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Simpson's paradox arises in statistics, when a trend in independent groups becomes different when they are combined. For instance, baseball player A might have a better average than player B in two successive seasons, but Player B could have a higher overall average; girls might have a higher test average in each of two math classes but boys could have a higher overall average when all results are pooled. Also, a big urban hospital might have a lower record of patient recovery after heart attacks, but rate higher because of the risks of the population they are serving.

It's a result of differing size groups in data collection.

I am working on a chemical application of it where an object sinks in liquid A, a different object sinks in liquid B, and when you mix the liquids, and strap the objects together, they float. The principle is analogous, involving differently sized objects. (I've actually created a demonstration but am working out a problem before I try to publish it.)

My question: can anyone think of completely different fields from statistics or buoyancy where Simpson's might lead to a dramatically counterintuitive result?

RobPratt
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  • This can happen whenever you are computing a weighted average of a quantity that is a ratio. That is "statistics" in your buoyancy example. Just look for other fields where you compute averages of averages. – Ethan Bolker Feb 21 '25 at 00:36
  • There was a famous example where graduate departments (at Berkeley?) were admitting a higher percentage of female applicants than of male applicants, but the university as a whole was accepting a higher percentage of male applicants than of female applicants (or maybe it was the other way around). But I guess that counts as "statistics". – Gerry Myerson Feb 21 '25 at 01:34
  • There are more examples at https://math.stackexchange.com/questions/83756/more-examples-of-simpsons-paradox-barring-the-ones-on-wikipedia-titanic-and but I suppose they all count as "statistics". – Gerry Myerson Feb 21 '25 at 02:07
  • I have posted two answers. Any thoughts on them, Nalacram? – Gerry Myerson Feb 23 '25 at 02:36
  • Thank you for your answers. As you concluded, they are also statistical; I’m looking for completely different manifestations, as the density example proved to be. E.G., perhaps examples within acoustics? Motion? Etc. –  Feb 23 '25 at 03:49

3 Answers3

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Here's an example with motion:

Alpha drives 90 miles at 30 mph on Monday, and 50 miles at 50 mph on Tuesday.

Beta drives 20 miles at 20 mph on Monday, and 200 miles at 40 mph on Tuesday.

So, Alpha drives faster than Beta each day.

But putting the two days together, Alpha drives 140 miles in 4 hours, for an average speed of 35 mph, while Beta drives 220 miles in 6 hours, for an average speed of a bit over 36 mph, so Beta drives faster overall.

Gerry Myerson
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  • Not bad…distinctly different I guess! Thank you for giving so much attention to this one question. –  Feb 24 '25 at 04:54
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The vector $u=(1,0)$ has a smaller slope [meaning, slope of the line segment connecting the origin to $u$] than the vector $x=(5,1)$; the vector $v=(3,5)$ has a smaller slope than the vector $y=(1,2)$; but the sum $u+v=(4,5)$ has bigger slope than the sum $x+y=(6,3)$.

$0/1<1/5$
$5/3<2/1$
$(0+5)/(1+3)=5/4>3/6=(1+2)/(5+1)$

Gerry Myerson
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Wikipedia says,

Berman, S.; DalleMule, L.; Greene, M.; and Lucker, J. (2012), "Simpson's Paradox: A Cautionary Tale in Advanced Analytics," give an example from economics, where a dataset suggests overall demand is positively correlated with price (that is, higher prices lead to more demand), in contradiction of expectation. Analysis reveals time to be the confounding variable: plotting both price and demand against time reveals the expected negative correlation over various periods, which then reverses to become positive if the influence of time is ignored by simply plotting demand against price.

Gerry Myerson
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  • Correlation is not causation. In this case, it seems that both price and quantity sold had a tendency to decline over time, but in other cases you might expect higher demand could lead to higher prices and so overall demand could be positively correlated with price. – Henry Mar 02 '25 at 00:23