SIMPSON's Paradox

Assignment #40: Skills Check Project Module #5

In baseball statistics, a players batting average gives the average number of hits per time at bat. For example, a player who has gotten 84 hits in 250 times at bat has a batting average of 84 250 = .336. This average can be interpreted as the empirical probability of that players getting a hit the next time at bat. The following are actual comparisons of hits and at-bats for two major league players in the 2007, 2008, and 2009 seasons. The numbers illustrate a puzzling statistical occurrence known as Simpsons Paradox. The example below, involving Mike Lowell and Jacoby Ellsbury, was provided by the alert Boston Red sox fan Carol Merrigan.

Mike Lowell Jacoby Ellsbury Hits At-bats Batting Average

Hits At- bats

Batting Average

2007 2008 2009 Combined (2007-2009) 191 115 129

___________ 589 419 445

___________

___________ ___________ ___________

___________ 41 155 188

___________ 116 554 624

_____ ___________ ___________ ___________

___________

1. Fill in the twelve blanks in the table. Batting averages must be rounded to three decimal places. 2. Which player had a better average in 2007? 3. Which player had a better average in 2008? 4. Which player had a better average in 2009? 5. Which player had a better average in 2007, 2008, and 2009 combined? 6. Did the results above surprise you? How can it be that one players batting average leads anothers for each of the three years, and yet trails the others for the combined years?