Simpson's ParadoxDate: 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 http://www.ma.iup.edu/~zhang/simpson.html 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 www.google.com). - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ 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 http://mathforum.org/dr.math/ 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? Thanks. 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 http://mathforum.org/dr.math/ |
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