The Fallacy of the Sheep
…[T]he operator of a diner…offered rabbit sandwiches at a remarkably low price. When questioned about it, he admitted that he used some horse meat to keep his costs down. "But I mix 'em fifty-fifty," he avowed. "One horse to one rabbit."2
This fallacy gets its name from the following traditional riddle: Why do white sheep eat more than black ones?
Categories of things or events are compared in terms of absolute numbers or percentages of some characteristic without taking into consideration the relative sizes of the categories.
Figures provided by the National Highway Traffic and Safety Administration reveal some interesting facts about car accidents. Of all collisions that occur in the United States, approximately 52% occur within a 5-mile radius of home while an astounding 69% occur within 10 miles. …
Why do so many accidents occur so close to home?
… Broadly speaking, drivers tend to have a false sense of security when driving close to home. For example, drivers are less likely to wear their seatbelts when driving to the neighborhood Jiffy store. Another big factor is distractions. Whether it's talking on a cell phone, scanning the radio/Ipod or eating while driving, any little thing that diverts your attention from the road can open the door for a collision. …
Just because you're close to home doesn't mean the danger of a car accident is lower. In fact, you should be twice as cautious when driving in your neighborhood or down to the corner mini-mart.4
The fallacy of the sheep occurs when a comparison between classes―the white sheep and the black―fails to take into consideration the differences in size of the two classes. For instance, suppose that we compare the prevalence of a disease in two groups of people: R, which includes only right-handed people, and L which is comprised of left-handed ones. Suppose that we discover ten cases of the disease in R but only one case in L: what should we conclude? Perhaps there's something about being left-handed that's protective against the disease.
Now, suppose that we find out that there are a hundred people in class R, but only ten in L. Clearly, then, there's no reason to think that left-handedness is protective, nor even that handedness has any effect on the disease. If you failed to take into account the differing sizes of the classes and wrongly concluded that left-handedness protects against the disease, you would have committed the fallacy of the sheep.
There's a special danger of committing this fallacy when comparing statistics from the past and the present. The population of the United States, and of the entire planet, has been growing steadily for the past couple of centuries. As a result, comparisons between absolute numbers from the past and now are bound to be misleading unless the change in population is taken into account.
For instance, if you compare the absolute numbers of poor people between now and, say, fifty years ago, it will appear that poverty in America has grown. However, the rate of poverty has substantially decreased in the last half century5.
In this example, there are actually two comparisons:
- In the first comparison, the two classes being compared are car accidents that occur within five miles of home and, implicitly, those that occur farther away. The conclusion that we're expected to draw from the comparison is that it's "interesting" that 52% of accidents occur within five miles of home, while only 48% occur farther away.
- In the second, the two classes being compared are car accidents that occur within ten miles of home and those that occur farther away. It's supposed to be "astounding" that 69% of accidents occur within ten miles of home, while only 31% occur farther away.
Why should it be "astounding"? What's missing from these comparisons is any information on the amount of driving done in the four different classes. If roughly half of driving is within five miles of home and two-thirds within ten miles, then the former comparison would be barely interesting and the latter scarcely astounding.
Stephen Campbell cites a similar example:
The National Safety Council [NSC] informs us that…65 percent of all accidents occur within 25 miles of home. Using these facts, much public service advertising points out that accidents are not…limited to long trips. … What such advertising doesn't tell us, however, is that most driving is done within 25 miles of home…. No wonder there are more accidents under such conditions; there is more opportunity for accidents to occur. This omission could lead one to misconstrue such well-intentioned advertising and conclude that the wise thing to do is…always make certain you are more than 25 miles from home when driving…. Granted, that would be a pretty dumb conclusion to draw….7
An even dumber one, suggested by an accompanying cartoon8, is that whenever you're driving within 25 miles of home, you should drive as fast as possible to limit the amount of time you're in this frightening danger zone. Still dumber is another suggestion I've come across: you should move at least 25 miles away to get out of this dangerous area. However, the dumbest of all is based on a related statistic: 80% of crashes occur at less than forty MPH9, so you should drive over forty at all times, but of course especially when you're within 25 miles of home.
The problem with these statistics is that by themselves they tell us nothing about safety, for we need to know how much of our driving takes place within 5, 10, or 25 miles of home. If about 70% of driving took place within ten miles of home, that would suggest that distance from home doesn't really affect safety. In contrast, if three-quarters or more of driving is that close to home, that would indicate that it's actually safer to drive near home than farther away. Only if significantly less than 70% of one's travel by car were close to home could you conclude that it was less risky to drive far away.
To judge the comparative riskiness of driving close to home or far away, what we need to know is not what percentage of accidents take place at these distances, but the rate, which would be measured in something like number of accidents per mile/kilometer driven. To figure the rate, divide the total number of accidents close and far from home by the total mileage driven at those distances. In this way, the amount of driving done at varying distances would be taken into account in the denominator of the fraction. By comparing the rates we could determine whether it is riskier driving in our own neighborhood, far from it, or it just doesn't matter.
So, without this other figure to compare, we really can't conclude anything about the safety, or lack of it, of near-home driving. Despite this fact, the example treats the figures it cites as though they require some special explanation. It claims that drivers have a false sense of security and are more easily distracted while close to home, and that you should be twice as cautious when driving in your own neighborhood, which suggests that it must be twice as dangerous to drive there than away from your neighborhood. However, the statistics cited don't support such a conclusion.
Answer to the Riddle: White sheep eat more than black sheep because there are more of them.
- Stephen K. Campbell, Flaws and Fallacies in Statistical Thinking (1974), pp. 39-41, 174-175 & 198.
- Darrell Huff, The Complete How to Figure it: Using Math in Everyday Life (1996), p. 391.
- For another example, see: Ron Paul on Drugs, 11/26/2011.
- "Nearly 70% of Car Accidents Occur Within 10 Miles of Home", The Babcock Law Firm LLC, 10/21/2012. Paragraphing suppressed.
- For an example of this mistake, see: The Riddle of the Sheep, 10/13/2011.
- In the Analysis, I'm assuming that the percentages cited in the Example, which seem plausible, are correct.
- Campbell, pp. 103-104.
- Campbell, p. 104.
- Wayne Tully, "Seat Belt Use Statistics", National Driver Training, 6/7/2011.