,000) divided by the number of salaries ($1,000,000 ÷ 10 = $100,000). So here too the “average” salary is $100,000. However, $100,000 is not a “typical” salary in any meaningful way. If you were to pick a person at random from the company, you’d probably find that he earns a measly $1. So in this case, it’s deceptive to pretend that “average” is “typical.”14 If the CEO were to recruit new employees by highlighting the company’s average salary of $100,000, he would be apple-polishing. The new hire would be shocked when he gets his first paycheck.
Whenever a politician announces a tax cut, it’s almost guaranteed that he’ll pull the exact same stunt to make the tax breaks look larger than they actually are. He’ll give a speech that talks about the “average” refund—the mean tax break