This post has many empirical problems, but let's start by looking at the male/female deck by Terri Oda. The only data in the entire deck is on slide 21, and the caption reads:
Two normal distributions that are 0.15 standard deviations
apart (i.e d=0.15. This is the approximate magnitude of
the gender difference in mathematics performance,
averaging across all samples.)
In other words, what is plotted there is actually NOT data but simply the textbook Gaussian curves for two distributions.
So let's take a look at actual data. Here is a significantly more rigorous analysis:
This actually uses data from three different tail populations: Female mathematicians in the NAS, Fields Medalists, and Putnam Competition winners. Lo and behold, a simple Gaussian model predicts that small differences in average mathematical ability produce significant sex differences in the tail[1]. And these predictions tally with reality (e.g. the empirical proportion of females in the NAS).
Ms. Oda also does not consider two other crucial aspects, which are:
1. The large difference in spatial ability between men and women:
http://goo.gl/SGhDw
On the whole, variation between men and women tends to be
smaller than deviations within each sex, but very large
differences between the groups do exist–in men’s high
level of visual-spatial targeting ability, for one.
2. The large difference in preferences between men and women:
http://goo.gl/ccKyj
A study by Lubinski and Benbow followed the careers of
mathematically precocious youth from age 13 to 23. All
were in the top 1% of mathematical ability. At age 23 less
than 1% of the girls were pursuing doctorates in
mathematics, engineering, or physical science, while
almost 8% of the boys were. Equal aptitude not
withstanding, girls pursued doctorates in biology at more
than twice the rate of boys, and in the humanities at
almost three times the rate of boys.
The asymmetric part of this whole debate is that someone who voices the above counterarguments in public runs the risk of being browbeaten like Larry Summers or Michael Arrington.
[1] This is of course assuming that distributions remain Gaussian, though it is well known that correspondence to Gaussianity drops off considerably as you move into the tails.
Mathematical achievement doesn't fit a gaussian, and neither do the gendered groups of your conception. And if mathematical ability were distributed as a Gaussian, and genetic, women couldn't be ascending so rapidly.
See for example Lisa Sauermann, the all time best performer in the international math olympiad:
The thing is that intelligence isn't some kind of nice, statistically normed quantity. There's more to most variables than a mean and a standard deviation -- so I don't know why people seem to always think that you can restrict a discussion of intelligence to such concepts.
...
But intelligence isn't a gene. Researchers have, since the time of Galton, tried to find a simple, biological basis for genius. You know, memory capacity, reaction times, brain size, brain structure convolution, etc. They haven't found anything -- literally everything has turned out to be a false start, even brain size, which, has been shown, within families does not even predict g.
In the mid ranges, there are greater standard deviations, yes. But every single normed test is normed on a sample on the order of 1000. They're designed for regular people. The designer of the Weschler has adamantly opposed the use of IQ tests for anything other than clinical settings, for this reason. It's just no good at drawing conclusions on the extremes of ability.
A better guide might be actual performance. The IOI this year had more girls than ever before -- 11. That's nearly 300% more than last year, where they had 4, and one medal. They are:
Emina Bukva (Bosnia and Herzegovina) Constanza Contreras (Chile) Anna Currel (Spain) Romina Huenchunao (Chile) Vaiva Imbrasaite (Lithuania) Taksapaun Kittiakrastien (Thailand) [Silver] Sepideh Mahabadi (Iran) [Gold!] Radwa Metwali (Egypt) Katie O'Mahony (Ireland) Phitchaya Phothilimthana (Thailand) [Silver] Ye Wang (USA) [Silver]
Sepideh Mahabadi had one of the best performances of anyone. If you're familiar with IOI scoring, only the top 1/12 are to get golds, and 2/12 to get silvers, thus 1 gold and 3 silvers out of 11 implies that they did at least as good as the boys, and in fact somewhat better.
Were the 'standard deviation' explanation correct they should have, instead, had 0.4 girls earning maybe 0.01 medals. It just doesn't work.
Why is it sexist to say we show up more frequently in science departments because we have also been designed by evolution to be better at math?
Because compared with bench-pressing, claims of mathematically ability being better in men (and partially ordered, to boot) is seriously jumping the gun.
We know what's involved in a bench press. We understand how testosterone stimulates the production of muscle. We are nowhere close with mathematical ability. We have no theory of mathematically ability -- we really don't know what it means, or if the simplest metrics are even useful for higher level math. We have no experimental results, because we have no controlled variables. We have few pieces of data, none of which are conclusively disentangled from cultural and historical influence.
In the past two decades, the number of women scoring highly on the IMO, the IOI, the Putnam, and SMPY has gone up by roughly a factor of six. Doubtful that the number of girls with 'math talent genes' have sextupled that quickly. Isn't this evidence that we should hold off on our conclusions?
"Brain size does not predict general cognitive ability within families"
http://www.pnas.org/cgi/content/abstract/97/9/4932
You can't discount that IOI statistic becuase it's an outlier. Every single participant at the IOI is an outlier in cognitive ability. Do you know about the theory of outliers?
There's this thing called the central limit theorem. It says that if you have a lot of small independent variables, randomly assigned some value, then the mean of all these variables (or, by the same token, the sum of the variables) is distributed approximately normally. But suppose the variables are not small, or they're not independent. Then the central limit theorem doesn't hold, and what you have, almost all of the time, is an outlier -- that's why there are often many more outliers than you'd predict in a given population, using a small sample.
Now, I'm not saying that g is zero. I said that psychometrics is a non-science, in the same sense that a lot of the social sciences are non-sciences (you can find papers which try to show a causal effect of insurance regulation on premium prices, ignoring profits entirely, for example).
The fact that g is non-zero can be readily explained by the following simple observation -- most academic subtests, including IQ's, rely on skills that are either practiced as a group, or on skills that are shared between subtests. One example is focus, in general. Another is visualisation. Another is working memory. And so on and so on.
Many of these skills are also practiced in situations, like school, where if one does well in one area, they do well in another. If you're the teacher's pet, you get more attention. If you're known as the bad kid, you're immediately discounted (and I've been on both sides). If you're poisoned against a learning environment, you just won't put any effort in.
So it's no mystery to me that g is non-zero. The point is that the field of psychometrics is totally absent of content. There's no objective test for the validity of a test, for example -- the best they have is g-loading. Over the years, this means that tests have become higher and higher g-loaded. Now this could mean that the tests are getting better, or it could be that the subtests only look different, they are becoming more similar in content.
I've been studying these tests, the actual tests, since I was twelve. It didn't take long before I figured out how poor they were at answering research questions, or questions of individual ability. If you get the chance, try to look up the history of the Stanford-Binet, or Terman's kids, or actually take a look at the scoring method behind most of these things. They're totally full of crap...
"And if mathematical ability were distributed as a Gaussian, and genetic, women couldn't be ascending so rapidly."
This is false. It could be that mathematical ability is genetic and normally distributed, but some component of women's underrepresentation is caused by factors other than this (e.g., discrimination, personal preferences).
"It's just no good at drawing conclusions on the extremes of ability. A better guide might be actual performance. The IOI this year had more girls than ever before -- 11... Were the 'standard deviation' explanation correct they should have, instead, had 0.4 girls earning maybe 0.01 medals."
Come on. I know you know better than to draw statistical conclusions from a single event with a sample size of 11. So why do it?
"The fact that g is non-zero can be readily explained by the following simple observation -- most academic subtests, including IQ's, rely on skills that are either practiced as a group, or on skills that are shared between subtests. One example is focus, in general. Another is visualisation. Another is working memory. And so on and so on."
You've come up with a plausible theory of what g is. It might be f(focus, visualization, working memory). This does not make the theory of psychometrics "totally absent of content", it just means they don't understand everything yet.
The fact that pressure is non-zero can be readily explained by the following simple observation -- air is made of particles obeying Newtonian mechanics which impart a force upon a vessel when they collide with it. Does this make thermodynamics "totally absent of content"?
The fact of the matter is that a variety of seemingly unrelated tests are correlated with each other. They are also strongly correlated with various life outcomes in a manner which more or less corresponds to our intuitive intuition about the idea of "intelligence". Stuff like this rarely happens by chance, and it is highly likely there is something behind it.
It could be that mathematical ability is genetic and normally distributed, but some component of women's underrepresentation is caused by factors other than this (e.g., discrimination, personal preferences).
I mean this to be evidence against a genetic explanation of under representation.
Come on. I know you know better than to draw statistical conclusions from a single event with a sample size of 11. So why do it?
Performances like this are more the norm now than ever, but the main reason is that the counterexample is such an extreme outlier that it blows a hole in the gaussian theory of distribution explanation for female under-representation.
They are also strongly correlated with various life outcomes in a manner which more or less corresponds to our intuitive intuition about the idea of "intelligence".
The Terman study of the gifted missed both future Nobel Prize winners in their sample.
The methods used to 'validate' IQ tests are not at all sufficient for equating their results with intelligence, or for arguing that it represents an immutable, genetic, predisposition of groups.
Q: Would you put on your right-thinking left-liberal educated-in-Berkeley-and-Madison hat for a moment?
A: I'd find nothing easier. (You left out the dirty hippyprogressive Montessori school where they taught me Pirandello and Diderot.)
Q: Very good. (It didn't fit the rhythm, and anyway they get the picture.) How would you react to the idea that a psychological trait, one intimately linked to the higher mental functions, is highly heritable?
A: With suspicion and unease, naturally.
Q: It's strongly correlated with educational achievement, class and race.
A: Worse and worse.
Q: Basically nothing that happens after early adolescence makes an impact on it; before that it's also correlated with diet.
A: Do you work at the Heritage Foundation? Such things cannot be.
I mean this to be evidence against a genetic explanation of under representation.
It is evidence against such an explanation, but it's very weak evidence. It shows other factors are involved, which no one disputes, but it does not rule out genetics as a factor.
Incidentally, your Q&A argument is incomplete. We have data which shows accent is not genetic - correlation between the accent of genetic parent and child is gone if you look at adopted children. If there is a twin study on the topic, I'd give 1/p value of the study odds that identical twins raised apart have minimal correlation of accent.
In contrast, identical twins raised apart have a 75% correlation in intelligence and adoptees have a 25% correlation with genetic parents. (I'm working from memory here since I don't have the book with me. The numbers are far from zero, but might be 70% and 20% or 80% and 30%. http://www.amazon.com/Genome-Autobiography-Species-23-Chapte... )
The source you cite is simply being dishonest by leaving this part of the dialogue out of his conversation with a straw man.
Also, your source has a very different philosophical basis for knowledge than most people. He believes that aggregate quantities (e.g., pressure, temperature, possibly g) are statistical myths. See here http://news.ycombinator.com/item?id=2210600 , which is a response to http://cscs.umich.edu/~crshalizi/weblog/523.html (which he cites in the article you link to).
It is evidence against such an explanation, but it's very weak evidence. It shows other factors are involved, which no one disputes, but it does not rule out genetics as a factor.
The core of my argument is that other factors have been shown to dominate underrepresentation for some time. While other factors persist and dominate, it is scientifically incorrect, and ethically irresponsible, to continually make reference to innate differences and outmoded, procrustean interpretations of how talent fits a curve.
It is genuinely too early, if not simply too brutal, to claim a group's intellectual inferiority. The evidence is flimsy, the mathematical framework naive, the conceptual underpinnings insufficiently examined, the data is simply not in.
Worse, people use tiny differences in group performance as justification for wildly unfair stereotypes, as applied to individuals. Even if those supposed group differences withstood criticism at the level of rigor exceeding that accorded to physics or mathematics, it still would be ethically irresponsible to make it publicly and commonly known, or the focus of so much discussion.
This statement: The core of my argument is that other factors have been shown to dominate underrepresentation for some time...
Directly contradicts this one: The evidence is flimsy, the mathematical framework naive, the conceptual underpinnings insufficiently examined, the data is simply not in.
You can't have it both ways. Either intellectual inferiority of some groups is a possibility (i.e., "the data is simply not in"), or else it has been accurately measured and shown not to be the case. Pick one.
Even if those supposed group differences withstood criticism at the level of rigor exceeding that accorded to physics or mathematics, it still would be ethically irresponsible to make it publicly and commonly known, or the focus of so much discussion.
It would be ethically irresponsible to make true facts public? Um, ok.
Have you considered the possibility that you don't actually object to the data/methodology, but rather you are seeking justification to reject conclusions that contradict your moral values?
I am totally confused by your misunderstanding. It is possible there are intrinsic differences, and anyone insisting that this has been shown through appeals to science is being insufficiently rigorous, and scientifically irresponsible.
Where did you get the idea that I was saying it was impossible?
"What is not surrounded by uncertainty cannot be the truth. - Richard Feynman."
Appeals to a supposed scientific justification of the inferiority of women in mathematics are extremely weak, and certainly damaging if untrue. The supposed scientific proof is thus pseudo-science - thinly veiled politics. I'm not going to spell it out again.
people often claim that greater variance in mathematical ability accounts for lower numbers of women in high level scientific positions. this is doesn't match the data: one can't mathematically fit a gaussian curve to explain the distributions. thus the supposed scientific explanation for why women are underrepresented is actually faulty science.
likewise, genetic explanations cannot account for the recent and rapid ascendancy of woman in such technical areas (though we have a long way to go). participation at the highest levels of academic competition has increased dramatically.
given the holes in these, and other, explanations, it is both scientifically and morally irresponsible to posit innate differences as the governing factors. yet despite these arguments, many people still feel they have the station to proclaim some scientific basis for what would otherwise be evidence of continued unequal opportunity.
the scientific claims of a human group's inferiority must withstand scientific scrutiny far beyond that expected in physics for, say, proving faster than light neutrinos. yet the scientific rigor of studies of human inferiority is actually far below that of physics. i ascribe the difference to politics.
an actual big problem preventing women from advancing: being successful, as a woman, is anticorrelated with being liked. this is far from the fault of men alone -- it is a pervasive cultural bias.
see Sheryl Sandberg's TED Talk: why we have too few women leaders.
likewise, academic success among some african american communities is occasionally dismissed as 'acting white.' social pressure is a powerful, powerful thing.
So let's take a look at actual data. Here is a significantly more rigorous analysis:
http://www.lagriffedulion.f2s.com/math.htm
This actually uses data from three different tail populations: Female mathematicians in the NAS, Fields Medalists, and Putnam Competition winners. Lo and behold, a simple Gaussian model predicts that small differences in average mathematical ability produce significant sex differences in the tail[1]. And these predictions tally with reality (e.g. the empirical proportion of females in the NAS).
Ms. Oda also does not consider two other crucial aspects, which are:
1. The large difference in spatial ability between men and women:
2. The large difference in preferences between men and women: The asymmetric part of this whole debate is that someone who voices the above counterarguments in public runs the risk of being browbeaten like Larry Summers or Michael Arrington.[1] This is of course assuming that distributions remain Gaussian, though it is well known that correspondence to Gaussianity drops off considerably as you move into the tails.