THERE IS NO such thing as a batter's slump in baseball. The chance of a batter getting a hit next time at bat is the same when he has been hitting poorly as when he has been hitting well. But because managers believe that slumps exist, they fail to use their best line-ups.
The "hot hand" in basketball is another illusion. The chance of a shooter scoring is the same after she has just missed a flock of shots as when she has just sunk a long string of baskets. Yet even professional teams design clutch plays for the shooter who supposedly has a hot hand.
Of course you don't believe what you have just read. Almost no one does, even after a long lecture. And that disbelief is the most important part of the story. We humans have an amazing capacity to put our faith in patterns we think we see, even without reason to trust in them -- and even when there is scientific evidence to distrust them.
Our understanding of such chimeras -- and their counterparts in our superstitious thinking about such phenomena as interest rates, the stock market and prices of raw materials -- arises from the confluence of two scientific developments: the influential statistical theory of "efficient financial markets" (i.e., immediate reaction to changing information); and a body of psychological studies about how people estimate the likelihood of uncertain events. In turn, these new lines of research have been aided by the newly developed machinery of the "Bootstrap" method in mathematical statistics.
Statistician Harry Roberts of the University of Chicago -- in connection with an ongoing study of "efficient markets" -- compared the batting records of major-league players against the behavior of random numbers. If slumps really existed, actual players' records during a single season would shift from a string of hits to a string of outs less frequently than would the random-number sequences. But in fact they don't.
(Of course, a batter feels as if he has a better chance of getting a hit at some times than at other times. After a series of successful at-bats, sandlot players and professionals alike feel confident -- just as gamblers often feel that they're on a "streak." But there seems to be no connection between a player's performance and whether he feels hot or cold, astonishing as that may be.)
Averages over longer periods may vary systematically, as Ty Cobb's annual batting average varied non-randomly from season to season, Roberts found. But short-run analyses of day-to-day and week-to-week individual and team performances in most sports have shown results similar to the outcomes that a lottery-type random-number machine would produce.
In a typical investigation, a scientist programs a computer to simulate a slumpless player. For example, if the player has a .250 season-long batting average, the computer acts like an urn containing three white balls and one black ball. For each trial at-bat, the computer shuffles the four balls, draws one, and records whether the result is black or white. For a season with 400 at-bats, a ball is drawn 400 times.
The result is compared to the player's real season. If there truly is a non-random slump or streak, there will be fewer but longer "runs" of hits or outs in the real record than in the simulated record. On the other hand, if there is no connection between how a player does from one at-bat to the next, the actual record will change from hit to out and vice versa as often as does the simulated record.
We're not talking far-out ideas, but solid evidence based on accepted techniques. In a 1969 text on research methods, I suggested this sort of test for slumps to illustrate an approach I had devised for handling problems in probability and statistics. Since then, Stanford statistician Bradley Efron has independently developed and analyzed this approach under the name "Bootstrap method." It is the most exciting recent development in mathematical statistics. (See Box)
Psychologists Thomas Gilovich of Cornell and Robert Vallone and Amos Twersky of Stanford recently performed a similar study of basketball. First they determined that 91 percent of the fans at their universities believe that a player has "a better chance of making a shot after having just made his last two or three shots than he does after having just missed his last two or three shots." Then they examined the records of shots from the floor by the Philadelphia '76ers, foul shots by the Boston Celtics, and a shooting experiment using Cornell University teams. They found that though "players and fans alike tend to believe that a player's chance of hitting a shot are greater following a hit than following a miss on the previous shot," their analyses "provided no evidence for a positive correlation between the outcomes of successive shots." That is, knowing whether a shooter has or has not scored on the previous shot -- or in any previous sequence of shots -- is useless for predicting whether he will score again.
The species homo sapiens apparently has a powerful propensity to believe that one can find a pattern even when there is no pattern to be found. Two decades ago I cooked up series of random numbers that looked like weekly prices of publicly-traded stocks. Players in the experiment were told to buy and sell stocks as they chose. Then I repeatedly gave them "another week's prices," and allowed them to buy and sell again. The players did all kinds of fancy calculating, using a wild variety of assumptions -- although there was no possible way that the figuring could help them.
When I stopped the game before completing the 10 buy-and-sell sessions they expected, subjects would ask that the game go on. Then I would tell them that there was no basis to believe that there were patterns in the data, because the "prices" were just randomly-generated numbers. Winning or losing therefore did not depend upon the subjects' skill. Nevertheless, they demanded that the game not stop until the 10 "weeks" had been played, so they could find out whether they "won" or "lost."
Price and Prejudice
Belief in non-existent patterns is costly. Hundreds of thousands of investors pay high prices for "market letters" that purport to be systems for finding patterns in the stock market. Others purchase the analyses of "chartists" who see in stock prices patterns like "head and shoulders" and "necklace" that they claim tell you when to buy and sell. But according to the massive body of statistical analyses labeled "efficient market theory," rules based on such chartist patterns have zero validity.
"But," you might object, "there are lots of individuals -- even very ordinary people with systems who have made fortunes on the market." Yes, indeed. But the success of those individuals does not prove what it seems to prove. Confusion arises because most of us lack an adequate intuitive sense of the power of random variability.
Picture 65,000 people in Yankee Stadium, each with 16 pennies. Each person shakes his coins, dumps them on the floor and counts the number of heads. On average, one person -- though it might be zero, two, or three -- will find that all 16 tosses come up heads. Such a person is likely to think that there is something remarkable about the event, and to wonder whether he or she did something special to obtain the remarkable result, even though the person really had no control over the coins' outcomes.
Now picture the same 65,000 people buying and selling stocks. Again chance will on average lead one person to be right rather than wrong on 16 stocks in a row. (One in 2 will be right rather than wrong by chance with one stock, one in 4 will be right twice in a row, one in 8 will be right three times in a row, and so on, 16 times. This is how we estimating the probability of one in 64,992 by the classical "multiplication rule.")
Imagine how the lucky one will think. If she or he has been following the advice of a broker, the advice and the broker will get credit. If the person has been following some personal system such as "Buy on bad news"or "Buy a stock whenever its price has gone down for four straight weeks," the investor will credit the system.
Many government officials misunderstand chance variability, at great cost to the public. Agency personnel infer trends in the price of, say, energy, grain or timber from only a few years' data, and make forecasts that mislead entire industries. The Department of Energy wreaked havoc on airplane manufacturers and airlines with its forecast that the price of oil would continue to rise after 1979. Similarly, the American auto industry and purchasers of American-made cars have suffered from the gas-mileage standards imposed due to those same forecasts.
The farmers of the country have suffered terribly for their mistaken faith in forecasts that land and food prices would continue going up, based on short-run observations during the 1970s.
After a sharp price rise in the late 1970s, federal "experts" predicted timber shortages. Then in 1983 timber prices fell to only a quarter of their peaks, devastating lumber companies. This caused grief for lumber companies that had contracted for government timber at the high prices. And of course the industry then applied to the federal government for a bailout on the grounds that their troubles were the fault of the government and its "official" forecasts.
Studies by William Ascher of Duke University and Scott Armstrong of the University of Pennsylvania show that official forecasts are no more accurate than private forecasts, and may well be systematically worse because of being constrained to be "middle of the road," and being subject to manipulation for political purposes. Examining a variety of private forecasts rather than just a single official forecast alerts one to the existence of variability and hence unreliability. Official forecasts cost taxpayers dollars for something worse than we can get for free.
Looking at the Long Run
Though it is not possible to predict short-run behavior on the basis of the immediately preceding observations, one can accurately predict some long-run economic changes. You can confidently forecast that the batting average of Reggie Jackson five years from now will be lower than his average this year -- if Jackson will be playing at all. Similarly, we can confidently forecast that the prices of raw materials in 25 or 50 years will be lower than they are now, because this has been the long-run trend throughout human history and because a solid theory explains how this can happen. In fact, if you predict that the price of a raw material will be lower next year than at present, you stand a better-than-random chance of being right -- but only very slightly better, because in the short run the variability dominates the trend.
Hot-hand and slump-type thinking has far-reaching relevance. The basis for the Japanese system of quality-controlled production is the statistically- based theory of W. Edwards Deming. Those industrial procedures are based squarely on the control of variability using the same logic that we use to analyze a batter's performance for slumps.
Such ideas bear on the future of Soviet economic policies and the chance that glasnost will have much effect economically. Party leader Mikhail Gorbachev and other Soviet leaders fail to understand that central planning based on an intricate web of government economic forecasts, no matter how "open" and reformed, is doomed to failure.
Interestingly, Gorbachev and other confirmed socialists also err in not understanding that there is pattern where they fear there will be none -- in the interactions of undirected individuals and firms. In our society, too, doing business freely is anathema to those who continually worry about "chaos" and "anarchy" and "loss of control" in the economy. Some think that way because they have never contemplated an anthill and absorbed the central lesson of the "unseen hand." For others, as for Gorbachev, it is distasteful or frightening to let everyone do what he thinks best, subject only to general rules of market fair play.
Can we -- policymakers, journalists, athletic coaches and citizens at large -- learn the importance of looking at long-run historical evidence, not just at the recent past? Can we accept the impossibility of finding patterns where there are none? Too often, these lessons seem unlearnable. Or do you now agree that there is no such thing as a batter's slump?