K-12 Math Curriculum — Where's the Statistics?by StarbuckOver the long stretch of time spent in primary and secondary school, children are typically not taught the skills that would enable them to assess the validity or invalidity of the ocean of statistical arguments they will encounter over the course of their life. In the areas of medicine and financial investment, people are presented with statistics designed to influence important personal decisions. Newspaper and magazine polls abound and gradually hack away at our sense of judgment. In court trials, juries are presented with statistical arguments that may or may not be sufficiently disputed during the proceedings. Government policies on economics, education, social welfare, pollution control, infrastructure repairs, and military operations are sometimes defended on the basis of statistical claims. Individual citizens need to be able to understand the use of statistics in the real world, or at least be able to understand it better than they do at the present time. You might object that statistical theory is too hard for people to learn. Well, learning to read and write one's mother tongue is wildly difficult, but children still learn to do it. Let me make a parallel. A few hundred years ago, some of the few people who could read and write, the scribes, wrote down the laws, became experts in the law, and greatly influenced the detailed development of laws. (From these scribes descend our modern-day lawyers.) The majority of the people at that time, not being literate, were relatively powerless when confronted with the legal system. People nowadays are much better off: they at least are able to read contracts and consult the wording of written laws. The parallel (obviously) is that presently those who can handle statistics are able to use it to create a justification (or pseudo-justification) of their claims. But modern-day citizens, being relatively innumerate, find themselves defenseless in many important areas when faced with claims based on the use of statistics. We want to advance to the point where ordinary people can take apart and check out the statistical arguments thrown at them. Let's look at a statistics problem. Statistical Pill-pushingYou're old. You go to the doctor for a checkup. He brandishes his
calculator and says, "Let's look at this question scientifically.
I enter your age and your category based on your medical history
and There's a point-of-view problem here. So of 100 non-X-takers, about 25 will die within 5 years, and of 100 X-takers, about 20 will die. Who would care about these numbers anyway? Conceivably, a doctor with a lot of patients. Maybe an insurance company or a funeral parlor. But what good does that information do for an individual patient like you? Not much. The information given is the answer to the wrong question (how many of some group will die). What you want to know, to balance off the known risks, is what taking X would do for you: if you take X, how many more years will you live than if you don't take X? So let's look at some raw data on this business of dying vis-a-vis taking or not taking X. Do I have this data? No, and it might be hard to get hold of. So I'll make up numbers. But they'll obey the required 25%/20% split. This will at least give us something concrete to think about. ------------------------------ 100 People Not Taking X: 1 year : 1 person 2 years : 2 people 3 years : 4 people 4 years : 8 people 5 years : 10 people 6 years : 18 people 7 years : 29 people 8 years : 18 people 9 years : 7 people 10 years : 3 people ------------------------------ 100 People Taking X: 1 year : 0 2 years : 1 person 3 years : 2 people 4 years : 7 people 5 years : 10 people 6 years : 20 people 7 years : 30 people 8 years : 20 people 9 years : 6 people 10 years : 4 people------------------------------ First of all, note that But what if we calculate the average life expectancy in the two cases? For the 100 non-X-takers, the average life expectancy is For the 100 X-takers, the average life expectancy is So if you decided to take X, your increase in average life
expectancy would be On that basis, despite the risk of side effects, would you take X for the rest of your life in the hope of gaining 12 weeks? I don't know, but at least you would be making the decision based on a result that reflects your point of view — what would X (probably) do for you, not how many people might live or die in some group. Beyond that, there are a couple of other things to look at. First of all, the numbers in the above tables obey the 25%/20% restriction, but beyond that, they're all made up. As long as you satisfy the 25%/20% restriction and keep the number of people involved the same (100), you can change all the numbers in the two tables. Then of course the two calculated life expectancies will come out different. What does that tell us? That the doctor's 25%/20% announcement, by itself, tells you nothing about your increase in average life expectancy. You would need to see the raw data, or at least you would need for the computer program massaging the numbers in the raw data to answer the right question. (By the way, I was personally handed an x%/y% declaration like this a few weeks ago — this basic scenario is not made up.) The other thing to look at is that 5-year-point dividing line. What if a different break point had been chosen? From the data in the tables we can calculate the following different total deaths for non-X-takers and X-takers after a certain number of years: 3 years : 7%/3% 4 years : 15%/10% 5 years : 25%/20% 6 years : 43%/40% 7 years : 72%/70% The 3-year point has dramatically different percentages (after all, 3% is less than half of 7%), but both the actual death rates at that point are pretty small, so that wouldn't impress the patient much. And the 7-year point would be a bad choice from a marketing point of view, since 72% and 70% don't look much different. So 5 years looks like the best break point for selling purposes, with its 25%/20% contrast. If by chance you're a mathematician, you're unhappy with the above analysis: the data were fake, the time span was short, all patients conveniently died within the allotted interval, and there was no analysis of the uncertainties in the two averages. Well, you're right. But the point here was to contrast the question answered by the physician with the question that should have been answered. Point-of-view difficulties (like the one above) occur from time to time in statistics-based arguments. Another common example: when a school advertises its average class size, it will probably be the average class size a typical professor sees rather than the noticeably larger average class size a typical student sees. Final RemarksHow much statistics could be introduced at the primary-school level is a matter of argument, but certainly the groundwork has to be laid: working probability problems, graphing data, understanding and calculating averages and medians, understanding dispersion and its effects on the uncertainty of an average, and looking at concrete examples of errors created by small sample size. At the secondary level, a statistics course based on a wide range of case studies could prepare people to confront the statistical arguments they will see throughout their life in a manner that will give them more control over the decisions they're forced to make. For further reading on this issue, there's an analysis available on the web, Helping Doctors and Patients Make Sense of Health Statistics by G. Gigerenzer et al, which is worth reading. Gigerenzer concentrates on the public's lack of understanding of medical statistics, but of course that's one of the areas of major concern. The first 6 pages are worth reading for Gigerenzer's discussion of three concrete examples; in the remaining 38 pages he documents the incidence, consequences, and causes of statistical illiteracy and discusses educational cures for the problem. A very readable book on this issue is the recently published Proofiness: the Dark Arts of Mathematical Deception by Charles Seife (Viking 2010, ISBN 0670022160). It's true that this book deals with various procedural problems in addition to specific misuses of statistical arguments. Nevertheless, Seife presents highly readable accounts of statistical shenanigans in the areas of advertising, journalism, medicine, nutrition, polls, elections, economics, and courtroom trials, and the focus is always on real-world data. For more articles, see Want to learn how to play & win at chess, sudoku, poker, crossword puzzles, scrabble, checkers, or bridge? See
Last updated March 26, 2011
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