Today I learned about Simpson's paradox in statistics.

The Simpson's chalkboard gag with Bart Simpson writing “The Simpson's paradox isn't mine” repeatedly on a chalkboard.
The Simpson's chalkboard gag reading “The Simpson's paradox isn't mine”.

What is the Simpson's paradox?

The Simpson's paradox arises when the trends shown by several groups of data disappear, or reverse!, when the groups are combined.

Example of Simpson's paradox

Let me show you a good example taken from the Wikipedia.

The batting average is a statistic used in sports such as baseball, cricket, and softball. I have no intention of offending players of any of these sports, but for the sake of simplicity and brevity, let's assume that “batting average” is the number of times a player hits the ball divided by the number of times the player goes to bat.

With that out of the way, let me share some numbers with you:

Batter vs Year 1995 1996 '95 & '96
Derek Jeter .250 .314 .310
David Justice .253 .321 .270

The data shown in the table above is real: it shows that David had a higher batting average than Derek in both 1995 and 1996. However, if we combine the two years, Derek's batting average is better than David's.

To understand why this is the case, it is very important that I share the underlying data that allows one to compute the batting average:

Batter vs Year 1995 1996 '95 & '96
Derek Jeter .250 (12/48) .314 (183/582) .310 (195/630)
David Justice .253 (104/411) .321 (45/140) .270 (149/551)

As one can see, Derek's superior overall batting average is explained by the fact that he batted at .314 in a year with a total of 582 opportunities, which outweights the fact that David batted .321 in a year with only 140 opportunities.

Have you ever tripped up on Simpson's paradox before? Where?

That's it for now! Stay tuned and I'll see you around! (Thumbnail image with the Simpson's chalkboard gag courtesy of this webpage.)

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