This month, like every other month, there is no shortage of news stories involving the possible misinterpretation of numbers.
Here are three - on female happiness, containing fires, and satisfaction with medical insurance - along with, I hope, a bit of numerical clarification. There's also a little question on the World Series.
Declining Female Happiness?
A paper entitled "The Paradox of Declining Female Happiness" by Betsey Stevenson and Justin Wolfers, two researchers at the University of Pennsylvania, was recently published and its results were widely described in the media as showing that, in general, women were happier than men in the 1970's but that men are happier now.
This led some conservatives to blame feminism, some liberals to blame women's extra burdens, and others to plug into various stereotypes from men's reluctance to complain to women's hyperbolic emotionality. The only problem with these and many other speculations and assertions is that they purport to explain something that probably doesn't need much of an explanation other than statistical noise.
Most years between 1972 and 2008 the study chose a random collection at least 1,500 men and women of all ages and asked them a large number of questions. In particular, it asked them whether, all things considered, they were very happy, pretty happy, or not too happy.
If the results over the years are graphed, it's apparent that the percentages in the three categories jump up and down in no clearly discernible way. Commentators who pretend that the graphs indicate a steep and steady decline in happy women and a steep and steady increase in happy men ignore these fluctuations.
Differences Between Men's and Women's Percentages Are Minimal
In fact, the differences between the men's and women's percentages are minimal and could be attributable to almost anything at all or to nothing at all.
For example, for the years 1972, 1973, and 1974, the cumulative results (taken from the linguistics site language log and rounded to the nearest percent) were that 37 percent, 49 percent and 14 percent of the women declared themselves, respectively, very happy, pretty happy, and not too happy. The corresponding percentages for men during those years were 32 percent, 53 percent, and 15 percent.
As noted, the percentages have oscillated quite a bit since then, but for the most recent years, 2004, 2006, and 2008, the study found that 31 percent, 55 percent, and 14 percent of the women said they were, respectively, very happy, pretty happy, and not too happy.
The corresponding percentages for men during these years were 30 percent, 56 percent, and 14 percent. And if we compare the percentages in the late 80's to those today, there are essentially no differences between them.
Are these results really enough to warrant pundits' spouting off about the stress of too many choices, women's roles in the workplace, men's loutishness, and all manner of other issues?
Jumping out of the frying pan of sex differences into the fire of, well, fire, let's take a look at those ubiquitous news stories during the fire season that report fires consuming so many acres and being X percent contained.
I suspect that few people know what it means to say that a fire is 40 percent contained or how many acres are in a square mile, a unit of area with which most Americans would be more familiar.
The containment figure is determined by the fraction of the approximate perimeter of the fire that has a "fire line" cleared along it. So if the fire is estimated to have a perimeter of 20 miles and along 8 miles of that perimeter there is a fire line or trench that's been dug, then the fire is said to be 8/20 or 40 percent contained.
Often absurdly precise figures for the percent of containment are given, but the figure is very approximate since the fire and hence its perimeter could be growing rapidly.
And there are 640 acres in a square mile. My guess is that only a tiny fraction of Americans know this, which makes the media's use of the term unnecessarily, but possibly intentionally, alarmist.
What sounds worse: In the hills there are 23,000 acres ablaze or there is a roughly square area 6 miles on a side that is burning.
A World Series Question
It's October, so let me interpose here a question on the World Series. The answer appears at the end of this column. Imagine that the opposing teams are evenly matched and that the probability of either one of them winning any given game is 50 percent.
In a best of seven series, whichever team wins four games is the winner and so the series might end in 4 games, in 5 games, or may take 6 or 7 games. Without conducting any sort of calculation, determine if the series is more likely to end in 6 or 7 games.
Overstating Satisfaction with Medical Insurance
Moving on to another incendiary issue, I note that opponents of health care reform often cite surveys showing that most people are quite satisfied with their insurance and medical care.
The satisfaction rate is likely to be artificially high, however, since relatively few of the people surveyed are seriously ill and so they don't place any significant demands on the system.
When surveys are taken, it might be better to focus on this more telling reference class, those seriously ill people who've actually needed considerable help from their insurance companies or who were not insured at all.
By analogy people with no strong opinions about anything are likely to think that the right to free speech is secure. Just as it's crucial to uphold the right to free speech of someone who is actually saying something unpopular, so it is to guarantee the right to health care of someone who is actually very sick.
Relevant to this last is a recent study by researchers at Harvard who estimate that there are approximately 45,000 deaths annually due to lack of insurance coverage. There are a number of methodological problems in conducting such a study, but a previous commonly cited estimate was 18,000 annual deaths.
Numbers Are to Be Debated, Deconstructed
The Harvard study noted that the risk of death for those without health insurance was 25 percent greater than for those with insurance in 1993, but that the figure has risen to 40 percent now.
A partial explanation is the uninsured have a harder time getting treatment, which has improved over the years, because strapped hospitals have reduced their services to the poor.
The bottom line for these as well as most numerically flavored stories in the news is that numbers can and should be examined, deconstructed, debated, and put into context.
Answer to World Series question: For the series to end in 6 or 7 games, it must last at least five games, at which point, one team, call it A, will have won 3 games and the other team, B, just 2.
If A wins the 6th game, the series is over after 6 games, but if B wins the 6th game, the series moves on to the 7th game. Since these possibilities are equally likely (remember A and B each have a 50 percent probability of winning any given game), it's equally likely that the series will end in 6 or 7 games.
John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of the best-sellers, "Innumeracy" and "A Mathematician Reads the Newspaper," as well as (just out in paperback) "Irreligion: A Mathematician Explains Why the Arguments for God Just Don't Add Up." His "Who's Counting?" column on ABCNews.com appears the first weekend of every month.