What Sample Size Does to a Batting Average

When a broadcast says a hitter is batting .300, that is not his hitting ability. It is one estimate of it, drawn from however many at-bats have happened so far. The estimate carries a margin of error, and early in a season that margin is enormous.

Here is the same .300 average at four different sample sizes, with its 95% confidence interval:

• 40 at-bats: .158 to .442
• 100 at-bats: .210 to .390
• 400 at-bats: .255 to .345
• 600 at-bats: .263 to .337

At 40 at-bats, "hitting .300" is compatible with being one of the worst hitters in the league or one of the best. The number on the screen does not change; what it can support does.

Take a true .300 hitter and a true .270 hitter — in baseball terms, a meaningful gap. How many at-bats before their averages reliably tell them apart?

• 100 at-bats: the gap is 0.47 standard deviations. Invisible.
• 300 at-bats: 0.81. Still invisible.
• 600 at-bats: 1.15. Suggestive, not conclusive.
• 1,000 at-bats: 1.49. Nearly two full seasons, and still short of the usual bar.

This is not a flaw in the analysis. It is what batting average is: a noisy measurement of a small difference. Most of what looks like a hitter getting better or worse within a season is this noise.

A genuine .300 hitter goes 0-for-4 about 24% of the time — roughly one game in four. Over a week he will have two of them. That is not a slump; that is arithmetic.

The math is one line: a .300 hitter fails 70% of the time, so four failures in a row is 0.70 x 0.70 x 0.70 x 0.70 = 0.24.

Any story built on a single hitless game is a story about a coin landing tails twice. Ask how many chances the player got before you accept any explanation of what he did with them.