The physics and psychometrics of breakfast-cereal inequality

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Nicholas Watt, "Boris Johnson invokes Thatcher spirit with greed is good speech", The Guardian 11/27/2013:

Boris Johnson has launched a bold bid to claim the mantle of Margaret Thatcher by declaring that inequality is essential to fostering "the spirit of envy" and hailed greed as a "valuable spur to economic activity".

In an attempt to shore up his support on the Tory right, as he positions himself as the natural successor to David Cameron, the London mayor called for the "Gordon Gekkos of London" to display their greed to promote economic growth.

This speech bolsters Mr. Johnson's already-strong claim to be the most Dickensian of modern politicians. I was especially impressed by the following passage:

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… and I'm afraid that violent economic centrifuge
is operating on human beings who are already very far from equal
in raw ability
if not in spiritual worth.
Whatever you may think of the value of IQ tests
it is surely relevant to a conversation about equality
that as many as sixteen percent of our species
have an IQ below eighty five
while about two percent –
((about- anyway sixteen percent of you want to put up your hands?))
sixteen percent have an IQ below- uh uh below eighty five
uh two percent have an IQ above a hundred and thirty.
And the harder you shake the pack
the easier it will be for some cornflakes to get to the top.

Mr. Johnson raises two scientifically-interesting issues. One is the Brazil Nut Effect — as explained in Matthias Möbius et al., "Brazil-nut effect: Size separation of granular particles", Nature 2001,

Granular media differ from other materials in their response to stirring or jostling — unlike two-fluid systems, bi-disperse granular mixtures will separate according to particle size when shaken, with large particles rising […]

For particles of equal density, like corn flakes, the effect is generally due to smaller particles falling rather than to larger particles rising. But depending on other circumstances, larger particles can actually sink rather than rise — see e.g. Troy Shinbot, "Granular materials: The brazil nut effect — in reverse", Nature 2004. It seems that which breakfast-cereal fragments "get to the top" depends not only on relative size, but also on the distribution of densities, on air pressure, and on other parameters as well.

The question of whether granular separation is a Good Thing also has different answers depending on circumstances. In general, granular separation of breakfast cereals and other bulk products is not what either producers or consumers want.

The second issue is the quantification of corn-flake size intelligence. As Mr. Johnson explains,

Whatever you may think of the value of IQ tests
it is surely relevant to a conversation about equality
that as many as sixteen percent of our species
have an IQ below eighty five

Some might question the relevance of IQ to the question of economic inequality by citing studies like Jeffrey Zax and Daniel Rees, "IQ, Academic Performance, Environment, and Earnings", The Review of Economics and Statistics 2002, which found that IQ explained only about 5-8% of the variance in earnings:

But my point is a different one. Since IQ tests are carefully and elaborately calibrated to generate a normal distribution of scores, with mean = 100 and standard deviation = 15, it follows from the definition of "normal distribution" that about 15.87% of test-takers must score 85 or below. This would remain true if individual differences in the population being tested were ten times greater than they are now, or ten times less — the scores would simply be re-normed to conform to the definition.

[Update — As Ran Ari-Gur points out in the comments, this procedure ensures that even if the underlying distribution of "ability"  is far from normal, with quantiles arbitrarily crowded together or spread apart in different ranges, the test scores (and perhaps the test procedures) will simply be re-adjusted to fit the stipulation that the results should fit a normal distribution with mean 100 and standard deviation 15. So the observation that "sixteen percent of our species have an IQ below 85" is exactly as informative as the observation that "half of our species are below median height".]

A few years ago, I commented on another attempt to derive socio-political theorems from statistical definitions:

"There are serious problems in the legislation, and that was recognized when Congress passed the bill," said Education Prof. Fred Hess, director of NU's Center for Urban School Policy.  […]

Hess said some of the act's problems go beyond funding. The tests being used are formulated so that 50 percent of the test-takers will fall below the median score — in effect setting school districts up for failure no matter how much preparation students receive, he said.

Mr. Johnson's rhetorical flourish about the distribution of IQ scores is similarly meaningless, though in a slightly more complex and better-disguised way.





  1. Rodger C said,

    November 28, 2013 @ 1:47 pm

    I'm interested in the phrase "very far from equal in raw ability if not in spiritual worth." Did he mean "but not in spiritual worth" or "and quite possibly in spiritual worth as well"? I think I know which he'll say he meant.

    [(myl) "If not" leaves it open, as you suggest. Some random examples in each direction:

    We face a dim if not dismal future. = the future may well be dismal
    This violates the spirit if not the letter of the law. = the law is not literally violated

    If Mr. Johnson had wanted to state an unambiguous belief in equality of spiritual worth, he could have said something like "very far from equal in raw ability, though equal in spiritual worth". The Maxim of Quality tells us that his choice was a meaningful one.]

  2. Anthony said,

    November 28, 2013 @ 2:24 pm

    Ages ago, I read a commentary riffing on FDR's comment about "one third of the nation is ill-fed, ill-housed, and ill-clothed", in which the author blamed that circumstance on statisticians. The article was about the primarily relative measures used for poverty during the post-war era.

  3. Ran Ari-Gur said,

    November 28, 2013 @ 2:25 pm

    > This would remain true if individual differences in the population being tested were ten times greater than they are now, or ten times less — the scores would simply be re-normed to conform to the definition.

    What's more, it would remain true even if the difference between 85 and 100 were minuscule and all-but-undetectable while the difference between 100 and 115 were so great as to be miraculously apparent to the naked eye. (Or vice versa.) So it's not that IQ is an expression of some normally-distributed variable, with just the mean and standard deviation being arbitrarily assigned certain values (100 and 15); rather, it's that IQ is an expression of some variable of unknown distribution, with all percentiles being arbitrarily assigned values according to what they would be if the variable were normally distributed with μ = 100 and σ = 15.

  4. hector said,

    November 28, 2013 @ 4:23 pm

    Not to mention that high intelligence is hardly a predictor of success in a greed-based society.

    Really, his argument is quite weak. If it's Gordon Gekkos he's after, then his argument against equality should be based on the unequal distribution of sociopathy in human societies.

  5. Rubrick said,

    November 28, 2013 @ 5:17 pm

    The likelihood that someone will mention the News From Lake Wobegon sign-off in this post's comments is 100%.

  6. Stephen said,

    November 28, 2013 @ 5:22 pm


    Well, it is NOW.

  7. Eric P Smith said,

    November 28, 2013 @ 8:06 pm

    I read Boris as meaning, “They are very far from equal in raw ability: I say nothing about whether they are equal in spiritual worth.” He is not asserting that they are equal in spiritual worth, but he has sought to defend himself against any accusation that he is denying it.

    [(myl) Exactly. If I write "Eric P Smith appears to be correct, if perhaps not insightful", I'm not asserting that he is not insightful. But by bringing the issue up without making such an assertion, though it would have been easy and relevant for me to do so, I imply something.]

  8. Aygül Özkaragöz said,

    November 29, 2013 @ 5:48 am

    Judging by the distribution of the utterly and astoundingly inept politicians and bureaucrats in important posts in Turkey, the identity of the "shaker" is also important in determining the process of floating-to- the-top.

  9. D.O. said,

    November 29, 2013 @ 11:43 am

    I am a bit surprised by the results of Zax and Rees. It seems to be quite a firm conviction of some right-thinking people that IQ is the best predictor of the lifetime earnings (something related to The Bell Curve). Of course, labor market earnings is different from the earnings from all sources. Also, if distribution of earnings is not even approximately normal, what linear regression can tell us about the relationship?

    Anyways, what do you think about putting a bunch of people in a giant bowl, shaking it violently and then correlating relative position of people with their IQ?

    [(myl) Zax and Rees survey the literature on "IQ and Earnings" at some length in Section II of their paper. And in their conclusion, they say:

    In sum, adult economic performance is related to several different adolescent contextual levels. The true effects of context almost surely arise from complicated and subtle social interactions that are only crudely approximated by the contextual measures available here. From this perspective, this analysis may still understate them.

    At the same time, previous analyses have overstated the role of intelligence in economic success. Controls for family and high school context dramatically reduce the estimated income effects of IQ. […]

    Perhaps the most striking result here is, however, the limited scope of all effects included in this analysis. At least 85% of the variation in earnings at age 35, and 75% of that at age 53, is orthogonal to everything measured at age 18.

    As for the issue of earnings distribution, I believe that they regress against log income, which is much closer to being normally distributed.]

  10. D.O. said,

    November 29, 2013 @ 6:46 pm

    Thank you for the reply. I also see this, footnote 4:

    IQ measurements late in high school are appropriate here because the issue is whether subsequent interventions can compensate for cognitive deŽficits in the determination of earnings during the peak years. In addition, these measurements can incorporate only limited experience effects. In contrast, Bishop (1989, p. 180) requires “that GIA [general intellectual achievement] be measured long after the completion of schooling and as close as possible to the date of the wage rate observations ” because “the more recent test is by far the more powerful predictor of earnings.” Achievement test scores that are contemporaneous with earnings measures must derive some of their predictive power from their relationship with work experience (Bishop, 1989, p. 179; Neal & Johnson, 1996, p. 873).

    The latest statement can be interpreted as "high paying job is good for you IQ". Isn't it nice.

    Also, they use IQ data from 1957 Wisconsin, and report that in their sample average IQ was 101 with standard deviation 5.12. Huh? Never mind that average and standard deviation have to be reported with the same number of significant figures. What, something happened with the magical s.d. 15? Wisconsin was an epitome of average back then? Anyways, sorry for all these ramblings, because I really don't plan to go deeper in figuring out what is in this paper.

    [(myl) Those numbers are from Table 1, which "presents averages of the contextual variables used here, for the sample of 2,959 male respondents analyzed in section IV". These are "Community Characteristics", so that the IQ values (for example) are means and standard deviations not for individuals, but for schools. That's why the sd is so low. Table 2, which presents "Household and Individual Variables" for (apparently the same) 2,959 men, gives an average IQ of 102 and a standard deviation of 14.9, which is close to the nominally expected values.]

  11. Lane said,

    November 30, 2013 @ 6:06 am

    I did a little, slightly unscientific, poll and found that "if not" was analyzed by 44 respondents the one way, and by 17 the other, and so with my advice hat on, deemed it too confusing to be recommended for use.

  12. Pat said,

    December 2, 2013 @ 12:09 pm

    If Mr. Johnson had wanted to state an unambiguous belief in equality of spiritual worth, he could have said something like "very far from equal in raw ability, though equal in spiritual worth". The Maxim of Quality tells us that his choice was a meaningful one

    There's also a principle of charity, is there not? Given the hypotheses that a major British politician either used an awkward construction, or publicly denied the "spiritual equality" of human beings, do folks here seriously entertain the latter possibility? Before the point was raised in the comments, it seemed clear to me that "if not" was intended as "though not". Upon rereading, it still strikes me as needlessly uncharitable to reject that interpretation.

    Ultimately I don't quite understand the animus against Johnson, who is basically arguing for a sort of pop-Rawlsianism:

    The mayor added: "It seems to me that though it would be wrong to persecute the rich, and madness to try and stifle wealth creation, and futile to stamp out inequality, we should only tolerate this wealth gap on two conditions. One, that we help those who genuinely cannot compete; and two, that we provide opportunity for those who can."

    This is admittedly not a verbatim quotation of the Difference Principle, but it's pretty orthodox welfare-state liberalism.

    People do differ in their raw cognitive abilities, and it would be highly surprising if their life outcomes did not reflect that. (As other commenters have pointed out, this correlation between IQ and income is far from unity, but it is not zero either.) The folks in this thread snarking about the definitional normality of the IQ distribution seem rather uncurious about what a difference of one or two s.d.s makes in terms of one's ability to navigate the world, on average: there is a smooth gradation in human cognitive ability which ends in a left tail of people who are unable to live independently. It seems strange to argue (correctly, in my view) that the state has a responsibility to care for the severely congenitally cognitively impaired without acknowledging that there is such a thing as severe congenital cognitive impairment.

    My apologies if I've been uncharitable in my own readings of other folks' comments, but I'm having a hard time locating Johnson's misstep.

  13. Will said,

    December 7, 2013 @ 10:13 am

    Something similar was noticed by dilbert a few years ago:

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