## Saturday, March 23, 2013

### Two averages, the median and mode for batting averages

Question: The Table below contains data on at bats, hits and batting averages for Tony Gwynn over a 16 game period at the end of 1996.  What is the average of the game batting averages for the 16 games?  What is the aggregate batting average based on all at bats in this time period?  What is the median of the game batting average?  What is the mode of the game batting averages?

How do these statistics differ regarding the information that they provide?

 Batting Performances of Tony Gwynn Over 16 Games in 1996 Game # At Bats Hits Game BA 1 5 2 0.400 2 4 1 0.250 3 4 1 0.250 4 4 1 0.250 5 4 4 1.000 6 5 1 0.200 7 3 0 0 8 2 2 1.000 9 4 2 0.500 10 3 0 0 11 4 0 0 12 4 1 0.250 13 5 3 0.600 14 4 3 0.750 15 5 1 0.200 16 4 0 0 Total 64 22 0.347

Answer:  The simple average of the 16 game batting averages is the sum of all 16 game batting averages divided by 16, which is 0.353.  Tony Gwynn’s overall batting average over this 16 game period is simply total hits divided by total at bats or 0.347 (22/64).

The median of these 16 observations is obtained by averaging the two observations in the middle, the eight and ninth largest numbers, both of which are 0.250 in this data-set.  Hence, Tony Gwynn’s median batting average over these 16 games is 0.250.

Over these 16 games, Tony Gwynn had a 0.000 batting average in 4 games, a 0.200 batting average in 2 games, a 0.250 batting average in 4 games, a 0.400 batting average in 1 game, a 0.500 batting average in 1 game, a 0.600 batting average in 1 game, a 0.750 batting average in 1 game and a 1.000 batting average in 2 games.  The observations with the highest frequency, 0.000 and 0.250, are the modes of this sample.

In this sixteen game sample, Tony Gwynn’s overall and simple batting averages are both very high and the difference between these averages is small.  Both averages are pulled up by a small number of good days in which Tony Gwynn gets 4 or 5 hits.  I am a bit surprised the average of the game batting averages and the batting average based on total hits and total at bats are so similar.  I expected a small number of great games to increase the average of the game batting averages.    Perhaps my expectations are wrong because the batting average is truncated on the upside at 1000.

The median and the mode are both substantially lower than the averages largely because these statistics are not affected by Tony Gwynn’s great performances.  On a typical day Tony Gwynn, a great hitter, has a batting average of 0.250.    A 0 batting average is not uncommon. Baseball really is a tough game!!!!

Author's Note:

This problem first appeared in my book Statistical Applications of Baseball, published in 1996.  It is available at a very low price on kindle.

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