## Statistics: Love �em or Leave �em

September 21, 2009

�Rose of the Crimean War.� Florence Nightingale�s famous 1858 �coxcomb� diagram showing the variation in mortality over the months of the year, as redrafted by Hugh Small; http://www.florence-nightingale-avenging-angel.co.uk/Coxcomb.htm. From Picturing the Uncertain World.

Book Review
Philip J. Davis

Picturing the Uncertain World: How to Understand, Communicate, and Control Uncertainty through Graphical Display. By Howard Wainer, Princeton University Press, Princeton, New Jersey, 2009, 280 pages, \$29.95.

Is the world really uncertain, or is it merely our view and interpretation of the world that are uncertain? After all, the world is what it is. This may just be philosophical nit-picking on my part. Whatever the world is, there is no doubt that who, what, and where we are in it seem now to be defined statistically. The Statistical Abstract of the United States has been compiled yearly since 1878, and in recent years statistics have become our national and personal IDs.

I received my introduction to the world of mathematical statistics in the classroom of Professor Edward V. Huntington, a logician, statistician, table maker, and proposer of a method for reapportioning seats among the states in the U.S. House of Representatives after a new census---a topic that is still controversial. Huntington used no textbook; he passed out mimeographed notes. His notes were very, very long on formulas and their derivation and short on real-world examples. We students cut our teeth on Huntington's analysis of the weights of twenty sea-serpents. Perhaps the sea-serpents were covers for Washington opponents of his reapportionment algorithm.

Judging from its title, Wainer's book is a description of how graphics---meaning graphs, charts, other kinds of visuals---are able to provide enormous clarification of statistical data that might originally have been compiled in long lists or tables. One can agree promptly that this is so. The difficulty comes in interpreting what one sees and in wondering who gathered the data, how it was gathered and for what purpose or agenda, what surrounding factors were ignored or omitted, how much was faked, and so forth. Faked? Yes, the reader learns about the practice of "curbstoning" among census takers,

"a practice by which a census enumerator fabricates a questionnaire for a residence without actually visiting it. This practice is sufficiently widespread to warrant inclusion in the glossary of The 2000 Census Interim Assessment [National Academy Press]."

In brief, what are the objective and what are the subjective aspects of a statistical investigation?

But take comfort from the optimistic subtitle of the book: How to Understand, Communicate, and Control Uncertainty through Graphical Display. Presentation, as the maitre d' of an up-market restaurant once told me, is 50% of the game. According to Howard Wainer, if you can picture uncertainty appropriately, you'll be on the right interpretive track. But, backpedaling a bit, he admits that pictures can also be misleading.

Wainer is a member of the National Board of Medical Examiners and an adjunct professor of statistics at the Wharton School of the University of Pennsylvania. He has written many books and received numerous honors. In this book he has pulled together data relating largely to social or medical situations, often historical (Napoleon's march to Moscow, the Holocaust), often with the original historical graphics beautifully re-produced in color.

Wainer cuts a very wide swath: household incomes, voting dilemmas, SAT scores, mathematical proficiency, morbidity data, crowding in homes in different countries, economic issues, the Boston Marathon, to name just of few of the issues pictured and discussed.

On one level, the book functions well as an adjunct to the books on the visual display of quantitative information by E.R. Tufte, the guru of good graphics. It contains bar graphs, 3-D pie charts, Florence Nightingale's "Rose of the Crimean War" diagram (Nightingale, a student of J.J. Sylvester, was the first female member of the British Statistical Association). We see the use of color gradations and, of course, graphs and scatter diagrams fitted out with straight line approximations. There are panty hose charts, box-and-whisker plots---I had probably seen instances of the latter but never knew that they had been thus baptized. Wainer discusses the uses, misuses, and appropriateness of each form, along with its history.

Though graphics may illuminate some dark statistical areas, they are not a presentational "must." Consider baseball stats, to which salaries and strategies are said to be linked and for which fans seem to have an insatiable appetite. In a televised game, stats fill the gaps when the action on the diamond is momentarily suspended. Baseball stats are not effectively presented in pie charts, panty hose charts, or whatever.

On a second level, apart from his pictures, Wainer comments incisively and at some length on various statistics-generated controversies. What are we to make of SAT tests and scores? Or to take an example from an earlier time: In 1936, Sir Ronald A. Fisher, a giant figure in biostatistics, said that Mendel's results on his peas were just too good to believe. In 1968, Bartel van der Waerden absolved Mendel, inferring

"that the too-good fit was not due to any dishonesty on Mendel's part, but rather to the arbitrary stopping rule that he was thought to have employed; he ran the experiment . . . until it matched his expected value and then he stopped."

Wainer concentrates on social statistics, and his book leaves me with the impression that this area of statistics---which goes back to Adolf Quetelet (1796�1874), if not earlier---can be a can of worms. Consider the following:

• The recent flap about Herrnstein and Murray's The Bell Curve, a book that presented a statistical explanation of variations in intelligence.
• The flap accompanying the well-publicized remarks about sex differences made by Larry Summers as president of Harvard. Summers was rebuked by biologist and 1995 Nobelist Christiane N�sslein-Volhard, who, in turn, has been rebuked by Wainer.
• The flap surrounding the views of Sir Ronald A. Fisher, a pipe smoker, relative to the cigarette/cancer association, a flap that is still alive and well on the Web. I believe that the point Fisher wanted to make is simply that correlation is not causation.
• The amusement that can arise from linear extrapolation. For instance: The Boston Marathon data for men for the period from 1890 to 2000 and for women from 1972 to 2000, when extrapolated linearly, lead us to conclude that by 2005 women should have run faster than men. Spline approximations of the same data, by contrast, show evidence of the times of both sexes flattening out (asymptosis!) to about 155 minutes for women and 130 minutes for men. (The winning time in the 2009 men's race was 2:08:42 = 128.7 minutes. Hooray for splines!)

For me, the real eye-opener in the assessment of social statistics is that the procedures used have often been questioned, which in some cases has resulted in litigation. Here are a few examples: The U.S. Supreme Court (well before settling the 2000 presidential election) had ruled on the kinds of census figures that can be used: direct head counts or sampling-based adjustments. The U.S. District Court for the Eastern District of Pennsylvania recently passed judgment on the time that can be allowed to disabled test-takers.

Statistical methods sometimes become enshrined in statutory law. The method of least proportions is used to allocate seats to the states in the House of Representatives. It was approved by the Supreme Court in Dept. of Commerce v. Montana, 503 U.S. 442 (1992). Another example is a multiple regression model used in an employment discrimination class action suit; it was approved by the Supreme Court in Blakemore v. Friday, 478 U.S. 385 (1986).

Jurimetrics, a journal in the relatively new and flourishing field that I recently dubbed "jurimath,"* includes discussions of issues that hang on statistics, issues that could lead to litigation and court decisions: When are racial and ethnic statistics relevant? When can causation be inferred from probabilities? What can be said about evidence obtained from large-scale screening of people?

The methods developed by jurimetricians, along with established precedents, can all be digested by legal eagles who know some mathematics. But litigation involving statistics can also be pushed forward by lawyers for whom a variance is the permission sought to open a liquor store next to a school, and a case might come before a judge for whom skewness and kurtosis are medical conditions.

Statistics! It's hard to live without them, and it's hard to live with them. Pictures, graphs, formulas notwithstanding, this is the conclusion that I reached after reading Wainer's stimulating book.

*See "Jurimath: the Flowering of an Ancient Field," SIAM News, May 2008.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at [email protected].

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