Arnold Kling  

The Bell Curve or The Bimodal Distribution?

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Charles Murray writes,


As recently as half a century ago, Americans across all classes showed only minor differences on the Founding virtues. When Americans resisted the idea of being thought part of an upper class or lower class, they were responding to a reality: there really was such a thing as a civic culture that embraced all of them. Today, that is no longer true. Americans have formed a new lower class and a new upper class that have no precedent in our history.

Murray is (in)famously the co-author of The Bell Curve. Its thesis is that intelligence matters for many outcomes. If intelligence is normally distributed and it largely determines one's position in the class structure, then we should observe a class structure that also is normally distributed. Yet Murray today seems to be saying that the class structure is bimodal. At some point, I would like to directly ask Murray how he reconciles his new view with the bell curve.

The story I tell for bimodalism is mating behavior. When high earners marry high earners, class divisions will emerge. But this has implications for the IQ distribution. One would expect bimodalism to appear in the IQ distribution, with the children of high-IQ parents tending centered around one mode and the children of low-IQ parents centered around another.

What does the distribution of IQ scores among today's Americans of age, say 15-24, look like? Are the 75th and 25th percentiles farther apart in terms of IQ than they were fifty years ago? If so, then I would think it is big news. If not, then I do not think that we can tell a story of class divergence in which IQ is a major factor.


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COMMENTS (25 to date)
Dave Schuler writes:

How would you go about diaggregating that from the effects of increased housing segregation, particularly by income level, and the way we finance schooling?

A century ago rich people lived side by side with their servants and their servants' kids. Now they have fewer servants and their servants can live far away.

Glen S. McGhee writes:

Salary distributions for lawyers are famously bifurcated -- but this is tri-modalism when you take into account those without a job or underemployed.
http://www.nalp.org/2008jansalaries
And the trend is worsening, but doesn't seem to be having any effect on law school enrollment.

What many people miss when discussing bimodalism is that it is emergent, and indicates an underlying instability. This is because non-linear bifurcations generate bimodalism and trimodalism.

krulong writes:

How does regression to the mean fit into your story?

Alex J. writes:

1) Price ceilings/floors that matter a incomes below a certain point, but not above it. If your IQ is below 80, say, you can't get a job above the de facto minimum wage and hence you get a black market job and hence you get a criminal record and have a no legitimate work history. Now you are on a different track.

2) Those with access to political power use it for signaling that they care about (or execrate) those low status people who lack it. Because it is a very noisy signal, you don't actually have to call for policy which actually helps anybody, and in fact, people needing help provides more opportunities for you to call for their aid. So, dysfunctional policy hurts those out of power. Those in power can make policy at least adequate for themselves.

Foseti writes:

The thesis of the Bell Curve was that intelligence is normally distributed AND and that we increasingly sorting people by intelligence. For example, on a daily basis, I don't associate with anyone who did not attend a top 25 college. This was not the case 40 years ago.

The latter trend was the part that worried Murray, but everyone had to reach for the smelling salts because he mentioned black people.

An increasingly bimodal intelligence distribution follows from his Bell Curve thesis if intelligence is sufficiently heritable. Smarts are having kids together and dumbs are having kids together - never the two shall meet.

Alex Godofsky writes:

Aren't IQ tests to have constant mean and variance, making your question technically unanswerable? The 75th percentile will always be the same IQ? You'd need to administer the same uncalibrated IQ test to both groups.

Floccina writes:

Interesting and related poor white Americans are no longer attending church. This goes along with them interfacing less with middle class & rich people, marrying less, being more likely to be single parents, lacking wisdom etc.

Floccina writes:

Also are smart high earning guys less likely today to marry a pretty but poor unintelligent girl? If so why?

Paul Rain writes:

Murray has never claimed that the within-population normal distribution of IQ will translate perfectly to some normal distribution of social results. The within-population 75th and 25th percentiles are no further apart- but the advantage that is gained from being in the 75th percentile rather than the 25th percentile is likely a lot easier to take advantage of nowadays than in a less mobile society.

D writes:

The thesis of the Bell Curve was that intelligence is normally distributed AND and that we increasingly sorting people by intelligence.

Correct. Murray already dealt with this. It was one of the book's main ideas. He's now just following up on it, confirming his prediction/observation.

Paul Rain writes:

OTOH there is the additional point to consider that Asian Americans, who are primarily high-IQ East Asians, are rapidly increasing as a percentage of the population. The spread of IQ throughout the population, particularly at the top end, has already changed as a result.

Linda Seebach writes:

To Foesti (who is right about the smelling salts) and D: The second part of Murray's thesis is that individually measured IQ has a more significant effect on life outcomes, good or bad, than family SES. So all the educators who tell you that family income and parental education are the most important factor in children's academic performance are simply wrong; those are proxies for the (deliberately) unmeasured variable of IQ.

James Lee writes:

Suppose that a test consists of item with equal pass rates that are all perfectly correlated. What will be the distribution of the scores on this test look like?

It will be bimodal - everyone will get either a perfect score or a score of zero. If you get the first item right, you will get every single item right. Conversely, if you get the first item wrong, you will get every single item wrong.

This is an extreme example, but it shows that the distribution of test scores is somewhat under the control of the test constructor. Thus it is not straightforward to use the distribution of observed scores to infer anything about the distribution of the trait that the test measures.

It is usually assumed that IQ tests measure a normally distributed trait. Therefore IQ is rescaled to be normally distributed in some reference population *no matter what the distribution of actual scores looks like.*

There are several hand-waving justifications for this assumption. One is that the "true score," from which the observed score can be thought of as differing by an "error of measurement," is the score that would be obtained asymptotically on a test of increasing length, sampling from the infinite set of items a priori known to measure the trait (vocabulary, reading comprehension, arithmetic, geometry, algebra, spatial visualization, and so on). But without an explicit measure on this infinite set, it is difficult to say anything about what the distribution of true scores converges to.

Another justification is that the trait measured by IQ tests must be affected by very many genetic and environmental causes, no single one of which has a large influence on the outcome. The problem with this argument, as I see it, is that many traits of which this is undoubtedly true are not normally distributed. Weight, for example, is not normally distributed, although I believe it is after a log transformation--probably because on the original scale the causes of weight do not act additively. But if we are free to transform to apply any monotonic transformation, what substantive considerations favor one scaling over another?

The totality of the actual and potential knowledge packed into a person's head is an inherently invisible, intangible, odorless attribute. Whether individual differences in mental attributes can be said to have a distribution is a rather deep question that psychometrics and psychology have not fully grappled with.

Chris Koresko writes:

Since nobody else has mentioned it here, let me inject a simple explanation that appeals to me as a conservative.

The distribution of intelligence probably can't change enough in 50 years to make much difference in income levels. So I don't buy the idea that IQ alone can account for bimodally-distributed personal incomes.

What has changed is that our steeply progressive tax rates and our huge social welfare 'safety net' have created a powerful poverty trap. Those who manage to escape it do well, while those who don't are dragged down by it.

Mr. Econotarian writes:

The distribution of intelligence probably can't change enough in 50 years to make much difference in income levels.

Until genetic engineering (or possibly genetic therapy) can influence IQ, that is...

Mark Little writes:

This post disappointingly misses Murray's point I'm afraid. Commenter Foseti nails the main defect, although I doubt the bimodal inference.

The main thesis of Herrnstein and Murray was that the increasingly meritocratic structure of society creates socioeconomic segregation by cognitive ability, resulting in rule by a "cognitive elite" that is out of touch with the non-elite, and so adopts policies often contrary to the interests of the majority. (They found it necessary first of course to spend many pages refuting the tendentious claims--stemming from members of that same elite class--denying the reality of human variability in cognitive ability or that it is measurable.) This is a thesis that one would think that the author of the excellent and not-completely-unread Unchecked and Unbalanced would find in sympathy with his own views.

The resulting assortative mating by cognitive ability (or by the associated social status and educational attainment) is a corollary of Herrnstein and Murray, not an original insight of Kling. But in any case it does not necessarily imply a bimodal distribution of IQ scores.

The impact of assortative mating on the IQ distribution of children should depend on the degree of assortative mating, the heritability of IQ, and on the extent of regression to the mean. (No one, least of all Murray, claims that the segregation of the "cognitive elite" is anywhere near complete; no one, least of all Murray, claims that the heritability of IQ is anywhere near 100%; even if IQ were 100% genetically determined, we would still have regression to the mean--the generation of genetic variability in offspring is the whole evolutionary point of sexual reproduction.)

My intuition is that even with a fairly high level of assortative mating the distribution in the offspring would remain unimodal, and also remain roughly Gaussian in shape. (Although I expect the variance would increase.) My confidence in that intuition is not high. It would be nice to see a formal statistical model of the effect.

Robert writes:

"If intelligence is normally distributed and it largely determines one's position in the class structure, then we should observe a class structure that also is normally distributed."

This is obviously false.

If I divide individuals up into three classes based on their IQ so that individuals are in class 1 if their IQ is less than t1, in class 2 if their IQ is between t1 and t2, and in class 3 if their IQ is greater than t2, then it would be straight forward to pick t1 and t2 so that 40% of people are in class 1, 20% in class 2, and 40% in class 3, so the distribution of class would be non-normal and bimodal.

This can clearly be generalized so that essentially whatever the distribution of IQ and however many classes there are, I can pick any distribution I like for the class structure where an individual's class is completely determind by IQ and it is never the case that a higher IQ individual is in a lower class than a lower IQ individual (ie class is a monotonic function of IQ).

I also disagree with this:
"The story I tell for bimodalism is mating behavior. ...One would expect bimodalism to appear in the IQ distribution."

I haven't thought about this in great depth, but I would conjecture that even if you started out with an extreme bimodal distribution - say 50% with an IQ of 140, and 50% with an IQ of 60 - so long as there is a non zero amount of inter-marriage between High-IQs and Low-IQs, and some random variability in the IQ children inherit from their parents then the distribution of IQ in the population would tend towards a unimodal distribution under fairly mild assumptions.


Steve Sailer writes:

"The story I tell for bimodalism is mating behavior. When high earners marry high earners, class divisions will emerge."

Actually, that was Murray's future co-author Richard Herrnstein's hypothesis way back in his famous September 1971 article "I.Q." in The Atlantic Monthly. Murray's latest book "Coming Apart" on growing class divisions among white Americans shows that Herrnstein's prophecy has turned out increasingly right.

Steve Sailer writes:

James Lee makes a number of excellent points about how much we don't know about I.Q. distributions.

Let me add that there's an 80/20 rule for I.Q. tests: You can an answer about 80% of the questions most people want to use an IQ test to find out about after investing only about 20% of the work into devising the test. But to answer more subtle questions, such as whether there have been shifts of a few points over time, requires a lot more work in testing and sampling.

Steve Sailer writes:

The number of kids scoring 700 or higher on the Math SAT has gone up a lot over the last 15 years. But what are the causes? Assortative mating? Legal immigration? Tiger Mothering? Test prep? Paying some nerd to take the test for you? All of the above? And if the causes are many, how do you divvy them up?

These are good questions and I'd like to see somebody try, but it's a challenging assignment.

John Ray writes:

I think Murray is just saying that the distribution is less leptokurtic now

Hugh writes:

@ Mark Little

I agree with all you have written. The only thing I would add is that Murray found that people at the lower end of the curve were having more children than those with higher levels of ability.

To the extent that IQ is heritable, this will change the position of the curve (over a number of generations).

Glen S. McGhee FHEAP writes:

Don't forget that standardized tests of all kinds are specifically designed to produce normal distributions.

If they did not, no one would use them. There is nothing magical about this.

The whole range of tests using nominal variables, as opposed to raw scores, are predicated on continuous distributions, which is also quite artificial. Item Response test designs go to great lengths to produce these kinds of distributions.


Rastus writes:

Another issue arises from the fact that incomes do not vary in a linear fashion with IQ. Increasingly, those with IQs north of 130 are in a position to reap a huge share of income. Likewise, in a global economy, workers with an IQ below 100 have skillsets that have become commoditized, meaning that the price paid for those skillsets gets driven down to the costs of the workers supplying them. For tasks that can be performed anywhere on the planet (eg, assembling an iPhone), this means that the jobs will go to countries with the lowest-priced labor. This leads to unemployment among low-iq workers, thereby tending to increase the gap between thier incomes and those of the cognitive elite. I don't see this problem doing anything but getting worse.

Curt Doolittle writes:

It looks like it's sort of bi-modal. There are really just two classes: middle class and proletarians, with little genetic overlap. Studies, at least on UK students as they age, seem to bear that out. Upper middle and upper class individuals are lottery winners, and move in and out of the middle class. It also looks like there is a rotation between the lower middle class and upper proletarians. But the proles, the lower proles, and the 'out of sights' (per Fussell) seem to remain constant. And we no longer have the environmental conditions that remove them from the gene pool. IQ, time preference (impulsivity) and morphological symmetry seem to determine who is in each class.

And what's happening (Murray is right) is that classes are becoming internationally oriented due to 'globalization', and competing internationally against their peers. Our upper classes are doing well against their international peers. Our lower classes are not. And we aren't helping them by failing to educate them so that they are able to compete internationally.


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