Batting .400 and the Law of Large Numbers

Rod Carew, one of the few to make a serious run at .400 since Williams, has studied the .406 season and contends that Williams’s absences were a blessing.

“The fewer at-bats any hitter has over the required number of plate appearances, the better his chance is of hitting over .400,” Carew wrote in an e-mail responding to questions about 1941. “When I hit .388 in 1977, I had 694 plate appearances and 616 at-bats (239 hits). Ted had something like 450 at-bats in 1941 when he hit .406, and I think George Brett and Tony Gwynn had fewer then 450 at-bats when they made their runs at .400.

“All in all, the less at-bats, the better.”

He’s trying to articulate the Law of Large Numbers.  Anyone hitting near .400 is deviating from the expected proportion of hits.  If you flip a coin 10 times, you just might get 7 tails.  If you flip if 1000 times, there’s no chance you’ll ever get 700 tails.  Many people may bat .400 during a single game (a handful of at-bats), but almost no one does as the number of at-bats increases.  Their average converges to the more realistic season average.