Innumeracy: Mathematical Illiteracy and its Consequences
by John Allen Paulos
Innumeracy refers to mathematical incompetence. Given the frequency of misleading social media memes that simply don’t add up, I’d say the book is as relevant today as it was when published in 1988.
“If the headline reads that unemployment declined from 7.1 percent to 6.8 percent and doesn’t say that the confidence interval is plus or minus 1 percent, one might get the mistaken impression that something good happened. Given the sampling error, however, the ‘decline’ may be nonexistent, or there may even be an increase.”
One of the topics covered in this book is probability. Obviously, if you had 367 people in a room, there would be a 100 percent probability that at least two of those people share a common birthday. How many people would you need to have a 50% probability of this happening? “The surprising answer is that there need be only twenty-three.” The book explains how this was calculated.
Also on the topic of probability, Paulos explains the gambler’s fallacy, “the mistaken belief that because a coin has come up heads several times in a row, it’s more likely to come up tails on its next flip.” The probability of heads on the next flip is always 50 percent. “The gambler’s fallacy should be distinguished from another phenomenon, regression to the mean, which is valid.”
Here’s a puzzler. Babe Ruth has a higher batting average than Lou Gehrig in both the first half and second half of the season, but Gehrig has a higher batting average than Ruth for the season overall. How could this be? “What can happen is that Babe Ruth could hit .300 the first half of the season and Lou Gehrig only .290, but Ruth could bat two hundred times to Gehrig’s one hundred times. During the second half of the season, Ruth could bat .400 and Gehrig only .390, but Ruth could have come to bat only a hundred times to Gehrig’s two hundred times at bat. The result would be a higher overall batting average for Gehrig than for Ruth: .357 vs. .333.”
There is some math geek humor sprinkled in the pages of this book: “A man who travels a lot was concerned about the possibility of a bomb on board his plane. He determined the probability of this, found it to be low but not low enough for him, so now he always travels with a bomb in his suitcase. He reasons that the probability of two bombs being on board would be infinitesimal… Isolated but vivid tragedies involving a few people should not blind us to the fact that myriad prosaic activities may involve a much higher degree of risk.”
Paulos cites research by psychologists Amos Tversky and Daniel Kaheman which provides insight into some mathematically irrational decisions. In one study where participants were to imagine they are military generals, they made contradictory decisions simply based on whether otherwise identical questions were phrased in terms of lives saved or lives lost. “Tversky and Kahneman conclude that people tend to avoid risk when seeking gains, but choose risk to avoid losses.”
The author explains a survey method which makes it safe for people to answer anonymously. “Assume we have a large group and want to discover what percentage of them have engaged in a certain sex act, in order to determine what practices are most likely to lead to AIDS…. If the coin lands heads, the person should answer the question honestly… If it comes up tails, the person should simply answer yes… Since the experimenter can’t know what yes means, people presumably will be honest… Let’s say 620 of 1,000 responses are yes… Approximately 500 of the 1,000 people will answer yes simply because the coin landed tails. That leaves 120 people who answered yes out of the 500 who replied to the question honestly (those whose coins landed heads). Thus, 24 percent (120/500) is the estimate for the percentage of people who engage in the sex act.”
He also explains the capture-recapture method for estimating the number of fish in a lake. “We capture one hundred of them, mark them, and then let them go. After allowing them to disperse about the lake, we catch another hundred fish and see what fraction of them are marked. If eight of the hundred fish we capture are marked, then a reasonable estimate of the fraction of marked fish in the whole lake is 8 percent. Since this 8 percent is constituted by the one hundred fish we originally marked, the number of fish in the whole lake can be determined by solving the proportion: 8 (marked fish in the second sampling) is to 100 (the total number of the second sampling) as 100 (the total number marked) is to N (the total number in the lake).” So, we can estimate there are 1,250 fish in the lake.
A common error is assuming that correlation indicates causation. “Quite often, two questions are correlated without either one being the cause of the other. One common way in which this can occur is for changes in both quantities to be the result of a third factor… There are many purely accidental correlations. Studies reporting small nonzero correlations are often merely reporting chance fluctuations… Too much research in the social sciences, in fact, is a mindless collection of such meaningless data.”
In the chapter on pseudoscience, the author skewers such topics as psychics, astrology, and numerology. He explains the Jeane Dixon effect, “whereby the relatively few correct predictions are heralded and therefore widely remembered, while the much more numerous incorrect predictions are conveniently forgotten or deemphasized… People read into the generally vague astrological pronouncements almost anything they want to… They’re also more likely to remember true ‘predictions,’ overvalue coincidences, and ignore everything else… Both innumeracy and defective logic provide a fertile soil for the growth of pseudoscience.”
“There’s always enough random success to justify almost anything to someone who wants to believe.”
Paulos, John Allen. Innumeracy: Mathematical Illiteracy and Its Consequences. New York: Hill and Wang, 1988. Buy from Amazon.com