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At first glance, that may seem contradictory. But the improvement in performance since 1941 may not be as apparent in baseball as it is in other sports because batting average has remained relatively stable, at around .260–.270, for decades. But the stable average masks two important developments. First, batting average reflects not individual skill but rather the interaction between pitchers and hitters. It's like an arms race. As pitchers and hitters both improve their skills on an absolute basis, their relative relationship stays static. Although both pitchers and hitters today are some of the most skillful in history, they have improved in lockstep.8 But that lockstep was not ordained entirely by nature. The overseers of Major League Baseball have had a hand in it. In the late 1960s, for example, when it appeared that pitchers were getting too good for the batters, they changed the rules by lowering the pitcher's mound by five inches and by shrinking the strike zone, allowing the hitters to do better. Thus the rough equilibrium between pitchers and hitters reflects the natural evolution of the players as well as a certain amount of intervention from league officials.
Gould argues that there are no more .400 hitters because all professional hitters have become more skillful, and therefore the difference between the best and worst has narrowed. Training has improved greatly in the last sixty years, which has certainly had an effect on this convergence of skills. In addition, the leagues began recruiting players from around the world, greatly expanding the pool of talent. Hungry kids from the Dominican Republic (Sammy Sosa) and Mexico (Fernando Valenzuela) brought a new level of skill to the game. At the same time, luck continued to play a meaningful role in determining an individual player's batting average. Once the pitcher lets go of a ball, it is still hard to predict whether the batter, however skilled, will connect with it and what will happen if he does.
The key idea, expressed in statistical terms, is that the variance of batting averages has shrunk over time, even as the skill of the hitters has improved. Figure 3-4 shows the standard deviation and coefficient of variation for batting averages by decade since the 1870s. Variance is simply standard deviation squared, so a reduction in standard deviation corresponds to a reduction in variance. The coefficient of variation is the standard deviation divided by the average of all the hitters, which provides an effective measure of how scattered the batting averages of the individual players are from the league average. The figure shows that batting averages have converged over the decades. While Gould focused on batting average, this phenomenon is observable in other relevant statistics as well. For example, the coefficient of variation for earned run average, a measure of how many earned runs a pitcher allows for every nine innings pitched, has also declined over the decades.9
FIGURE 3-4
Reduction in standard deviation in Major League Baseball batting averages
Source: Analysis by author.
This decline in variance explains why there are no more .400 hitters. Since everybody gets better, no one wins quite as dramatically. In his day, Williams was an elite hitter and the variance was large enough that he could achieve such an exalted average. Today, the variance has shrunk to the point that elite hitters have only a tiny probability of matching his average. If Williams played today and had the same level of skill that he had in 1941 relative to other players, his batting average would not come close to .400.
Hitting a baseball in the major leagues is one of the hardest tasks in all of sports. A major league pitcher throws a baseball at speeds of up to one hundred miles an hour with the added complication of sideways or downward movement as it approaches the plate. The paradox of skill says that even though baseball players are more skillful than ever, skill plays a smaller role in determining batting averages than it did in the past. That's because the difference between success and failure for the batter has come to depend on a mistake of only fractions of an inch in where he places his bat or thousandths of a second in his timing in beginning an explosive and nearly automatic swing. But because everyone is uniformly more skillful, the vagaries of luck are more important than ever.
You can readily see how the paradox of skill applies to other competitive activities. A company can improve its absolute performance, for example, but will remain at a competitive parity if its rivals do the same.10 Or if stocks are priced efficiently in the market, luck will determine whether an investor correctly anticipates the next price move up or down. When everyone in business, sports, and investing copies the best practices of others, luck plays a greater role in how well they do.
For activities where little or no luck is involved, the paradox of skill leads to a specific and testable prediction: over time, absolute performance will steadily approach the point of physical limits, such as the speed with which one can run a mile. And as the best competitors confront those limits, the relative performance of the participants will converge. Figure 3-5 shows this idea graphically. As time goes on, the picture evolves from one that looks like the left side to one that looks more like the right side. The average of the distribution of skill creeps toward peak performance and the slope of the right tail gets steeper as the variance shrinks, implying results that are more and more alike.
We can test this prediction to see if it is true. Consider running foot races, especially the marathon, one of the oldest and most popular sports events in history. The race covers 26 miles and 385 yards. It was introduced as an original Olympic event in 1896, roughly fifteen hundred years after—legend has it—Pheidippides ran to his home in Athens from the battlefield of Marathon, where his countrymen had just defeated the Persians. When Pheidippides arrived, he proclaimed, “We have won!” He then dropped dead.
FIGURE 3-5
The paradox of skill leads to clustered results
Source: Analysis by author.
John Brenkus, host of Sports Science for the television network ESPN, speculates on the limits of human performance in his book The Perfection Point. After giving consideration to a multitude of physical factors, he concludes that the fastest time that a human can ever run a marathon is 1 hour, 57 minutes, and 58 seconds.11 As I write this, the world record, held by Patrick Makau of Kenya, is 2 hours, 3 minutes, and 38 seconds. So Makau's record is 5 minutes and 40 seconds slower than what is theoretically possible, according to Brenkus.
Figure 3-6 shows two results from each men's Olympic marathon from 1932 to 2008. The first is the time of the winner. That time dropped by about twenty-five minutes during those years. This translates into a pace that is almost one minute faster each mile, which (as you runners out there know) is a substantial increase, even considering that it was achieved over three-quarters of a century. The figure also shows the difference between the time of the gold medalist and the man who came in twentieth. As the paradox of skill predicts, that time has narrowed from close to forty minutes in 1932 to around nine minutes in 2008. So as everyone's skill has improved, the performance of the person who finished in twentieth place and the winner has converged.
FIGURE 3-6
Men's Olympic marathon times and the paradox of skill