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Simpson's Paradox

Date: 02/27/2002 at 11:16:05
From: Emily Wagman
Subject: Simpson's paradox

Adam and Kate are professional cricket batters. Lauren and Leora are 
professional cricket pitchers. During the regular season, Adam has a 
higher batting average against Leora than Kate does.

If Lauren is a better pitcher against Adam than Leora is, and Lauren 
is a better pitcher against Kate than Leora is, can Leora be a better 
overall pitcher against Adam and Kate than Lauren is?

Please help.

Date: 02/27/2002 at 13:48:23
From: Doctor Schwa
Subject: Re: Simpson's paradox

Hi Emily,

I'll answer the second question, which is the most extreme: how can 
Lauren be a better pitcher against Adam, a better pitcher against 
Kate, and yet worse overall?

Well, suppose Adam is a much better hitter (since you said Adam has a 
higher batting average against Leora). Then, if Lauren has to pitch 
against Adam a lot more often, she might come out worse overall than 
Leora, who gets to face Kate a lot more often.

For instance:

   Lauren vs Adam, Adam gets 30 hits out of 100 batting attempts,
   Leora vs Adam, Adam gets 4 hits out of 10 batting attempts

   Lauren vs Kate, Kate gets 2 hits out of 10 batting attempts
   Leora vs Kate, Kate gets 25 hits out of 100 batting attempts.

Then, Lauren is better vs Adam (30% vs 40% for Leora),
Lauren is better vs Kate (20% vs 25% for Leora),
yet Leora is better overall (29 hits out of 110 vs 32 for Lauren).

I hope that helps make Simpson's paradox more sensible!

For another example, try 

   What is Simpson's Paradox? - John Zhang   

which shows the importance of Simpson's paradox: looking at summary 
statistics can give you a very misleading picture of what's going on.

You can find more similar pages by searching for 
  simpson paradox
at your favorite search engine (I used

- Doctor Schwa, The Math Forum   

Date: 12/01/2001 at 14:07:13
From: Micah
Subject: Simpson's Paradox (a.k.a. Stein's Paradox)

Dear Dr. Math,

What exactly is Simpon's Paradox (a.k.a. Stein's Paradox) and how is 
it possible? All that I could find relating to it are examples, (i.e. 
"My batting average against right-handed pitchers is higher than 
yours, and my batting average against left-handed pitchers is higher 
than yours, but your overall batting average is higher than mine."). 
These examples helped a little, but I'm still a confused about it.

Date: 12/01/2001 at 19:07:48
From: Doctor Mitteldorf
Subject: Re: Simpson's Paradox (a.k.a. Stein's Paradox)

The best way to get a feel for it is to invent some numerical examples 
of your own, and calculate them out. Think of how many RH pitchers 
I've faced and how many hits I've got, how many LH pitches I've faced 
and how many hits, and the same for you. Invent a set of numbers that 
satisfies the given conditions. Hint: Suppose we both do a whole lot 
better against RH pitches than against LH pitchers, and you were just 
lucky enough to face LH pitchers most of the time, while I faced RH 
pitchers most of the time.

- Doctor Mitteldorf, The Math Forum   

Date: 02/11/2002 at 15:48:36
From: Emily
Subject: Simpson's paradox

There are two batters. Batter one has a better first and a better 
second half than batter two. Can batter two have an overall better 
batting average?

   batting ave. = numbers of hits/how many times up at bat

The two batters don't have to be up at bat the same number of times.

I know the answer is yes, but why? I don't know how to generalize it.
Can you help?


Date: 02/11/2002 at 17:24:50
From: Doctor Tom
Subject: Re: Simpson's paradox

It's pretty easy.  Here's an example:

In the first half, A gets 1 of 1; B gets 9 of 10.
In the second half, A gets 1 of 10; B gets 0 of 1.

Totals: A gets 2 of 11; B gets 9 of 11.

- Doctor Tom, The Math Forum   
Associated Topics:
High School Statistics

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