## Fluctuating Batting Averages

When Miguel Cabrera came up to the plate in the fifth inning of last night’s Tigers-Rays game, he was 0-for-1 in the game and his up-to-the-minute batting average was announced as .349. I found this strange because, when the game started, Cabrera’s batting average was .350.

A player’s batting average is equal to (total hits) / (total at-bats). Thus the effect of one more at-bat without a hit dropped his average by .001, or 1/1000 (**Note: rounding **probably plays an important role here).

I wondered if this information uniquely determined both Cabrera’s hits and at-bats this season. Or maybe some combination of mathematics, baseball knowledge, and guessing could help me get those numbers. I did get the numbers–unfortunately, they were wrong.

An interesting question here is “What is the smallest possible number of hits such that one more hitless at-bat results in one’s rounded batting average dropping by .001?”

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Did you mean smallest number of hits, or smallest number of at-bats? I found that starting a game with 1 hit in 23 at-bats (a rounded average of .043) will become 1 in 24 (.042) after one hit-less at-bat. So the smallest number of hits is 1 and the smallest number of at-bats is 23.

Given a set number of at-bats, the fewer overall hits one has always means less of an effect on the overall batting average after one hit-less at bat. This makes sense intuitively, because the closer your running average is to zero, the less it will be affected by “adding” a zero average (0 for 1). Taking this to an extreme (excepting the trivial case of 0 hits) leads to the fact that we should consider cases where the batter has only 1 hit heading into the game. Then we can play with the total number of at-bats, increasing them until we get to the case of 23 at-bats heading into the game.

This post made me think of a counter-intuitive baseball batting average problem that I have seen:

Break down a full season into two half-seasons. Is it possible for player A to have a better batting average than player B for the 1st half-season and also beat player B’s average in the second half-season (just looking at stats from that half-season), and yet have player B’s overall season average be higher than player A’s?

You can probably guess that it is possible, but can you create such a scenario?