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NUMBERS RULE YOUR WORLDTHE HIDDEN INFLUENCE OF PROBABILITY AND STATISTICS ON EVERYTHING YOU DO
By KAISER FUNG
McGraw-HillCopyright © 2010 Kaiser Fung
All right reserved.
Chapter OneFast Passes / Slow Merges
The Discontent of Being Averaged
Meter mystery If no one likes, why obey? One car per green, please —Haiku about the Minneapolis–St. Paul commute by reader of the Roadguy blog
Heimlich's Chew Chew Train Good film, big buildup, nice queue Twenty-second ride —Haiku about Disney by Anonymous
In early 2008, James Fallows, longtime correspondent at The Atlantic, published an eye-popping piece about America's runaway trade deficit with China. Fallows explained how the Chinese people were propping up Americans' standard of living. The highbrow journal has rarely created buzz on the Internet, but this article beat the odds, thanks to Netizens who scrapped Fallows's original title ("The $1.4 Trillion Question") and renamed the article "Average American Owes Average Chinese $4,000." In three months, Internet readers rewarded the piece with more than 1,600 "diggs," or positive responses, which is the high-tech way of singing praise. Evidently, the new headline caught fire. Our brains cannot comfortably process astronomical numbers such as $1.4 trillion, but we can handle $4,000 per person with ease. Simply put, we like large numbers averaged.
The statistical average is the greatest invention to have eluded popular acclaim. Everything has been averaged by someone, somewhere. We average people ("average Joe") and animals ("the average bear"). Inanimate things are averaged: to wit, after the terrorist attacks of September 11, 2001, a security dispatch demonstrated how to "weaponize the average water cooler." Economic processes are averaged, as when a market observer in early 2008 proclaimed "the new hope: an average recession," presumably predicting a shallow one that would depart with haste. Even actions cannot escape: when Barack Obama's lawyer interjected on a Clinton conference call during the heated Democratic primary elections of 2008, the media labeled the occasion "not your average conference call."
Can rare items be averaged? You bet. Forbes magazine told us, "The average billionaire [in 2007] is 62 years old." Surely no one averages uncountable things, you think. Not so quick; the U.S. Census Bureau has devised a methodology for averaging time: on an "average day" in 2006, U.S. residents slept 8.6 hours, worked 3.8 hours, and spent 5.1 hours doing leisure and sporting activities. It is a near impossibility to find something that has not been averaged. So pervasive is the idea that we assume it to be inborn and not learned, nor in need of inventing.
Now picture a world without averages. Imagine having the average child, the average bear, and the average such-and-so-forth punched out of our lexicon. We are dumbfounded to learn that such a world did exist once, before a Belgian statistician, Adolphe Quetelet, invented the "average man" (l'homme moyen) in 1831. Who would have thought: such a commonplace idea is younger than the U.S. Constitution!
Before Quetelet, no one had entertained the import of statistical thinking to the social sciences. Up until that time, statistics and probability fascinated only the astronomers who decoded celestial phenomena and the mathematicians who analyzed gambling games. Quetelet himself was first a distinguished astronomer, the founding director of the Brussels Observatory. It was in midlife that he set the ambitious agenda to appropriate scientific techniques to examine the social milieu. He placed the average man at the center of the subject he named "social physics." While the actual methods of analysis used by Quetelet would strike modern eyes as hardly impressive, historians have, at long last, recognized his impact on the instruments of social science research as nothing short of revolutionary. In particular, his inquiry into what made an able army conscript earned the admiration of Florence Nightingale (it is little known that the famous nurse was a superb statistician who became an honorary member of the American Statistical Association in 1874). In this body of work also lay the origin of the body mass index (BMI), sometimes called the Quetelet index, still used by doctors today to diagnose overweight and underweight conditions.
Since the concept of the average man has been so firmly ingrained into our consciousness, we sometimes fail to appreciate how revolutionary Quetelet really was. The average man was literally an invention, for the average anything did not, and does not, physically exist. We can describe it, but we cannot place it. We know it but have never met it. Where does one find the "average Joe"? Which "average bear" can Yogi Bear outsmart? Which call is the "average" conference call? Which day is the "average" day?
Yet this monumental invention constantly tempts us to confuse the imaginary with the real. Thus, when Fallows calculated an average of $4,000 debt to China per American, he implicitly placed all Americans on equal footing, spreading $1.4 trillion evenly among the population, replacing 300 million individuals with 300 million clones of the imaginary average Joe. (Incidentally, the Netizens mistakenly fabricated only 300 million Chinese clones, rhetorically wiping out three-quarters of China's 1.3 billion people. The correct math should have found the average Chinese lending $1,000 to America.) Averaging stamps out diversity, reducing anything to its simplest terms. In so doing, we run the risk of oversimplifying, of forgetting the variations around the average.
Hitching one's attention to these variations rather than the average is a sure sign of maturity in statistical thinking. One can, in fact, define statistics as the study of the nature of variability. How much do things change? How large are these variations? What causes them? Quetelet was one of the first to pursue such themes. His average man was not one individual but many; his goal, to contrast different types of average individuals. For him, computing averages was a means of measuring diversity; averaging was never intended to be the end itself. The BMI (Quetelet index), for good measure, serves to identify individuals who are not average, and for that, one must first decide what the average is.
To this day, statisticians have followed Quetelet's lead, and in this chapter, we shall explore how some of them use statistical thinking to battle two great inconveniences in modern living: the hour-long commute to and from work and the hour-long wait to get on a theme park ride. A reasonable person, when trapped in traffic or stuck in a long queue, will suspect that whoever was in charge of planning must have fallen asleep on the job. To see why this reaction misplaces the blame, we need to know a little about the statistics of averages. Working with engineers and psychologists, statisticians are applying this knowledge to save us waiting time.
* * *
To label Dr. Edward Waller and Dr. Yvette Bendeck Disney World die-hards would be an understatement. On October 20, 2007, they toured every last open attraction in the Magic Kingdom in just under thirteen hours. That meant fifty rides, shows, parades, and live performances. Buzz Lightyear's Space Ranger Spin, Barnstormer at Goofy's Wiseacre Farm, Beauty and the Beast—Live on Stage, Splash Mountain, Mad Tea Party, Many Adventures of Winnie the Pooh, you name it—everything in the park! Nice work if you can manage it, no? Disney buffs know this to be a mission impossible; they feel lucky to visit four major rides on a busy day, not to mention the nonstop walking required within the hundred acres of park area. Waller and Bendeck had help from Len Testa, who devised the Ultimate Magic Kingdom Touring Plan. Testa's plan lays out precise directions for reaching every attraction in the shortest time possible. He warns unsuspecting novices that it "sacrifices virtually all of your personal comfort."
Len Testa is a thirty-something computer programmer from Greensboro, North Carolina. As the patron saint of disgruntled Disney theme park-goers worldwide, he brought the gift of touring plans, which prescribe routes that guide patrons through a sequence of attractions in the shortest time possible. While the Ultimate Plan grabs attention, Testa creates touring plans for just about every need: for small kids, families, tweens, active seniors, grandparents with small children, and so on. He is mainly looking after rabid Disney fans, ones who are the most loyal—and easily the most demanding—customers. Sampling their typically breathless trip reports, posted on fan websites or relayed to journalists, one frequently comes across affectionate gripes like these:
"Going to Disneyland in the summer months is kind of like cruising to the Bahamas during hurricane season. You're just asking for it."
"You haven't lived until you've stampeded to Space Mountain as the opening rope drops, alongside thousands of stroller-wielding soccer moms at a full run." "When those gates spring open at 8 A.M., the weak and the semi-comatose will be left in the dust."
"We felt we spent more time in lines than on rides—the fact is, we did! When a wait in the line is ninety minutes and the ride is only five minutes, you have to question your sanity!!"
"I've never really forgiven my brother for that one time he slowed us down with an untimely bathroom break at Disney's Epcot Center five years ago."
These park-goers have plenty of company. Disney's own exit polls reveal long lines as the top source of customer unhappiness. Industry veterans say the average guest dawdles away three to four hours in queues during a visit lasting eight to nine hours; that's one minute of standing around out of every two to three minutes inside the park! Amusement Business estimated that the national average for wait time at major attractions in a theme park during the summer was sixty minutes—after which patrons get to spend two minutes on the ride. Since a family of four can spend $1,000 or more in a single trip, it is no wonder why some guests are irritated by seemingly interminable lines.
These trip reports leave vivid images of heroic maneuvers to avoid lines. A suitable attitude is required:
"When I'm in the parks, I'm a Daddy on a mission.... In the course of the afternoon, I'll go from one end of the park to the other and ride more rides, wait less in lines, and see more shows and parades than many other park patrons, with or without kids."
So are small sacrifices ...
"We manage to avoid long lines with an occasional early morning, and hitting popular attractions during parades, mealtimes, and late evenings."
... and knowing how to play the system ...
"The mother behind me told me that they had waited three hours to ride Dumbo during their last visit. [This time,] she took advantage of early admission to let her kid ride three times in a row with no waiting."
... and sweet-talking teachers into granting special permission ...
"Taking your kids out of school [to go to Disney]. Is it worth it? Yes!"
... and spotting opportunities that others give ...
"It does rain in Florida, especially during summer afternoons. The good news is that this tends to scare off some people. My advice: Buy bright-yellow ponchos for $5 each from any of the gift shops. Then keep those kids walking."
... while always adapting tactics:
"We are starting to think that reverse-reverse psychology might work: Disney opens one park earlier for all their guests so all the guests go to that park.... [Everyone else avoids that park, and] therefore we can go to that park because people think it is going to be packed and they avoid it."
Queues happen when demand exceeds capacity. Most large rides can accommodate 1,000 to 2,000 guests per hour; lines form if patrons arrive at a higher rate. If Disney accurately anticipated demand, could it not build sufficient capacity? Did the appearance of long lines reflect negligent design? Surprisingly, the answer to both questions is no. The real culprit is not bad design but variability. Disney constructs each theme park to satisfy the "design day," typically up to the ninetieth-percentile level of demand, which means, in theory, on nine out of ten days, the park should have leftover capacity. In reality, patrons report long lines pretty much any day of the year.
Worse, statisticians are certain that queues would persist even if Disney had designed for the busiest day of the year. To understand this piece of anti-intuition, we must realize that the busiest-day demand merely conveys the average park attendance, and this number ignores the uneven distribution of patrons, say, from one attraction to another or from one hour to the next. Even if Disney correctly predicted the total number of patrons riding Dumbo on the peak day (which itself would have been a tough assignment), a line would materialize unavoidably because the patrons would appear irregularly during the day, while Dumbo's capacity does not change. Statisticians tell us that it is the variable pattern of when guests arrive, not the average number of arrivals, that produces queues on all but the peak days. Capacity planning can cope with large and static demand, not fluctuating demand. (A theme park that guarantees no lines would require capacity wildly disproportionate to demand, ensuring substantial idle time and unviable economics.)
The engineers who figured out these secrets are hailed as heroes by the Disney die-hards, and they work for the Imagineering division, based in several nondescript buildings in Glendale, California, near Los Angeles. They also design new rides, handling not only the thrill factor but also operations management. In the realm of waiting lines, scientists rely heavily on computer simulations as the mathematics of queuing are super complex and frequently irreducible to neat formulas. Think of simulations as turbocharged what-if analyses, run by farms of computers. Thousands, perhaps millions, of scenarios are investigated, covering possible patterns of arrival and subsequent movement of guests around the park. The summary of these scenarios yields reams of statistics, such as the likelihood that the Dumbo ride will reach 95 percent of its capacity on any given day. This creative approach to working around intractable mathematical problems was invented by the Manhattan Project team while building the atomic bomb and also forms the basis of Moneyball statistics featured in Michael Lewis's account of how the Oakland Athletics outwitted powerhouse baseball teams with much bigger budgets.
* * *
Wouldn't you know it? The same script plays itself out on our highways: the bane of commuters is not so much long average trip time as it is variable trip time. The statistics paint a harsh reality indeed: the average American worker spent 25.5 minutes traveling to work in 2006, and in addition, more than ten million people endured commutes of over an hour. In total, traffic delays cost $63 billion a year while wasting 2.3 billion gallons of fuel. But these scary numbers miss the mark. Just ask the pileup of readers who sent grievances to Minneapolis Star Tribune. Those truly put off by a long trip to work every day either practice avoidance ...
"I chose to live in Minneapolis for transportation-related reasons: great access to transit and reverse commutes.... If people chose to live in Eden Prairie [an edge city southwest of Minneapolis], then I don't have much sympathy for their complaints about traffic problems."
... or have made peace with the inevitable:
"Every day, no matter how much traffic there is, it slows down right by McKnight [Road near Maplewood on I-94].... There have been times when we have stopped and had a Coke somewhere because it gets so miserable sitting on the highway."
Commuters know what they are in for, and they take charge of the situation.
If average trip time is not the source of bother, what is? Julie Cross, another Star Tribune reader, articulated this well:
"Picking the fastest route for my morning commute from Apple Valley is a daily gamble. Should I chance Cedar Avenue, hoping to hit free-flowing traffic for a 5-minute trip to work in Eagan? Or would Cedar be stop-and-go, making the reliable 10-minute trip on Interstate Hwy. 35E the better bet?"
Pay attention when Cross used the word reliable. She knew well the average length of her trip to work; what troubled her was the larger variability, and thus unreliability, of the Cedar Avenue option. The highway route required ten minutes, with hardly any day-today variation. Now, if the Cedar Avenue option took five minutes without fail, Julie would never consider taking I-35E. Conversely, if the Cedar Avenue stretch took fifteen minutes without fail, Julie would always take I-35E. The only reason Julie Cross anted up each morning was that the Cedar Avenue route might take less time, even though she knew on average it would take longer. In general, if but one of two routes has variable trip times, then the bet is on. (See Figure 1-1.)
Excerpted from NUMBERS RULE YOUR WORLD by KAISER FUNG Copyright © 2010 by Kaiser Fung. Excerpted by permission of McGraw-Hill. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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