31 March 2012

Women are more average than men


The table below is taken from the American National Assessment of Educational Progress (NAEP). It collects male and female results over fifteen years and five subjects.  NAEP is a true survey, so problems of self-selection (rife in college entrance exams, licensing exams and so on) are substantially reduced. (American Scientist volume 95, May-June 2007, p. 255)

On average, males score higher in math, science, geography and history, females in reading. But whatever the subject, the standard deviation of males is from 3 to 9 percent greater than females.


It does appear that on many, many, different human attributes—height, weight, propensity for criminality, overall IQ, mathematical ability, scientific ability— there is relatively clear evidence that whatever the difference in means—which can be debated—there is a difference in standard deviation/variability of a male and female population. And it is true with respect to attributes that are and are not plausibly culturally determined. (Lawrence Summers, Remarks at NBER Conference on Diversifying the Science & Engineering Workforce, 2005.)


From Howard Wainer, The Most Dangerous Equation (American Scientist volume 95, May-June 2007, p. 249-256).

For many years it has been well established that there is an over-abundance of males at the high end of academic test-score distributions. About twice as many males as females received National Merit Scholarships and other highly competitive awards. Historically, some observers used such results to make inferences about differences in intelligence between the sexes. Over the past few decades, however, most enlightened investigators have seen that it is not necessarily a difference in level but a difference in variance that separates the sexes. (...)

The males' score distributions are almost always characterized by greater variance than the females'. Thus while there are more males at the high end, there are also more at the low end.(...)

Both inferences, the incorrect one about differences in level, and the correct one about differences in variability, cry out for explanation. The old cry would have been "why do boys score higher than girls?" The newer one should be "why do boys show more variability?" (...) Obviously the answer to the causal question "why?" will have many parts. Surely socialization and differential expectations must be major components—especially in the past, before the realization grew that a society cannot compete effectively in a global economy with only half of its workforce fully mobilized. But there is another component that is key—and especially related to the topic of this essay. In discussing Lawrence Summers's remarks about sex differences in scientific ability, Christiane Nusslein-Volhard, the 1995 Nobel laureate in physiology and medicine, said: "He missed the point. In mathematics and science, there is no difference in the intelligence of men and women. The difference in genes between men and women is simply the Y chromosome, which has nothing to do with intelligence." But perhaps it is Professor Nusslein-Volhard who missed the point here. The Y chromosome is not the only genetic difference between the sexes, although it may be the most obvious. Summers's point was that when we look at either extreme of an ability distribution we will see more of the group that has greater variation. Mental traits conveyed on the X chromosome will have larger variability among males than females, for females have two X chromosomes, whereas males have an X and a Y. Thus, from de Moivre's equation we would expect, all other things being equal, about 40 percent more variability among males than females. The fact that we see less than 10 percent greater variation in NAEP scores demands the existence of a deeper explanation. First, de Moivre's equation requires independence of the two X chromosomes, and with assortative mating this is not going to be true. Additionally, both X chromosomes are not expressed in every cell. Moreover, there must be major causes of high-level performance that are not carried on the X chromosome, and still others that indeed are not genetic. But for some skills, perhaps 10 percent of increased variability is likely to have had its genesis on the X chromosome. This observation would be invisible to those, even those with Nobel prizes for work in genetics, who are ignorant of de Moivre's equation. 

It is well established that there is evolutionary pressure toward greater variation within species—within the constraints of genetic stability. This is evidenced by the dominance of sexual over asexual reproduction among mammals. But this leaves us with a puzzle. Why was our genetic structure built to yield greater variation among males than females? And not just among humans, but virtually aU mammals. The pattern of mating suggests an answer. In most mammalian species that re- produce sexually, essentially all adult females reproduce, whereas only a small proportion ot males do (modern humans excepted). Think of the alpha-male lion surrounded by a pride of females, with lesser males wandering aimlessly and alone in the forest roaring in frustration. One way to increase the likelihood of offspring being selected to reproduce is to have large variance among them. Thus evolutionary pressure would reward larger variation for males relative to females.


It's a simple observation that men and women are very different creatures. The average man is taller, faster and stronger than the average woman; his brain is larger and functions differently. No matter how many exceptions you happen to know, they're statistically irrelevant. (Einstein's brain, for instance, was below female average.) So it's not to be expected that the average man and average woman would miraculously have the same IQ. After all, if evolution came up with mammals in two sexes, it's not to have them as identical as possible. You may have noticed that both Summers and Wainer are very careful in stating anything about statistical means for men and women. Both know it may be dangerous to refer to scientific facts not in line with PCD (Politically Correct Dogma). Summers, prudent as he was, was toppled as Harvard President after street protests by an angry mob calling him a sexist. He was knowledgeable and right, they were ignorant and wrong, but he was the one that had to go. Scary, isn't it? Makes one think of "Dark Ages", when dogma prevailed over facts.

To return to the point made by Summers and Wainer: a male sample has more variation than a female one, and Wainer even provides an evolutionary explanation for that fact. So, yes, geniuses are predominantly male. Idiots too.

By the way, the "most dangerous equation" according to Wainer is de Moivre's: the standard error of the mean is the standard deviation of the sample divided by the square root of the size of the sample. Read the paper to see how more than a billion dollars were lost on a theory based on ignorance of this equation.