What the Numbers Say: A Field Guide to Mastering Our Numerical Worldby Derrick Niederman
A decade ago, computer scientist Douglas Hofstadter coined the term innumeracy, which aptly described the widespread ailment of poor quantitative thinking in American society. So, in What the Numbers Say, Derrick Niederman and David Boyum present clear and comprehensible methods to help us process and calculate our way through the world of “data smog” that we live in. Avoiding abstruse formulations and equations, Niederman and Boyum anchor their presentations in the real world by covering a particular quantitative idea in relation to a context–like probability in the stock market or interest-rate percentages. And while this information is useful toward helping us to be more financially adept, What the Numbers Say is not merely about money. We learn why there were such dramatic polling swings in the 2000 U.S. presidential election and why the system of scoring for women’s figure skating was so controversial in the 2002 Winter Olympics, showing us that good quantitative thinking skills are not only practical but fun.
–Arthur Levine, President, Teachers College, Columbia University
“Entertaining, interesting, and educational. A real winner!”
–Brian L. Roberts, President and CEO, Comcast Corporation
“This book is full of insights. With examples ranging from baseball to Social Security to the stock market, the authors make the numbers we live with everyday understandable and accessible. Read this book and you will probably double your numerical literacy. What’s more, you will enjoy the read!”
–Joseph S. Nye Jr., Dean, John F. Kennedy School of Government, Harvard University, and former Assistant Secretary of Defense for International Security Affairs
“It’s almost always about the numbers! What the Numbers Say is a witty and engaging book that highlights just how critical understanding the numbers is to our daily life.”
–David Bach, best-selling author of Smart Couples Finish Rich and Smart Women Finish Rich
“Numbers have always been important in daily living, but now the computer and the Internet have made them more indispensable than ever. The authors of this splendid volume show us how to cope with the numbers that bombard us from all directions. Do you understand sports statistics? Can you spot the flimflams in statistical claims about health and medicine? If you don’t want to be a dummy about quantitative information, then read this book. It will tell you all you need to know, in clear and entertaining prose.”
–Martin Gardner, author of The Colossal Book of Mathematics, and former “Mathematical Recreations” columnist, Scientific American
From the Hardcover edition.
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Read an Excerpt
The Quantitative Information Age
Whether we realize it or not, quantitative information pervades our professional and personal lives. Every time a doctor reads a patient's chart, a farmer weighs a hog, a businesswoman reviews a budget, or a driver glances at a speedometer, someone is seeking quantitative information. And thanks to computers and the Internet, numbers are spreading faster than deer populations. A few years back, a car shopper found it hard to get much in the way of guidance--a magazine article here, a friend's opinion there. Now, a little web surfing reveals sticker and invoice prices, dealer holdbacks and incentive packages, depreciation estimates, customer satisfaction ratings, reliability statistics, expected insurance costs, crash test results and various accident, injury, and fatality rates, not to mention a wealth of financing data. We may say we live in the "Information Age," but it might be more accurate to say we live in the "Quantitative Information Age."
As technology builds this crescendo of numbers, our ability to make wise decisions, whether at work or at home, increasingly depends on proficiency with quantitative information. Unfortunately, Americans seem much better at producing numbers than making sense of them. It was to underscore this point that cognitive scientist Douglas Hofstadter coined the term "innumeracy," a word brought into the national spotlight by John Allen Paulos's pathbreaking book of the same name.1 But identifying educational failures is far easier than remedying them, and while countless books, articles, and government reports have diagnosed the widespread ailment of poor quantitative thinking, they have not provided therapy. Or at least not much of it. That's where this book comes in. Our goal is to introduce you to the quantitative concepts, skills, and habits you need for success in work, and success in life.
Sounds dreadful, doesn't it? Yet another remedial math book. Well, if that's what you're expecting, we've got good news. It turns out that formal mathematics is not the best way to teach quantitative thinking, and superior quantitative reasoning is not restricted to those who aced high school math.
Imagine, if you will, the work of an accountant, green eyeshade and all. Few jobs involve greater contact with numbers, and quantitative skills are an obvious prerequisite. If an accountant isn't good with figures, he might not notice a liability that has been excluded from a company's balance sheet, or a depreciation charge that has been miscalculated. Yet how much advanced mathematics is required to assess numbers on financial statements? These numbers get added, subtracted, multiplied, divided, and displayed as fractions, decimals, and percentages. They don't get expressed as Gaussian integers. The last time we looked at a financial statement, we didn't see a --, a , a u, or a $.
Think about other professional occupations--architects, doctors, management consultants, financial planners, marketing executives. Again, large amounts of quantitative information are a feature of many such jobs, and good quantitative thinking is critical to doing the jobs well. But matrix algebra is not required. Nor are high school staples such as quadratic equations, analytic geometry, and imaginary numbers.
This is why high school math teachers fear nothing more than the question, "Why do I need to learn this?" The conventional response, "Because you'll need the skill in your job," is dishonest, because even if the class contains some future engineers, they aren't the ones asking the question. The more honest reply, "Because you'll need it for the SAT," wins points for candor but loses them right back to overt cynicism. Thomas Jefferson's endorsement of mathematics--"The faculties of the mind, like the members of the body, are strengthened and improved by exercise"2--is closer to our idea of why the study of math is so essential, but we admit that such an argument will hardly motivate apathetic teenagers.
But if math isn't the key to good quantitative thinking, what in the world is inside this book? Here's a sneak preview. What distinguishes good quantitative thinkers is not their skill with pure mathematics, but rather their approach to quantitative information. Effective quantitative thinkers possess certain attitudes, skills, and habits that they bring to bear whenever they need to make decisions based on numbers. And with rare exception, these attitudes, skills, and habits are not taught in math classes or textbooks. For example, good quantitative thinkers demand empirical evidence instead of conventional wisdom. At the same time, they assume that figures are often wrong or misleading. Good quantitative thinkers don't examine quantitative information until they have a strategy for doing so. They know that some numbers are far more important than others, and they systematically pare data in an effort to find the most revealing figures. They also make lots of rough estimates, scribble on the backs of envelopes, and use arithmetic shortcuts like the Rule of 72. These are the skills we intend to convey.
Now for the most welcome news of all: In showing you the ways of good quantitative thinkers, we are confident that you already know all the math you need. We may jog your memory from time to time, but we're talking about the basics--arithmetic, percentages, fractions, decimals, square roots, and exponents. If you're still stumped every time you hear someone say, "That's six of one, a half dozen of the other," then perhaps you do need a remedial math book. But otherwise you've come to the right place and we're delighted to have you aboard. Be prepared to see numbers in a brand new light.
The Ten Habits of Highly Effective Quantitative Thinkers
When Shaquille O'Neal accepted the NBA's Most Valuable Player award in 2000, he quoted Aristotle: "Excellence is not a singular act, but a habit. You are what you repeatedly do." Classicists might quibble with Shaq's translation of The Nicomachean Ethics, but in our view he was right on the money. Excellence in anything is the product of practice. That's especially true of quantitative reasoning, which doesn't come naturally to any of us. It seems to be our fate to enter this world with lousy quantitative instincts, as if Adam miscounted the fruit on the tree of knowledge and forced all of us to suffer for his arithmetic sin. Or, to put the same thought in more secular terms, the remote ancestors whose struggle for survival shaped our genetic makeup faced an environment where quantitative skills were not especially important in solving the problem of finding a meal without becoming a meal.
Like the young tourist asking how to get to Carnegie Hall, someone asking the way to good quantitative thinking must be told, "Practice, practice, practice." But what are you practicing? Ideally, habits that foster effective quantitative thinking. Thanks to Stephen Covey's best-selling books (The 7 Habits of Highly Effective People and its offshoots), seven has become the canonical number for lists of good habits. However, for those born into a decimal system, seven is an awkward number. Neurologists test for dementia by asking patients to serially subtract 7, starting from 100, precisely because 7 doesn't create simple patterns. Highly effective quantitative thinkers understand the advantages of working with round numbers, particularly powers of ten: quicker calculations, fewer errors, better recall. That, and the examples of Moses and David Letterman, has led us to make a list of ten--or, perhaps, the top ten--habits of effective quantitative thinkers. Some of these habits are intertwined, but that's no problem. Because 10 is a composite number (unlike, for example, 7), we can display our 10 habits in five groups of two. Get ready to practice.
Attitude Is Everything
What makes someone a skilled quantitative thinker, above all, is the ability to effectively use quantitative information when drawing conclusions or making decisions. In that sense, numbers are a form of evidence. And as any experienced judge would tell you, when you're weighing evidence, attitude is all-important.
Habit 1: Only Trust Numbers
Accepting numbers as evidence is more difficult than it sounds. We tend to trust what we experience, and seeing a number on a piece of paper is not much of an experience. Numbers, after all, are just symbolic representations of quantities, quantities that themselves are often more abstract than real. Do you really understand what it means when an air conditioner has an energy efficiency rating of 10.7, other than higher is better? We certainly don't. But if you want to be a good quantitative thinker, you must learn to make decisions on the basis of numerical information, even when that information conflicts with your instincts and perceptions. How? Try to raise your level of trust in careful quantitative analysis, and reduce your confidence in hunches, theories, and casual observation. Sublimate your impulse to leap to conclusions, transforming it into an urge to seek hard data. And keep an open mind when the data don't go your way. In short, only trust numbers.
Enough pop psychology. We'll try to demonstrate why you should only trust numbers.
One of your authors suffered recurrent ear infections as a young child. The infections promptly came to an end when he began taking Sudafed on the advice of a leading otolaryngologist (the fancy name for an ear, nose, and throat doctor). His parents think Sudafed is a wonder drug. He thinks his parents are crediting the rooster for the sunrise.
The otolaryngologist no longer recommends Sudafed for ear infections. He stopped doing so when the numbers came in. Studies that randomly gave Sudafed to some children and sugar pills to other kids found that Sudafed did not affect the frequency or duration of infections. Unlike the otolaryngologist, your author's parents have had a hard time accepting this quantitative evidence. "Look, I know what I saw," they both claim, which is another way of saying that hundreds of unwitnessed outcomes reported by medical researchers are no match for one firsthand observation. Good quantitative thinkers would demand the numbers, but for some reason your author's parents left the Sudafed research to plead its case in the manner of Groucho Marx: "Who are you going to believe, me or your lying eyes?"
So what did put a stop to your author's ear infections right after he started taking Sudafed? Development and coincidence. Most children grow out of ear infections as their middle-ear structures develop and as their immune systems strengthen. So once a kid's susceptibility to ear infections has peaked, any treatment regimen will tend to coincide with reduced incidence of infection. Sudafed, bee pollen, shark cartilage, an extra bowl of Cap'n Crunch every morning--it all works.
Only trusting numbers is especially important when quantitative analysis clashes with people's irrational tendencies--like shortsightedness. When the U.S. military downsized in the early 1990s, thousands of servicemen and women were given the choice of a lump-sum severance payment or an annuity (payments made over a period of years). In comparative terms, the annuity was so favorable that it effectively offered a guaranteed annual return of 17.5 to 19.8 percent to those who declined the lump sum. That's guaranteed, as in risk-free.
Yet despite receiving pamphlets demonstrating the superiority of the annuity at prevailing money market rates, 92 percent of enlisted personnel and 51 percent of officers chose the lump sum. Granted, a small number may have had pressing financial needs that forced them to pass up what was likely the best investment opportunity they would ever have. However, the rest of those who took the lump sum were simply bad quantitative thinkers. They were shown the numbers, but they could not bring themselves to trust them.
Interestingly, Department of Defense economists, naively underestimating the power of immediate gratification to defeat sound quantitative reasoning, had predicted that roughly half of enlisted personnel and almost no officers would pick the lump sum. It's hardly reassuring to learn that the armed forces are teeming with unskilled quantitative thinkers. But look on the bright side. By shortchanging themselves, those who grabbed the up-front cash saved the rest of us taxpayers an estimated $1.7 billion.
By the way, in case our hyperbole has escaped you, we don't literally mean that you should only trust numbers. Of course, other forms of information, as well as verbal, spatial, and other types of reasoning, are also critical to making intelligent judgments. But to counteract the almost universal tendency to undervalue quantitative information and reasoning, Only Trust Numbers is a good place to start. Moreover, when you're first learning a new practice, it's often helpful to overemphasize it. So the next time someone makes an unsubstantiated assertion, yell, as if you're Cuba Gooding Jr. in Jerry Maguire: "Show me the numbers!"
Habit 2: Never Trust Numbers
Sorry to contradict ourselves so early in the game, but you should never trust numbers. Before we reconcile our apparently inconsistent advice, first let us explain why numbers are not worthy of your trust: It's because numbers can be wrong, are frequently misleading, and all too often have an agenda.
There are a host of reasons why numbers can be wrong. For starters, people lie and cheat. Some quantitative deceit is obvious, as when a teenager says "21" to a bartender, or a used car dealer pitches you a run-down, twelve-year-old coupe showing 38,538 on the odometer. Other deceit is harder to spot, as when a scientist doctors his experimental results, or respondents give inaccurate answers to survey questions.
Numbers are also wrong for more innocent reasons. People are slow to update databases--so, for example, the apple juice marked "Sale $2.49" on the grocery store shelf adds $2.99 to your bill when it's scanned at checkout. People misremember figures, bungle arithmetic, and punch the wrong keys. Computer programs (and some of the processors they run on) have bugs. Scales, radar guns, barometers, and other measuring devices are also imperfect, even when used and calibrated properly.
Even when accurate, numbers can easily mislead. Quantitative data are seductive; they invite us to engage in the risky behavior of reading more into data than is warranted. Between 1980 and 1987 the rate of newly diagnosed breast cancer cases increased by 32 percent in the United States. This was bad news, right? No, it was probably good news. Epidemiologists believe that most or all of the increase was the result of improved detection of breast cancer, a beneficial development. On a lighter note, Wilt Chamberlain made 54 percent of his field goal attempts during his storied career, while Larry Bird sank fewer than half of his tries. Was Wilt the better shooter? No way--Bird took much harder shots.
Still another reason to distrust numbers is that numbers are used to advance agendas. Tell yourself this: Every number I see is generated and presented by people who have an interest in how that number is used or interpreted. You have our permission to suspend this presumption if you're looking at the periodic table--as far as we know, there is no scheming behind the atomic weight of tungsten--but any other suspensions are taken at your own risk. Clearly, a cynical attitude toward quantitative information is appropriate when a Weight Watchers ad sports a woman who has lost fifty pounds. The fine print--"results not typical"--shouldn't surprise you in the least.
From the Hardcover edition.
Meet the Author
DERRICK NIEDERMAN received a B.A. in mathematics from Yale and a Ph.D. in mathematics from MIT. He is the author of This Is Not Your Father’s Stockpicking Book, The Inner Game of Investing, and A Killing on Wall Street. DAVID BOYUM received a B.A. in applied mathematics and a Ph.D. in public policy from Harvard. He has been a visiting scholar at the Yale School of Management and a Robert Wood Johnson Scholar in Health Policy Research at Yale.
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