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 dataset. 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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