The casinos, of course, are happy to exploit this failure to understand randomness. Some of them even post the recent results of the roulette wheel spins, hoping to dupe gamblers: “Hey, it’s landed on odd five straight times. We’re due for even!”
Now the contradiction between the strong belief in the hot hand or momentum in sports and the lack of actual evidence starts to make sense. A basketball player who shoots 50 percent will not miss an attempt and then make an attempt. A batter may hit .300, but it’s only an average. It doesn’t mean that he’ll get three hits in every ten at-bats. He might go 0 for 10 and then 6 for 10. Over the 600 at-bats throughout a season, however, he probably will get 180 hits. The larger a sample, the more accurately it represents reality.
Kobe Bryant shoots free throws much better than Shaquille O’Neal does. For their respective careers, Kobe hits about 84 percent from the line, and Shaq only 53 percent. Take a sequence of only five shots, however, and it’s entirely possible they’ll shoot comparably. It’s even reasonably possible that Shaq will outshoot Kobe. In fact, the chances are about 22 percent; that means if Shaq and Kobe staged a five-shot free throw shooting contest, about one out of every five times Shaq would do at least as well as Kobe and might even beat him. Over ten attempts, it’s less likely. Over 100, it’s remote.
What does this mean with regard to David Wright’s hitting slump? A career .307 hitter, Wright expects to get a hit 30 percent of the time. Three weeks into the season, after getting a hit only 23 percent of the time, his performance is perfectly consistent with his .307 average. The same would be true on the other side. He could have hit .400 the first few weeks, and fans would be ready to declare him the first player since Ted Williams to bat .400 for a season. Yet over a short period, a .307 career hitter batting .400 is perfectly consistent with random chance, too. Some athletes get this better than others and try to avoid “getting too high or too low.” Wright’s former teammate Jeff Francoeur performs a self-assessment on hitting every 50 at-bats. But even that is woefully narrow.
Being fooled by chance can create seemingly unbelievable statistics. Consider the following, all of which are true. Over the last decade, in every single MLB season:
At least one National League pitcher has had a longer hitting streak than a starting All-Star (nonpitcher).
At least one National League pitcher has had a longer hitting streak than a designated hitter in the American League.
At least one batter hitting under .225 for the season has put together a hitting streak longer than that of a player hitting over .300.
At least one player who finished the season hitting over .300 has had at least one six-game or longer hitless streak.
These stats, surprising as they might seem on their face, hold up every year. Pitchers are not supposed to hit better than position players, much less all-stars. Players hitting under .225 aren’t supposed to have longer hitting streaks than .300 hitters. The best batters aren’t supposed to go six games—25 or so at-bats—without a hit, and on average, they don’t. But in isolated cases it happens, and it’s perfectly consistent with random chance.
We search for an explanation, but the true explanation is simple: Luck or chance or randomness causes streaks among even the best and worst players. It has nothing to do with momentum. When we consider the bigger picture and the larger numbers of players in Major League Baseball, this starts to make sense. How likely is it that Tim Lincecum, the star pitcher for the San Francisco Giants, will outhit the mighty Albert Pujols over any stretch of the season? Not very. How likely is it that any pitcher will outhit Pujols over a two-week stretch? More likely. How likely is it that at least one pitcher will outhit at least one all-star position player over those weeks? Very, very likely. The larger the sample, the more you can find at least one seemingly unlikely example.
If you were predicting the likelihood of an MLB player getting a hit in his next at-bat, which of the following do you think would be the best predictor?
a) His batting average over the last five plate appearances
b) His batting average over the last five games
c) His batting average over the last month
d) His batting average over the season so far
e) His batting average over the previous two seasons
Most people are tempted to select (a), on the grounds that it is the most recent and therefore the most relevant number: He’s streaking and will continue riding the wave. Or, he’s slumping and still mired. But to pick (a) is to be fooled by randomness, tricked into thinking there’s momentum.
We looked at all MLB hitters over the last decade and tried predicting the outcome of their next at-bats by using each of the five choices above. It turns out (a) is the worst predictor. Why? Because it has the smallest sample size. Choice (b) was the next worse, then (c), and then (d). The best answer was (e), the choice with the largest sample size.
The same thing is true at the team level. Heading into the postseason—and barring the unusual, such as a recent horrific injury to a star—which of the following is a better predictor of playoff success?
a) The team’s performance in its most recent game
b) The team’s performance in the last week before the playoffs
c) The team’s performance in the last month before the playoffs
d) The team’s regular season performance
Momentum would lead one to think that it’s (a) or (b) and, to a lesser extent, (c), yet those are actually the worst predictors. In every single sport (MLB, NBA, NHL, NFL, European soccer) we studied, we found (d) to be the best predictor of postseason or tournament success. The true quality of teams can be measured best in large samples. Small samples are more dominated by randomness and therefore are inherently unreliable.
Nor is this unique to sports. In the investment management industry, investors often “chase short-term returns,” flocking toward mutual funds that had a good quarter or year and fleeing from funds that didn’t. They ascribe success on the basis of a small sample of data. But as with the hot hand in sports, it turns out that one quarter or even one year of a fund’s performance has no special predictive power for the next year’s performance in the mutual fund industry. In fact, one year of performance for almost any fund is dominated by luck, not skill. Yet people usually don’t see it that way. Entire businesses have been built on selling short-term performance measures to investors to help them identify the best funds, and funds aggressively market their recent strong performance to investors (and hide or bury bad performance when they can). But the reality is that every year the top 10 percent of funds are just as likely to be among the bottom 10 percent of funds the next year. It’s just pretty much random from year to year.
Sports gamblers, too, are fooled by momentum. Colin Camerer, a Caltech professor of behavioral economics, found that winning and losing streaks affected point spreads. Bets placed on teams with winning streaks were more likely to lose, and bets placed on teams with losing streaks were more likely to pay off. In other words, gamblers systematically overvalued teams with winning streaks and undervalued those with losing streaks.
Just as an astute investor can take advantage of these misperceptions with potentially big gains, so can a savvy coach and player (and sports gambler). If the majority overvalues the recent winners and undervalues the recent losers, do the opposite.
The only problem is convincing people to go against their (and everyone else’s) intuition. After the initial study asserting the fallacy of the hot hand in basketball, Red Auerbach, the revered Hall of Fame coach and then president of the Boston Celtics, was presented with the findings. Auerbach rolled his eyes and waved the air with his hand. “Who is this guy? So he makes a study. I couldn’t care less.” Bob Knight, the volatile and decorated college coach, was similarly dismissive: “There are so many variables involved in shooting the basketball that a paper like this doesn’t really mean anything.” Amos Tversky, the famous psychologist and pioneering scholar who initiated t
he original research on momentum and the myth of the hot hand, once put it this way: “I’ve been in a thousand arguments over this topic. I’ve won them all, and I’ve convinced no one.”
DAMNED STATISTICS
Why “four out of his last five” almost surely means four of six
“There are three kinds of lies: lies, damned lies, and statistics.”—Mark Twain
At some point it became almost cartoonish, as though he wasn’t shooting the basketball so much as simply redirecting his teammates’ passes into the hoop. In the first half of the second game of the 2010 NBA finals, Ray Allen, the Boston Celtics’ veteran guard was … well, the usual clichés—“on fire,” “unconscious,” “in the zone”—didn’t do it justice. Shooting with ruthless accuracy, Allen drained seven three-pointers, most of them bypassing the rim and simply finding the bottom of the net. Swish. Swish. Swish-swish-swish. In all, he scored 27 points in the first half. Celtics reserve Nate Robinson giddily anointed Allen “the best shooter in the history of the NBA.”
As Allen fired away, the commentators unleashed a similarly furious barrage of stats, confirmed by the graphics on the screen. The shooting was cast in the most glowing terms possible. Allen, viewers were told at one point, had made his last four shots. When he missed a two-pointer (turns out he was only three for nine on two-point attempts), the stats suddenly focused only on the three-pointers.
It was inevitable that Allen would cool off. And he did in the second half, making only one three-pointer, although his eight treys for the game became a new NBA finals record and his 32 total points enabled Boston to beat the Los Angeles Lakers 103–94. But he really cooled off in his next game. This time he was ruthless in his inaccuracy, missing all 13 of his shots, including eight three-point attempts, as Boston lost 91–84. As Allen clanged shot after shot, the commentators were quick to note this whiplash-inducing reversal of fortune, framing it in the most damning terms possible. At one point viewers were told that between the two games, Allen had missed 17 straight attempts.
Inasmuch as sports fans are tricked by randomness, the media share in the blame. Statistics and data are the forensic evidence of sports, but like all pieces of evidence, they can be mishandled and tampered with. We are bombarded by stats when we watch games, but the data are chosen selectively and often focus on small samples and short-term numbers. When we’re told that a player has reached base in “four of his last five at-bats,” we should assume right away that it’s four of his last six. Otherwise, rest assured, we’d have been told that the streak was five out of six. Clearly, a team that “has lost three in a row” has dropped only three of its last four—and possibly three of five or three of six or … otherwise it would have been reported as a four-game losing streak.
Those of us in the sports media have an interest in selling the most extreme scenario. Collectively, they (we?) pick and choose data accordingly. Take, for instance, a September 15, 2009, game between the New York Yankees and the Toronto Blue Jays, a showdown between Alex Rodriguez and Roy Halladay, arguably the league’s best hitter and best pitcher at the time. The Yankees broadcasters might have framed the encounter along the following lines, using the most positive statistics at their disposal:
Rodriguez steps to the plate. He’s hitting .357 against Halladay this season, including five hits in his last 12 at-bats against the big righty, a .412 clip. Over his last 11 games, A-Rod is hitting .436. Remember that as trade rumors swirl, Halladay has lost 4 of his last 5 starts and 11 of his last 15.
Upon receiving this information, it sounds almost like a foregone conclusion that A-Rod is going to crush the ball. In fact, one almost feels pity for Halladay. It’s as though he should have taken the mound wearing a helmet and protective covering.
Now listen to how the Toronto broadcasters might have addressed the showdown, using the best available statistics to make their case:
Halladay comes in having pitched two straight complete games. Over those 18 innings, he struck out 18 men and gave up only four earned runs, a 2.00 ERA. Meanwhile, A-Rod is hitless in his last six at-bats against Halladay. Among all opposing teams, Rodriguez has his lowest average—and strikes out the most—against the Blue Jays.
After hearing this we’d be surprised if Rodriguez made contact with a Halladay pitch, much less reached base.
Both renderings would have been perfectly accurate. Both sets of statistics are true. Yet they paint radically different pictures. Incidentally, in that Yankees–Blue Jays game, Halladay pitched six innings, allowed two earned runs, and got the win; Rodriguez was one for three with a double against Halladay—pretty much what a neutral observer, ignoring the noise and looking at as much data as possible, would have predicted.
Teams are complicit in this selectivity, too. Check the scoreboard next time you’re at a baseball game. Had you attended a White Sox–Tigers game at U.S. Cellular Field in the summer of 2010, you could have learned that Chicago’s outfielder Carlos Quentin was “hitting .371 over his last nine games.” Although this was impressive and meant to convey a hot streak, it told us … what exactly? Not much, not with a sample size that small. If the White Sox were attempting to predict the outcome of Quentin’s next at-bat, they would have provided a more meaningful statistic, using a larger data set. But noting that Quentin was “351 for 1,420 (.247) for his career” doesn’t quite stir up passion.
When Nate Robinson declared Ray Allen the best shooter in the annals of the NBA, he may have been right, but not because Allen had one torrid shooting half. Otherwise, you could just as easily make the case that based on the following night’s game, Allen was the worst shooter in NBA history. Robinson’s more convincing evidence would have been this: For his NBA career Allen has taken more than 6,000 three-point attempts and made roughly 40 percent of them.
Those two games of extremes during the 2010 NBA finals? Unsexy as it might have been to use the largest available data set and note Allen’s career average, it would have helped the viewers. Between the two games, he was 8 of 19, or 42 percent, on three-point attempts, conforming almost exactly with his career mark.
ARE THE CHICAGO CUBS CURSED?
If not, then why are the Cubs so futile?
The ball collided with the bat of Luis Castillo and made a hollow thwock, the tip-off that it hadn’t been hit cleanly. It wafted into the autumn night sky and descended between the foul pole and the third-base line at Wrigley Field. The Cubs left fielder, Moises Alou, ambled over, tracking the ball.
It was a foregone conclusion that Alou would make the catch, completing the second out of the eighth inning in this, the sixth game of the 2003 National League Championship Series. Ahead of the Florida Marlins 3–0 this night and leading in the best-of-seven series three games to two, the Cubs would then be just four outs from reaching the World Series for the first time since 1945. Already champagne was nesting on ice in the Cubs’ clubhouse. The Marlins’ team president had just called his wife to tell her there was no need to come to Chicago because there would be no game seven. Alou, a capable fielder and a veteran of several all-star games, positioned himself under the ball. The crowd, already rock-concert-loud, thickened its roar. Alou extended his left arm, yelled “Got it, got it,” jumped up alongside the stands, unfurled his glove, and …
You probably know the rest of the story. A 26-year-old consultant had managed to score a sweet ticket for this game: aisle 4, row 8, seat 113, the first row before the field. A mishandled nacho and the cheese would have landed in the dirt of foul territory. For Steve Bartman, this was about as close to nirvana as he could get. A Chicago native, Bartman was the kind of long-suffering citizen of Cubs Nation whose moods moved in accordance with the team’s fortunes. Bartman had recently graduated from Notre Dame but had returned home, yes, because of his job and family but also because of the proximity to his beloved baseball team. His level of fandom was such that despite his prime seat, he still listened to a radio broadcast of the game on headphones as he watched.
When Castillo�
��s foul ball traced an arc and began its downward flight, Bartman rose to catch it, a reaction almost as instinctive as withdrawing one’s hand from a hot flame. A souvenir foul ball? What better way to garnish a magical night. In less time than it will take you to read this sentence, Bartman’s life—to say nothing of his magical night—was turned on its head. In his zeal to catch the ball, he interfered with Alou and knocked the ball away. After realizing an out had been lost, Alou popped away as if bitten by a snake. He shot Bartman a death stare and, in a gesture unbecoming a 37-year-old man, slapped his glove in the manner of a Little Leaguer throwing a tantrum. “Alou, he is livid with a fan,” intoned the television broadcaster. Mark Prior, the Cubs pitcher, turned to left field and also stared darts into Bartman.
Given new life, Castillo walked. It was around that time that Bartman was escorted from his seat by security. “It’s for your own safety,” he was told. Even then, he was heckled and cursed and doused in beer. Bartman buried his face in his sweatshirt as if doing a perp walk through Wrigley—the ballpark, incidentally, nicknamed “The Friendly Confines.”
It was a good thing security arrived when it did. Castillo’s walk catalyzed an eight-run rally. There were wild pitches and cheap hits and an error by the Cubs shortstop, Alex Gonzalez, on what should have been an inning-ending double play. The Marlins won the game. By then, Hollywood production companies were already angling for movie rights. According to the next day’s Variety, Fan Interference, starring Kevin James, would tell the story of “a [fan] who screws up an easy out, and then has to deal with the ramifications.” Thanks to the speed and power of the Internet, Bartman’s identity was revealed by morning. At his office at Hewitt Associates, a management consulting firm in the North Shore suburbs, his voice mail was clogged with profane messages. Bartman released a statement, stating that he was “sorry from the bottom of this Cubs fan’s broken heart.”
Scorecasting: The Hidden Influences Behind How Sports Are Played and Games Are Won Page 24