Middleton Cricket Club plays a triangular tournament against its two local rivals, Easton Cricket Club and Weston Cricket Club, throughout the season. It is a keenly-fought but friendly competition that has been played for decades, and thus it is steeped in history.
At Middleton Cricket Club there are two prestigious awards given to players, based on their performances in the triangular tournament. There is one trophy given to the player with the best batting average, and another for the player with the best bowling average. These trophies go back almost as far as the tournament itself, and the players are very keen to win them.
[For readers who are not completely up to speed on batting and bowling averages, here is a quick explanation.
A player's batting average is the number of runs they have scored divided by the number of times they are out. If a player scores 100 runs and is out four times the batting average is 25. The higher the batting average, the better.
Similarly, a player's bowling average is the number of runs scored off their bowling divided by the number of batsmen they get out. If a player has 100 runs scored off their bowling and gets five batsmen out the bowling average is 20. The lower the bowling average, the better.]
This season the trophy for the best batting average is closely fought between Sam Slogger and Neil Nurdler, with the winner in doubt until the final match. The players' records against both Easton and Weston have been displayed on boards throughout the season, so both players know how the other is performing.
After the final game has finished the boards are brought up to date, and the two players' records are as follows:
Sam Slogger is over the moon. He has the highest batting average for the season against both Easton and Weston, and immediately buys all of his team mates a drink in celebration.
But what happens when the prizes are given out at the end of the evening that leaves a bitter taste in Sam Slogger's mouth?
You would think that if a player has the best batting average against both the other teams that they must have the best batting average for the season. Sam Slogger certainly thought so, and he was wrong too.
The overall batting averages are:
Neil Nurdler has scored more runs and been out fewer times than Sam Slogger, and so must have the higher average.
This is an example of Simpson's paradox, where a trend that is evident in individual data sets reverses in when the data sets are combined.
In this particular case, it would appear from the batting averages that Easton is a stronger team than Weston. Sam Slogger played more games against Easton and Neil Nurdler more games against Weston, which is probably why the paradox happens here.