## Thursday, February 13, 2014

### 27: Batting Average

A major league baseball player can have a batting average of .333 on the first day of the season, but not a .334 average.

When can he have a .334 average?

1. After 287 at-bats, when he has 96 hits. His batting average, truncated after five digits, will be .33449, which rounds down to .334.

To determine this, I coded an Excel spreadsheet. In column A, I entered successive integers starting from 1. This was the AB column.

In column B I entered the formula =ROUNDDOWN(A#/3,0), and in column C I entered the formula =B#+1. (The # sign represents the number of that particular row.) Thus, column B represents the number of hits to obtain a BA of exactly .333... or the closest BA to .333... that does not exceed .333...., and column C represents the number of hits to obtain the next highest possible BA.

In columns D, I entered the formula =C#/A#, thus providing the BAs for the second of the two hit counts. (I do not need to look at the first, because it will be at most .333....)

In column E, I entered the formula =IF(AND(D#>=0.3335,D#<0.3345),"XXX",""). This formula is used to test if the BA is in the range that would round to .334. If it is, "XXX" is reported. Otherwise, the cell remains blank.

Copying the formulas in successive rows, the first row that displays an "XXX" is the row where the number of ABs is 287, corresponding to when the batter has 96 hits.

2. You lost me at "Excel spreadsheet." :-) I want to know how you can figure this out without technology.

3. .334 is 334/1000, which reduces to 167/500
500 divided by an average of 4 at-bats per game gets you 125 games. Or you can do the analysis above, except that MLB does not calculate to the fifth decimal place.

4. The 167/500 was easy to figure out. I've been monkeying around with it for a bit to see if there is a lower number of at bats that yields this average and I haven't found one yet. I might pose this question to my students by asking about the interval of at bats necessary to achieve a .334 average. Thank you for the question.

5. Does MLB calculate to the 4th decimal place and round to the third?

6. Does MLB round from the 4th to the 3rd decimal place?