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The decline of the .400 hitter: an explanation and a test.

Stephen Jay Gould (1986), an evolutionary biologist, studies the evolution of systems. That is, he studies the nature of variation and change in various phenomena. One of the "systems" he studies is baseball. He addresses the query, "What has happened to the .400 hitter?" In 1897 Wee Willie Keeler batted .432; in 1924 Rogers Hornsby batted .424; and in 1941 Ted Williams batted .406. No baseball player since Williams - over a half century ago - has hit at a .400 or better clip.

Why has the .400 hitter become extinct? Baseball aficionados have argued that night baseball, grueling schedules, dilution of talent, and improved and specialty pitching have all contributed to this decline. Gould (1986), while not denying these factors may have had an impact, argues that such reasoning is based upon a false assumption. He says that the .400 hitter is not a thing or a phenomenon and proposes two related reasons to explain this extinction: (1) human performance in some activities, e.g., hitting in baseball, is approaching the outer limits of human capacity, and (2) systems tend to an equilibrium as they improve. Another way of looking at the .400 hitter is to view it as representing the extreme right hand tail of a normal distribution. In order to understand why the .400 hitter is nonexistent today one needs to look at the entire distribution of batting averages over time.

To paraphrase Gould's (1986) argument: For the past 100 years the average (arithmetic mean) batting average has hovered around .260. What has happened is that the standard deviation (a measure of variability) has declined in almost law-like fashion. The standard deviations over the past one hundred years have, in fact, steadily declined. Importantly, the arithmetic mean has remained fairly constant but the standard deviation has gotten smaller.

Why has the standard deviation declined? According to Gould (1986), the standard deviation has decreased because all athletes - pitchers and hitters - have gotten better. Unlike clock sports that have an absolute standard, batting in baseball reflects the limits of human performance. That is, there is only so much a body can do to achieve perfection. Further, there is a dynamic and delicate balance between hitting and pitching in baseball. The batting feats of the greats of yesteryear, performed close to the limits of human performance and a wide gap existed between the poorest performers and the best ones at that time. Today that range in baseball hitting is less variable. [As an aside, Gould maintains that a Wade Boggs is close to the limit of performance and could have batted well over .400 during the earlier days of baseball.] The standard deviation has declined because the participants in the sport have gotten so much better. Paradoxically, the departure of the .400 hitter is a sign of better athletic performances overall. Gould (1986, p. 62) says:

Paradoxically, this decline [variation] produces a decrease in the difference between average and stellar performance. Therefore, modern leaders don't stand so far above their contemporaries. The myth' of ancient heroes - the greater distance between average and best in the past - actually records the improvement of play through time.

With statistics, through a transformation process, we can compare performances of top hitters at different times. The formula for such standardization is z = (raw score-arithmetic mean)/standard deviation.

In the 1910, 1940 and 1970 decades the means and standard deviations were .266 and .037, .267 and .033, and. 261 and .032, respectively. As examples, Cobb batted .420 in 1911, Williams batted .406 in 1941, and Brett batted .390 in 1980. The respective standard scores (z-scores) for these stellar performers were 4.16, 4.21, and 4.03, respectively. Since 99%+ of all batting averages fall within plus and minus three standard deviation units of the mean and since these three individuals are all at least four standard deviation units above the mean, there is indisputable statistical evidence that Cobb, Williams, and Brett were spectacular hitters.

Research Query

Gould's (1986) argument can be stated as a theoretical proposition: the greater the variation in the performance of baseball players, the higher the batting averages of the best baseball players. In the early days of baseball when both hitters and fielders were exploring the nuances of the game there were some athletes who genuinely appeared to be "giants" of their day. Illustratively, Wee Willie Keeler's .432 in 1897 was accomplished by "hitting 'em where they ain't." As defensive alignments and positions became adapted to his style these defensive gaps were plugged. In short, the differences between the worst and best performances narrowed as these and other points of the game became fine tuned.

His hypothesis, that as a system stabilizes, there will be decreasing variation in the system over time, is intriguing. Focusing exclusively on hitting in baseball, he says that batting averages have remained relatively stable (around .260) from the inception of the National League in 1876 until present. A chart of Gould's (1986, p. 61) Discover article tells me a different story. There are more than a half dozen peaks that are at least .10 points higher than .260. It is true that in more recent years, since 1940, the average does tend to stay fairly close to .260. If my conclusions are correct, it doesn't seem accurate for him to write of a consistent batting average. On the other hand, the standard deviation values graphed on page 65 (and also reported in numerical form on the chart) do, with one exception, become smaller and smaller over time (the exception is 1970). The fact that he also includes a computation of the variation ratio (which is typically used when mean values fluctuate from measurement to measurement) is confusing since one would ordinarily not use it if the mean is fairly stable (as he says when he alludes to the .260 figure). In other words, I don't understand why the coefficients of variation are computed if the mean is constant. Considering the sport of baseball exclusively, his data apropos the declining variation (via standard deviation values) is for one performance measure, batting average. If his general model applies "across the board," other' performance measures such as slugging averages, home runs, doubles, triples (hitting indicators); earned run averages, strike outs, bases on balls (pitching indicators); and errors and fielding averages (fielding indicators) would also be expected to conform to the general pattern. Whether or not they do is the empirical question this study explores.

Importantly, Gould's (1986) thesis focuses on declining standard deviation values as indicators of better performance. The issue is not what is happening to the central tendency measures.


In this inquiry team statistics on 12 hitting performance variables were compiled for the years of 1901, 1910, 1920, 1930, 1940, 1950, 1960, 1970, 1980, and 1990. Decimal placement for comprehending the discussion is important. No decimals (for space purposes) were used in the original data entry. This means that a batter with 4 hits in 8 at bats would have a value of 5 (not .5) entered. Only integer values were entered into the data file. Note that the unit of analysis was the team, not the individual player, and the use of aggregate data typically reduces the variances vis-a-vis individual data. The variables analyzed were: games, at bats, runs scored, hits (singles), doubles, triples, home runs, runs-batted-in, bases-on-balls, stolen bases, batting averages, and slugging averages. In addition, data were coded for league (National and American), team, and age.

Statistical Procedures

After producing a variety of descriptive statistical products (measures of central tendency and dispersion) to assure the accuracy of the data, several inferential statistics were computed to determine the statistical significance of the outcomes. Since Gould (1986) claims that the variation in performance (batting average in particular) has decreased over time, several tests for the homogeneity of variances - Cochran's C, Bartlett's box F, and the ratio between maximum and minimum variances - were computed These tests determined whether there were statistically significant differences in variances over time.

Then the one way analysis of variance was computed to test whether the means of the different time periods were significantly different from each other [note that the analysis of variance procedure assumes equality of variances and the homogeneity of variance tests enabled us to assess the appropriateness of the ANOVA procedure]. The descriptive outcomes are demonstrated in pictorial form [ILLUSTRATION FOR FIGURES 1 & 2 OMITTED].

The coefficient of variation was computed (for each variable) because it "adjusts" for differences in arithmetic means. In other words, statistically, if the arithmetic average varies over time so, too, will the standard deviation. By computing the coefficient of variation we obtain a clearer picture of the variability over time since this statistic takes into account differences ill means.


The statistical data for testing all variables appears in Table 1. The far left-hand column lists the variable under examination, followed by the tests of the null hypotheses of: (1) homogeneity of variances ([H.sub.0]: variance 1 = variance 2 = variance k) and (2) homogeneity of means ([H.sub.0]: mean 1 = mean 2 = mean k). The F-ratio value is listed along with eta, the latter providing an indicator of the association between the dependent variable (e.g., age, games, slugging average) and the independent variable (time, e.g., 1901, 1910, 1990). Because Gould's (1986) thesis entails the variance in a system, the results are graphically portrayed in two ways: (1) charting the variability about the mean across time, and (2) plotting the trend in the coefficient of variation over time. Technically, the variation ratio is computed by dividing the standard deviation by the mean; the coefficient of variation is produced by multiplying this ratio by 100. Here the two concepts are used interchangeably.

Batting Average

The inferential statistical data (as well as eta, a descriptive statistic) for the variable batting average are produced in Table 1 and two line graphs depicting central tendency (i.e., mean), dispersion (i.e., standard deviation), and variation ratio measures appear in Figure 1. The lower tier of Figure 1 depicts + and - 1 standard deviation units (batting averages are calibrated along the ordinate) from the mean for the different time periods (year is located along the abscissa). The greater the vertical distance between the line markers for a given year the greater the variation (and vice versa). For example, the maximum variability was in 1910 (s=18); the minimum was in 1990 (s=8). The upper tier of Figure 1 displays an overall steady decline in the variability of batting average. The tests for the homogeneity of variances indicate that, contrary to Gould's (1986) claim, the variances across time are not statistically significant. Since the analysis of variance technique assumes such homogeneity the F-test is legitimate. The F value produced by the test indicated that there is a statistically significant difference in mean batting averages across time (the high was .295 in 1930 and the low was .249 in 1910). In both cases Gould's proposition is questionable - the variation across time is not significant but the arithmetic average is! Unlike Gould, the standard deviations have not decreased in almost law-like fashion.

Slugging Average.

Variability in slugging averages has not been constant (Figure 2 depicts this trend). As can be seen, while slugging averages have been fairly constant the past fifty years (lower tier) there has been an overall decline in variability (upper tier). For example, the smallest s value (17) was in 1990, but the largest (37) was in 1930. Additionally, the highest mean slugging average (.434) was in 1930, the lowest (.326) was in 1910.


Not surprising (since the number of games played has increased over the past 100+ years), there are significant differences in both dispersion and central tendency [TABULAR DATA FOR TABLE 1 OMITTED] measures for this variable. Further, the coefficient of variation today is virtually zero since the number of games player per team is uniform. For space reasons the figures are not contained herein. However, they are available upon request from the author.

At Bats

There is a statistically significant difference in the mean number of at bats over time (the low was 4819 in 1901 and the high was 5544 in 1980. However, there were no statistically significant differences in the variances over time (the smallest was about 59 in 1990 and the largest was about 120 in 1930). After 50 years of near identical standard deviations, the coefficient of variation precipitously declined in 1950 and again in 1990.

Runs Scored

There is a statistically significant difference in the mean number of runs scored across the decades. The lowest mean (599) occurred in 1910, the highest (856) in 1930. Standard deviations also differed significantly from a high of 134 in 1930 to 51 in 1990. The plot of the variation ratios reveals a steady decline.


There were statistically significant differences in the mean number of hits (1600 in 1930 vs. 1271 in 1910) and standard deviation values for this variable (maximum was 124 in 1930 and minimum was 51 in 1990). The graph of the coefficient of variation likewise depicts a steady overall decline.


There was a statistically significant difference in the mean number of doubles across the decades. The low was 176 in 1910 and the high was 298 in 1930. The standard deviations ranged from a low of 15 in 1960 to a high of 40 in 1920. The plot reveals a steady decline until 1960 and, since then, a steady increase.


There was a statistically significant difference in the standard deviation in the number of triples across time. The low was 8.4 in 1990; the high was 20.7 in 1900. Mean values were also significantly different with a high of 80 in 1930 and a low of 33 in 1990. The plot reveals a bumpy ride for the coefficients of variation. Overall, there values remained fairly constant from 1901 to 1950, then increased the next two decades, then fell the last two decades.

Home Runs

The mean number of home runs has varied significantly over time. A low of 22.44 occurred in 1910 and a high of 143 occurred in 1970. The standard deviations ranged from 8 in 1901 to 38 in 1930 and the plot of the coefficients of variation show a steady decline over time.

Runs Batted In

The variability in the mean number of runs batted in significantly varied across time. The low of 49.26 occurred in 1990 and the high of 133 occurred in 1930. Means, too, varied significantly from 796 in 1930 to 491 in 1910. The coefficient of variation plot showed an overall steady decline in runs batted in from 1901 to 1960, then an increase to 1980, followed by a drop.


The variability in the mean number of walks was fairly homogeneous (ranged from 56 in 1901 to 85 in 1930) although the mean number of walks per team varied significantly (from a low of 342 in 1901 to a high of 622 in 1950). The plot of the coefficients of variation was mixed. A steady rise from 1901 to 1930, a steady decline from 1930 to 1960, then an increase to 1970, followed by a plateau.

Stolen Bases

The mean number of stolen bases has varied significantly, from a high of 204 in 1910 to a low of 41 in 1950. The variability has likewise significantly varied (from 15.33 in 1950 to 52.81 in 1980). The coefficient of variation plot shows a steady rise from 1901 to 1960 and then a decline.


Although not a performance variable per se, there has been very little variability in the average age of baseball players over time. The standard deviation range from 1.18 to 1.77 and the means from 28.44 in 1960 to 27.44 in 1910. The coefficient of variation plot has been constant for the past forty years.

Summary and Discussion

What do all these statistics mean with respect to Gould's (1986) hypothesis? Consider, first, the test of the null hypotheses regarding the arithmetic means. For all variables, save one (i.e., age), there were statistically significant differences across time. Gradually, the number of games, at bats, home runs, rbi's, walks and stolen bases increased over time; runs scored, triples, and batting average have tended to decline; and hits, doubles, and slugging averages produced mixed results. Central tendency measures, of course, are not his concern. Inspecting the coefficient of variation plots reveals data trends very much in line with Gould's notions, although they are not necessarily statistically significant.

Other questions arise. In baseball, for example, what pattern takes place with respect to pitching performance? Do ERA's decrease? Innings pitched? What about fielding statistics? Would individual sport performances in golf, tennis, bowling, track and field, swimming, etc. also fit the model? Would team sports such as football, basketball, and hockey - more specifically, performance measures in these sports - reveal the same tendencies?

In summary, Gould's (1986) thesis - that systems experience declining variation as they stabilize and improve while maintaining constant rules of performance across time - is fascinating and could be explored in sport sociology and other areas as well. He, himself (Gould, 1986, pp. 65-66), has suggested this may be a general framework for "understanding trends in time as an interaction between the location of bell-shaped curves in variation and the position (and potential for mobility) of the limiting right wall for human excellence."


Gould, S. J. (1986, August). Entropic homogeneity isn't why no one hits .400 anymore. Discover, pp. 60-66.

Bainbridge, W. S. (1992). Social Research Methods. Belmont, CA: Wadsworth.

Losing the edge. (1993, March). Vanity Fair, pp. 264-272.

Neft, P. B. & Cohen, R. M. (1987). The Sports Encyclopedia: Baseball. New York. NY: St. Martin's.
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Article Details
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Author:Leonard, Wilbert M., II
Publication:Journal of Sport Behavior
Date:Sep 1, 1995
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