John Paulos: INNUMERACY: Mathematical Illiteracy and Its Consequences

The calculation is similar to the one on false positive results in drug testing, and, like it, demonstrates that misinterpreting fractions can be a matter of life and death.

According to government figures released in 1980, women earn 59 percent of what men do. Though it's been quoted widely since then, the statistic isn't strong enough to support the burden placed on it. Without further detailed data not included in the study, it's not clear what conclusions are warranted. Does the figure mean that for exactly the same jobs that men perform, women earn 59 percent of the men's salaries? Does the figure take into account the increasing number of women in the work force, and their age and experience? Does it take into account the relatively low-paying jobs many women have (clerical, teaching, nursing, etc.)? Does it take into account the fact that the husband's job generally determines where a married couple will live? Does it take into account the higher percentage of women working toward a short-term goal? The answer to all these questions is no. The bald figure released merely stated that the median income of full-time women workers was 59 percent of that for men.

The purpose of the above questions is not to deny the existence of sexism, which is certainly real enough, but to point out an important example of a statistic which by itself is not very informative. Still, it's always cited and has become what statistician Darrell Huff has called a semi-attached figure, a number taken out of context with little or no information about how it was arrived at or what exactly it means.

When statistics are presented so nakedly, without any information on sample size and composition, methodological protocols and definitions, confidence intervals, significance levels, etc., about all we can do is shrug or, if sufficiently intrigued, try to determine the context on our own. Another sort of statistic often presented nakedly takes this form: the top X percent of the country owns Y percent of its wealth, where X is shockingly small and Y is shockingly big. Most statistics of this type are shockingly misleading, although, once again, I don't mean to deny that there are a lot of economic inequities in this country. The assets that rich individuals and families own are seldom liquid, nor are they of purely personal significance or value. The accounting procedures used to measure these assets are frequently quite artificial, and there are other complicating factors which are obvious with a little thought.

Whether public or private, accounting is a peculiar blend of facts and arbitrary procedures which usually require decoding. Government employment figures jumped significantly in 1983, reflecting nothing more than a decision to count the military among the employed. Similarly, heterosexual AIDS cases rose dramatically when the Haitian category was absorbed into the heterosexual category.

Adding, though pleasant and easy, is often inappropriate. If each of the ten items needed for the manufacture of something or other has risen 8 percent, the total price has risen just 8 percent, not 80 percent. As I mentioned, a misguided local weathercaster once reported that there was a 50 percent chance of rain on Saturday and a 50 percent chance on Sunday, and so, he concluded, "it looks like a 100 percent chance of rain this weekend." Another weathercaster announced that it was going to be twice as warm the next day, since the temperature would rise from 25 to 50.

There's an amusing children's proof that they don't have time for school. One-third of the time they're sleeping, for a total of approximately 122 days. One-eighth of the time they're eating (three hours a day), for a total of about 45 days. One-fourth of the time, or 91 days, is taken up by summer and other vacations, and two-sevenths of the year, 104 days, is weekend. The sum is approximately a year, so they have no time for school.

Such inappropriate additions, although generally not as obvious as that, occur all the time. When determining the total cost of a labor strike or the annual bill for pet care, for example, there's a tendency to add in everything one can think of, even if this results in counting some things several times under different headings, or in neglecting to take account of certain resultant savings. If you believe all such figures, you probably believe that children have no time to attend school.

If you want to impress people, innumerates in particular, with the gravity of a situation, you can always employ the strategy of quoting the absolute number rather than the probability of some rare phenomenon whose underlying base population is large. Doing so is sometimes termed the "broad base" fallacy, and we've already cited a couple of instances of it. Which figure to stress, the number or the probability, depends on the context, but it's useful to be able to translate quickly from one to the other so as not to be overwhelmed by headlines such as "Holiday Carnage Kills 500 Over Four-Day Weekend" (this is about the number killed in any four-day period).

Another example involves the spate of articles a few years ago about the purported link between teenage suicide and the game of "Dungeons and Dragons." The idea was that teenagers became obsessed with the game, somehow lost contact with reality, and ended up killing themselves. The evidence cited was that twenty-eight teenagers who often played the game had committed suicide.

This seems a fairly arresting statistic until two more facts are taken into account. First, the game sold millions of copies, and there are estimates that up to 3 million teenagers played it. Second, in that age group the annual suicide rate is approximately 12 per 100,000. These two facts together suggest that the number of teenage "Dungeons and Dragons" players who could be expected to commit suicide is about 360 (12 x 30)! I don't mean to deny that the game was a causal factor in some of these suicides, but merely to put the matter in perspective.

In this section are several addenda to earlier material in this chapter.

The urge to average can be seductive. Recall the chestnut about the man who reports that, though his head is in the oven and his feet in the refrigerator, he's pretty comfortable on average. Or consider a collection of toy blocks which vary between one and five inches on a side. The average toy block in this collection, we might assume, is three inches on a side. The volume of these same toy blocks varies between 1 and 125 cubic inches. Thus, we might also assume that the average toy block has a volume of 63 cubic inches [(1 + 125)/2 = 63]. Putting the two assumptions together, we conclude that the average toy block in this collection has the interesting property of having three inches to a side and a volume of sixty-three cubic inches!

Sometimes a reliance on averages can have more serious consequences than misshapen cubes. The doctor informs you that you have a dread disease, the average victim of which lives for five years. If this is all you know, there may be some reason for hope. Perhaps two-thirds of the people who suffer from this disease die within a year of its onset, and you've already survived for a couple of years. Maybe the "lucky" one-third of its victims live from ten to forty years. The point is that, if you know only the average survival time and nothing about the distribution of the survival times, it's difficult to plan intelligently.

A numerical example: The fact that the average value of some quantity is 100 might mean that all values of this quantity are between 95 and 105, or that half of them are around 50 and half around 150, or that a fourth of them are 0, half of them are near 50, and a fourth of them are approximately 300, or any number of other distributions which have the same average.

Most quantities do not have nice bell-shaped distribution curves, and the average or mean value of these quantities is of limited importance without some measure of the variability of the distribution and an appreciation of the rough shape of the distribution curve. There are any number of everyday situations in which people develop a good intuition for the distribution curves in question. Fast-food restaurants, for example, provide a product whose average quality is moderate at best but whose variability is very low (aside from speed of service, their most attractive feature). Traditional restaurants generally provide a product of higher average quality but of much greater variability, especially on the downward side.