I found a rather normal distribution from the real world — if you consider major-league baseball to be part of the real world. In a recent post I explained how I normalized batting statistics for the 1901-2015 seasons, and displayed the top-25 single-season batting averages, slugging percentages, and on-base-plus-slugging percentages after normalization.
I have since discovered that the normalized single-season batting averages for 14,067 player-seasons bear a strong resemblance to a textbook normal distribution:
How close is this to a textbook normal distribution? Rather close, as measured by the percentage of observations that are within 1, 2, 3, and 4 standard deviations from the mean:
Ty Cobb not only compiled the highest single-season average (4.53 SD above the mean) but 5 of the 12 single-season averages more than 4 SD above the mean:
Cobb’s superlative performances in the 13-season span from 1907 through 1919 resulted in 12 American League batting championships. (The unofficial number has been reduced to 11 because it was later found that Cobb actually lost the 1910 title by a whisker — .3834 to Napoleon Lajoie’s .3841.)
Cobb’s normalized batting average for his worst full season (1924) is better than 70 percent of the 14,067 batting averages compiled by full-time players in the 115 years from 1901 through 2015. And getting on base was only part of what made Cobb the greatest player of all time.