Are there instances of Simpson's Paradox in Physics?

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Main Question or Discussion Point

The so-called Simpson's Paradox arises frequently in medicine, economics, decision sciences, demography and many policy fields. Have there been any instances of it in physics (or chemistry)?

Perhaps the most newsworthy instance of the paradox was in connection with possible gender bias in graduate admissions at University of California, Berkeley. When computed separately, in both humanities and sciences, the rate of acceptance was higher for women applicants relative to male applicants. When the data were pooled, however, the acceptance rate was higher for male candidates.

The paradox comes about because, arithmetically,

a1/b1 > c1/d1
and
a2/b2 > c2/d2

need not imply
(a1+a2)/(b1+b2) > (c1+c2)/(d1+d2)

When the inequality reverses itself in that fashion, it raises serious problems of interpretation and decision-making. I'd expect it to arise whenever density is a key criterion and there's heterogeneity. I'm curious to know if the phenomenon has cropped up in the harder sciences. It would be even more interesting if physics (and chemistry) have been free of it!

The Stanford Encyclopedia of Philosophy has a good entry on the topic:
http://plato.stanford.edu/entries/paradox-simpson/
 

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  • #2
andrewkirk
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For any example of the paradox we can construct an example in physics.

Say there are four closed containers A, B, C and D of gaseous Helium, with volumes b1, b2, d1, d2 respectively, and containing respectively a1, a2, c1, c2 moles of Helium.

Then the densities ##\rho## obey the relations

##a1/b1=\rho_A>\rho_C=c1/d1##; ##a2/b2=\rho_B>\rho_D=c2/d2##.

Now say we use valves to connect container A to container B, and to connect C to D, then open the valves and let the gases reach a new equilibrium.

Then we will have ##\rho_{AB}=(a1+a2)/(b1+b2)<(c1+c2)/(d1+d2)=\rho_{CD}##.

That is, the density of the AB combo will be less than that of the CD combo, even though the density of A is greater than C and the density of B is greater than that of D.

EDIT: Another example would be an inelastic collision between objects A and B and between C and D, where b1,b2,d1,d2 are the masses and a1,a2,c1,c2 are the momenta of A,B,C,D before the collisions. The ratios of momentum to mass will be the speeds of the pairs post collision.
 
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Thanks. Your main example is almost identical to my first exposure to Simpson's paradox. For many years, the GDP per capita has increased for developing countries as a group; and for industrial countries as a group. But for the same years, it has declined for the world!

My query was primarily from the point of view of history of science: whether Simpson's paradox has arisen and led to interpretation issues and debates.
 
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I don't know if you call this physics, but since CAFE standards were put in place, car mileage is up and truck mileage is up, but vehicle mileage is down. It comes about because people who used to buy station wagons now buy SUV's - which are classified as small trucks.
 

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