Summers Vindicated (again)

by on July 28, 2008 at 7:44 am in Science | Permalink

For the past week or so the newspapers have been trumpeting a new study showing no difference in average math ability between males and females.  Few people who have looked at the data thought that there were big differences in average ability but many media reports also said that the study showed no differences in high ability.

The LA Times, for example, wrote:

The study also undermined the assumption -- infamously espoused by former Harvard University President Lawrence H. Summers in 2005 -- that boys are more likely than girls to be math geniuses.

Scientific American said:

So the team checked out the most gifted children. Again, no difference. From any angle, girls measured up to boys. Still, there’s a lack of women in the highest levels of professional math, engineering and physics. Some have said that’s because of an innate difference in math ability. But the new research shows that that explanation just doesn’t add up.

The Chronicle of Higher Education said:

The research team also studied if there were gender discrepancies at the highest levels of mathematical ability and how well boys and girls resolved complex problems. Again they found no significant differences.

All of these reports and many more like them are false.  In fact, consistent with many earlier studies (JSTOR), what this study found was that the ratio of male to female variance in ability was positive and significant, in other words we can expect that there will be more math geniuses and more dullards, among males than among females.  I quote from the study (VR is variance ratio):

Greater male variance is indicated by VR > 1.0. All VRs, by state and grade, are >1.0 [range 1.11 to 1.21].

Notice that the greater male variance is observable in the earliest data, grade 2.  (In addition, higher male VRS have been noted for over a century).  Now the study authors clearly wanted to downplay this finding so they wrote things like "our analyses show greater male variability, although the discrepancy in variances is not large."  Which is true in some sense but the point is that small differences in variance can make for big differences in outcome at the top.  The authors acknowledge this with the following:

If a particular specialty required mathematical skills at the 99th percentile, and the gender ratio is 2.0, we would expect 67% men in the occupation and 33% women. Yet today, for example, Ph.D. programs in engineering average only about 15% women.

So even by the authors' calculations you would expect twice as many men as women in engineering PhD programs due to math-ability differences alone (compare with the media reports above).  But what the author's don't tell you is that the gender ratio will get larger the higher the percentile.  Larry Summers in his infamous talk, was explicit about this point:

...if one is talking about physicists at a top twenty-five research university, one is not talking about people who are two standard deviations above the mean...But it's talking about people who are three and a half, four standard deviations above the mean in the one in 5,000, one in 10,000 class. Even small differences in the standard deviation will translate into very large differences in the available pool substantially out.

If you do the same type of calculation as the authors but now look at the expected gender ratio at 4 standard deviations from the mean you find a ratio of more than 3:1, i.e. just over 75 men for every 25 women should be expected at say a top-25 math or physics department on the basis of math ability alone (see the extension for details on my calculation).  Now does this explain everything that is going on?  I doubt it.  As Summers also pointed out it takes more than ability to become a professor at Harvard and if there are variance differences in characteristics other than ability (and there are) we can easily get a even larger expected gender ratio.

Does this mean that discrimination is not a problem?  Certainly not but we need the media and academia to accurately present the data on ability if we are to understand how large a role other issues may play.

Addendum: Andrew Gelman points out that perhaps alone among the media, Keith Winstein at the WSJ reported the story correctly.

———-

The authors show variance ratios of 1.11 to 1.21, I take a VR of 1.16.  If we set the female variance to 1 this implies the standard deviation for female ability is 1 and for male ability 1.077.  Using an online calculator for the Normal distribution you can find that given their standard deviation .0102% of males have ability of 4 or greater (4 female sds) but given their sd only .0032% of females can be expected to have the same level of ability, thus a gender ratio of 3.18.

Note that we are assuming that mathematical ability is normally distributed – we know the data fit this distribution around the mean but we don’t know much about what happens at the very top.

1 londenio July 28, 2008 at 8:02 am

Many of these reports talk about “innate Math ability”. But let’s not forget that mathematical ability, like chess-playing ability, is a convolution of other abilities and character traits. For instance, any subpopulation with higher level of obsessiveness (e.g. males) will be overrepresented in academia. Future studies should disentangle these things. We may all be equal in Math ability, but we may have different willingness to spend hours sitting in a chair solving mathematical problems.

2 Tom West July 28, 2008 at 8:09 am

*sigh* Haven’t had my coffee. “Girl’s can’t do math” -> “Girls can’t do math”.

3 Billare July 28, 2008 at 8:35 am

Yes, yes, commenters, this is the same Tom West who argues we should suppress such research because anyone who raises SCIENTIFIC questions over major social issues might be doing so for ends Tom West doesn’t like, and couldn’t possibly be inevitable in the new genetic age.

4 Renee July 28, 2008 at 8:48 am

I love XKCD.

I hear the following all the time from my fellow left-wing intellectuals when talking about the unpopular nuances in arguments: we know there’s more to it than that, but we can’t count on other people to know that. Therefore we should encourage them not to talk about it at all.

Drives me crazy.

5 Anonymous July 28, 2008 at 8:55 am

Regardless of what Gore said, Gore’s comments were interpreted by almost everyone as essentially, “I invented the Internet”.

Regardless of what Obama said, …

You are being far too disingenuous, Tom West. If someone you actually supported and liked was widely misinterpreted, you would be the first to protest indignantly at the injustice.

6 Keith July 28, 2008 at 9:14 am

“Someone in a public position speaking in a public venue is responsible not only for the content of what he says, but how that content will be interpreted and used by the greater public. Being “technically correct” doesn’t get you off.”

You’re not allowed to say anything smart in public, because dumb people exist!

Tom, if you think I’m misrepresenting your position, just remember – it’s your own damn fault for saying something that I could interpret that way.

7 Jason Malloy July 28, 2008 at 9:24 am

One more factually inaccurate and/or libelous Summers jab from the New York Times:

“Three years after the president of Harvard, Lawrence H. Summers, got into trouble for questioning women’s “intrinsic aptitude” for science and engineering a study paid for by the National Science Foundation has found that girls perform as well as boys on standardized math tests.”

8 Jason Malloy July 28, 2008 at 9:28 am

Meanwhile John Tierney continues to expose the flimsiness of sexual discrimination claims on his New York Times blog.

9 Thomas Purves July 28, 2008 at 9:50 am

hmmm, but my undergraduate engineering class was hardly the 4th-5th std dev of natural math ability (myself as case in point…) and yet the class was nonetheless 70-75% male.

10 joan July 28, 2008 at 9:59 am

The importance of math ability for science and engineering is overweighted by people in other fields. A certain minimum level of competence is required but only in theoretical physics is math ability is the determining factor in success and even there physical insight is probably more important.

11 Zach July 28, 2008 at 10:11 am

The importance of math ability for science and engineering is overweighted by people in other fields. A certain minimum level of competence is required but only in theoretical physics is math ability is the determining factor in success and even there physical insight is probably more important.

I agree with this statement, but there may also be other characteristics, as yet unexplored, which could further select between the two populations. If you needed to be in the 99th percentile for two unrelated characteristics (say, math and sheer bloody-mindedness), you could get further differences. Based on this study, I’d say the gender differences model is unproven, but very much alive.

12 bbartlog July 28, 2008 at 10:44 am

How did the 99th percentile make it into this discussion? 4 million high school students graduate every year.

The 99th percentile made it into the discussion when they decided to talk about Ph.D. Engineering, which they use for their example.

One interesting example of the triumph of variance even over aptitude and preference is in competitive Scrabble. This is a field where most of the competitors are women, and there is some evidence that the average woman is better at Scrabble than the average man. But at the very highest levels of the game, the top competitors are still mostly men.

13 Paul J. Reber July 28, 2008 at 11:46 am

The difference in variance argument requires a strong assumption that the variance is symmetric. There’s pretty good evidence that boys have some additional downside risk probably due to genetics (e.g. autism). It’s also possible that there’s some cultural downside risks for boys, e.g., the recent findings that boys are doing worse on average throughout schooling also seems unlikely to be entirely innate. So it’s not safe to assume that larger variance means more men in the upper tail.

We just simply don’t know yet. The problem with Summers’ claim wasn’t that it is heresy to study these issues or consider the hypotheses. The problem is that the president of Harvard is in a position over the policy that could be implemented to remedy what might be a long-standing deep unfairness to women in these fields.

There is no possible question that women have been historically discriminated against in academia. So even with the best possible guess about what the ratio would be in a perfectly fair system (e.g., even if it isn’t 50-50 because some part of the variance hypothesis is correct), it’s almost certainly still higher than what it is.

Where Summers was wrong was in seeming to indicate that he wouldn’t pursue policies that would try to find a way to improve on past unfairness without weakening his university (the fundamental challenge of affirmative action). Scientists working in this area can and should be considering every possible hypothesis and the relevant data. Policy guys need to be more cautious where the science is uncertain and the policy decisions have significant impacts.

14 Billare July 28, 2008 at 12:26 pm

Strange, I didn’t know it was prudent to decide on policy before the fact, if indeed, the data isn’t easily interpreted. I would suppose that the onus would be on the diversicrats to produce studies that show systemic discrimination in light of the repeated findings of greater male variance in intelligence. If we are really dealing with the unknown, shouldn’t there be some mention about probabilities, the relative costs of erring on one side of the affirmative action debate over the other?

Is it a good idea to mismatch the “movers and shakers” of society, prominent and talented researchers and academics, so that our foremost position in research science is sacrificed upon the alter of Feel-Good Political Correctness?

15 Joan July 28, 2008 at 12:32 pm

Some evidence that natural math ability does not account for small number of women in physics and engineering is that the percentage of women getting phd’s in math is double that in physics and engineering. See figure 7 in paper at
http://www.aip.org/statistics/trends/reports/women05.pdf
and the percentage of women getting phd’s in applied math (which is closer to physics) is even more. Also the steady increase in the number indicates that they were not held back by natural ability in the past but by social factors which may still exist. Summers is far from being vindicated. He is just a product of his generation and his observations on math ability has little more meaning that if he had ascribed the difference to the fact that men were taller and being tall made people more successful.

16 Allan July 28, 2008 at 12:41 pm

I’m in a top 10 cognitive neuroscience research program. Recently, a female post-doc in another lab was pregnant. She planned to return to work 6 weeks after having the baby. Earlier that year, she had already been offered and turned down faculty positions at pretty good universities b/c they didn’t also include offers for her partner. Both she and her partner had excellent publication and funding records, so their getting jobs the next year was a foregone conclusion.

After she had the baby though, a 180 degree turn-around occurred. She suddenly declared no more interest in her research and instead took a flexible job near her partner. This woman had done all the hard work, post-doc, grad school,….etc. and just decided not to go through with it at the end b/c she wanted more time with the baby. Now, first, good for her for making the choice she wanted to instead of falling prey to sunk costs type of thinking, but I submit that no man would make that same decision, for better or for worse.

17 juandos July 28, 2008 at 1:01 pm

Someone in a public position speaking in a public venue is responsible not only for the content of what he says, but how that content will be interpreted and used by the greater public. Being “technically correct” doesn’t get you off.“…

What?!?!

You need more than another cup of coffee West…

18 Albert Findley July 28, 2008 at 1:46 pm

Larry Summers’ comments were scientifically validated, and understated. It is virtually certain that light weight studies such as the one trumpeted by the media, will not have the power to invalidate Summers statements, and the science they are based upon. Feminized science illustrates the lengths to which leftists will go in academia and the media to warp reality. The ministry of truth in action.

19 Andrew July 28, 2008 at 2:11 pm

“no man would make that same decision, for better or for worse”

Well, though not as impressive as the person discussed, I’m considering it. It is so easy to get out of these highly competitive environments. Every turn begs you to quit, and your life will surely improve once you do. A very small thing, and having a kid is no small thing, can have a huge impact.

20 brown July 28, 2008 at 2:31 pm

what exactly do you mean by “symmetric variance”?

21 albatross July 28, 2008 at 2:46 pm

The misrepresentation is shameful, whether it’s because of the journalists’ innumeracy and laziness, or because of their dishonesty.

The whole idea that journalists or researchers should shade the results of their research to avoid undesirable political effects is pure poison. Nobody is smart or wise enough to decide what data the rest of the society should be permitted to have, and attempts to do so leave people in unrelated fields either relying on spun/cooked/dishonest results from other fields, or not feeling safe trusting the results from those fields.

22 albatross July 28, 2008 at 2:46 pm

I’ll point out at least one area I’ve seen where having kids makes men change their career goals–willingness to be involved in start up companies. The prospect of being out of work after a year and a half of 70 hour weeks looks very different when you’re a married man with three kids and a stay at home wife, instead of a married man with a professionally employed wife and a couple dogs.

23 Renee July 28, 2008 at 4:08 pm

“The problem isn’t your left-wing intellectual friends. Your problem is that friends of an intellectual feather flock together. Or perhaps, show me your friends and I’ll know who you are. Generalizing the qualities of other people seems to be a trait you share in common with your pinko friends.”

Wow, way to miss the point.

24 happyjuggler0 July 28, 2008 at 4:59 pm

In nations in which men and women are approximately equal, their scores are approximately equal

How is that not a tautology?

Followed by:

and the presence of women in math and science increases dramatically

Since I clearly don’t understand what countries you are referring to, what exactly is the M/F ratio in math and science employment in these putatively equal countries that clearly aren’t equal, noting that if they were equal then that last quote surely would have said so, yes?

25 Paul J. Reber July 28, 2008 at 5:37 pm

It still always surprises me how many smart people reason in this area with blinders on. The evidence that the variances differ does not imply that the difference must be genetic in basis.

I pointed out upthread that there is existing evidence for downside risk to boys that appears to be genetic. I suppose I should have pointed out that there’s plenty of evidence of upside risk that is cultural.

For a simple example, click through to the Science article and you’ll see that in White children, the M/F upper tail ratio is 2.0 — meaning twice as many white boys >99% than girls. However, with an admittedly smaller Asian sample, the ratio is 1.0 (same numbers of boys & girls >99%). It’s not impossible that this difference also has some genetic basis, but it’s also entirely possible that some (White) families nurture their boys with math aptitude more than their girls.

Both genes and culture have big effects on outcomes. It’s pretty stupid to ignore either of them.

And even if you’re going to reason by anecdote, it pays to consider the anecdotes —
* Tendency to monomania for boys is one hypothesis about chess. But consider: my 14yo daughter was better at chess than my 12yo son. I took them both to a tournament and she took one look at the sea of low-social-functioning boys and said, no way. My 12yo loves to hang out at the chess club and at tournaments. That doesn’t mean the culture is the only factor, but it’s silly to pretend it doesn’t matter.

* Why might women opt out of high-pressure academic
positions to raise a family? Because women are still expected to raise the kids and leaving your husband at home to do this means you are now spending your whole like fighting upstream culturally. Now add the fact that the tenure process over-weights your productivity during your most fertile years and the fact that a lot of universities are atrocious on childcare (like mine).

I have 4 kids and tenure. It was hard. I bet it’d be even harder for a woman. If I was a woman facing this apparent choice, I’d be pretty annoyed if the president of my university was telling me I wasn’t genetically predisposed to be far enough out on the tail of the performance distribution.

26 Skorri July 28, 2008 at 6:19 pm

Thanks Rachelle for clarifying what I meant — that where social equality increases, equality in mathematical ability increases. Happyjuggler, for the point about comparative advantage leading to women opting for fields requiring language processing ability over mathematics ability, read here.

This whole debate often gets framed in unfortunate ways. One side is portrayed like it’s shouting “Men are so too better than girls, it’s biological fact!” while the other gets derided as “I don’t care what reality is, I want everyone to be warm and fuzzy and equal.” And the characterizations have a mild degree of truth to them — people obviously feel threatened, even if irrationally, when they’re told that because of their gender they’re probably not more likely to be a math genius or if they’re told that, because of their gender, they shouldn’t bother with a math career because the odds are good they won’t make it. But, at heart, it’s really just a typical nature v. nurture debate.

If people had simply accepted in 1960 that men were genetically better at math, we would have been horribly wrong. Peer-reviewed and repeated studies have shown that a large portion of the discrepancies in math ability that were present in earlier generations are visibly NOT the result of biological differences, but social factors

We still haven’t found out how much is social and how much is biological. But unless you’re willing to argue non-Asians sucking at math is also a biological fact (see Paul’s comment) then differences in math aptitude found today continue to have a great deal to do with non-genetic factors.

27 scottynx July 28, 2008 at 6:40 pm

Paul J. Reber: “For a simple example, click through to the Science article and you’ll see that in White children, the M/F upper tail ratio is 2.0 — meaning twice as many white boys >99% than girls. However, with an admittedly smaller Asian sample, the ratio is 1.0 (same numbers of boys & girls >99%).”

Here is a question, are those asians the top 1% of asians, or are they the asians that made it into the general top 1% of students? Because those two sets are not the same due to higher average asian achievement compared to whites. If the study used asians in the top 1% of the general population, then it is entirely possible that that set includes far more than 1% of asians. Thus we would be talking about less standard deviations above the asian mean of achievement.

Also, even if the study used the top 1% of asians among asians alone, it is possible that the test is hard enough to discriminate among whites at that level but not hard enough to discriminate among asians at that level. Is anyone else familiar enough with the test used to know if that is possible or not? Is the 99th percentile reaching the limit of the tests discriminating abilities, making it possible that the test actually fails to discriminate among asians at that level?

28 Cliff July 28, 2008 at 8:31 pm

Skorri,

Here is a more accurate history of the debate:

1) Women stay at home and don’t get jobs. People get married after high school. Predictably, not many women are in math and science fields. There is no way to say how women compare to men in such fields. Jackasses say women can’t do math.

2) Women begin to work. As we reach equilibrium, more and more women enter all fields. Despite women outnumbering men in college and in many graduate schools, men overwhelmingly dominate the top positions in pretty much every field that exists. Data shows that on average, women are as good or better than men at most things. Data also shows that men have higher variance and hence dominate the top and bottom ranges.

I have never heard someone seriously challenge the idea that men have higher variance than women in pretty much every field up until now. I think you have a huge burden if you want to prove that this variance is due to cultural effects. What cultural effect could possibly cause this? You are positing a massive genetic disadvantage for men that causes their average to be far lower than women, followed by enormous societal effort to elevate men, that is so successful it propels the top men far past the far-superior (genetically) women?

29 Allison July 28, 2008 at 9:36 pm

1. Is this higher variance the result of nature or nurture?

2. I can’t believe that economics *requires* mathematical ability in the top one per cent. Economics is not pure math – to the extent that it seeks to answer questions about the real world, women’s perspectives are necessary.

30 albatross July 28, 2008 at 10:11 pm

I really like Skorri’s comment, above. In 1940, if you’d asked informed peoples’ opinions, they would have told you, correctly, that in 2008, relatively few women would be working in math or physics or engineering. They would have told you with equal certainty that relatively few women would be working in medicine, law, psychology, or biology. Their reasoning would have been equally plausible in both cases, it just would have been massively wrong in one of the cases.

I suspect there is some innate difference in mathematical ability between men and women, which drives the huge difference in representation in math-oriented fields. But I acknowledge that I could be completely wrong about this–maybe I’m like the guy in 1940 explaining that only very rare, freakish women would ever want to do (or be capable of) the hard study and tireless work required to become a doctor. (Now, I think more graduating MDs are women than men.)

31 Ricardo July 28, 2008 at 10:35 pm

32 darren July 28, 2008 at 10:53 pm

You are positing a massive genetic disadvantage for men that causes their average to be far lower than women, followed by enormous societal effort to elevate men, that is so successful it propels the top men far past the far-superior (genetically) women?

Straw man much?

33 Austin July 28, 2008 at 11:46 pm

Sorry to rain on the parade.

http://www.gifteddevelopment.com/What_is_Gifted/learned.htm

IQ testing in childhood clearly demonstrates the equality of intelligence between males and females. Until the IQ test was developed, most of society believed in the “natural superiority of males.† Even now, the fact that most of the eminent are men leads some to believe that males are innately more intelligent than females. On the contrary, we have found more than 100 girls with IQ scores above 180. The highest IQ score on record at our Center was attained by a girl, and four of the five highest scores were earned by girls. However, parents are more likely to bring their sons for assessment and overlook their daughters. From 1979 to 1989, 57% of the children brought for testing were male, and 43% were female, whereas 51% above 160 IQ were male and 49% female (see chart). Now, 60% of our clients are male and 40% female, which matches the distribution in the highest IQ ranges.

Gifted girls and gifted boys have different coping mechanisms and are likely to face different problems. Gifted girls hide their abilities and learn to blend in with other children. In elementary school they direct their mental energies into developing social relationships; in junior high school they are valued for their appearance and sociability rather than for their intelligence. Gifted boys are easier to spot, but they are often considered “immature” and may be held back in school if they cannot socialize with children their own age with whom they have no common interests.

34 Jake Young July 29, 2008 at 12:22 am

It may well be true that men have a higher variance than men in this data set, but that is not a robust finding. International comparisons show that is some countries women have a higher variance than men.

In general, I am skeptical of Summer’s comments with respect to upper-tail effects because they are predicated on some quantifiable (and heritable) trait that is the “right stuff” for science and math. I am curious what people think that trait is exactly. It is certainly not true that men and women in the sciences are drawn exclusively from the upper tail in SAT-math scores. So clearly whatever that “right stuff” is, SAT-math is not adequately measuring it.

35 phys July 29, 2008 at 1:44 am

Agree with brown/Alex that examining performance by percentile in the tails can be more useful than using the variance, if one has the data. I would also be interested in seeing how the distributions of scores change if time limits are removed from the exams.

Anecdotally, in response to the Niederle & Vesterlund paper brown mentioned, I can attest that the viciously competitive culture of the sciences has been a turn-off to many of the women I’ve known. One-up-manship, chest-pounding, and ego battles the whole way. It helps to have an unshakable amount of confidence in your own abilities if you’re going to play this game. Women tend to doubt themselves, whether their abilities are weak or strong. (As I’m sure agnostic can tell us.)

The physical sciences in particular would be better off with a more congenial atmosphere so that we can work together and focus on the science instead of the self-promoting, posturing bullshit that wastes so much damned time and energy. The likelihood of this happening is small, so I predict that at some point I’ll jump ship to save my soul rather than listen to one more roomful of fellow dipshits convinced that bellowing will, somehow, take the place of insightful analysis.

36 bgc July 29, 2008 at 4:06 am

Hans J Eysenck – from his autobiography Rebel with a Cause (Transaction Publishers (1997), ISBN 1-56000-938-1):

“I always felt that a scientist owes the world only one thing, and that is the truth as he sees it. If the truth contradicts deeply held beliefs, that is too bad. Tact and diplomacy are fine in international relations, in politics, perhaps even in business; in science only one thing matters, and that is the facts.”

37 lb July 29, 2008 at 5:19 am

I love how math has become the new phallic symbol. There are plenty of women doing applied maths and I know of plenty female PhD students in applied maths – many of them leave academia for jobs in the industry, many of them put family first, but this doesn’t mean they are less talented.
Of course, it is easier to say all cultural factors are irrelevant, and the difference is innate.
In the same way, it is easier to consider prizes as the absolute proof of quality, forgetting that there are famous cases of discriminated female scientists (as was the case with Rosalind Franklin).
I would say that, considering the way our society functions, women do OK and, who knows, maybe in the future there will be plenty of female Nobel laureates in physics and female Abel and Fields medalists.

38 Ron Hardin July 29, 2008 at 6:56 am

Vicki Hearne has a better explanation than standard deviations copied out here.

It has more to do with men and women are respectively willing to obsess on.

Merely being extremely smart pales by comparison.

39 lb July 29, 2008 at 8:46 am

@JL: it is a fact that there are less women studying maths than men. Some people try to relate this to innate abilities and imply that women do not study maths because they are not talented. This is obviously not true – one may choose a different career simply because one feels more comfortable among women, for example, or because they feel discriminated (though not necessarily are discriminated), or because they are practical and want a good paying job and maths are hard work without necessarily a higher salary at the end. All this is obvious, but seems to escape to the people who count Abel laureates.
@james alan sutherland: when “Jane Eyre” was published, it was published under a male pseudonym, for fear of discrimination (“while we did not like to declare ourselves women, because — without at that time suspecting that our mode of writing and thinking was not what is called ‘feminine’ — we had a vague impression that authoresses are liable to be looked on with prejudice”). At that time, in Europe, there wasn’t even one full professor who was a woman. This happened only in 1889. Guess what? She had a doctorate in mathematics (from 1874) – and she was the first woman to do this (with private lessons at one point, since U Berlin did not allow women to enroll).
I guess you want to say “oh, not this again!”. The point is that, what happens now in sciences is very similar with what happened before in literature.

cheers!

40 JL July 29, 2008 at 10:18 am

Some people try to relate this to innate abilities and imply that women do not study maths because they are not talented. This is obviously not true – one may choose a different career simply because one feels more comfortable among women, for example, or because they feel discriminated (though not necessarily are discriminated), or because they are practical and want a good paying job and maths are hard work without necessarily a higher salary at the end. All this is obvious, but seems to escape to the people who count Abel laureates.

lb, in your earlier post you claimed that some people here are arguing that women pursuing academic careers in mathematics are less talented than men who do the same. However, as far as I can see, no one here is of that opinion. I was just pointing out your straw man argument.

It is not “obvious” that one of the reasons for the scarcity of females among mathematicians is not that fewer women than men are truly great at math, as implied by standardized tests such as the study at hand. There are obviously other reasons for the female scarcity, for example many women who have the required math aptitude may prefer some other line of work than academia, or prioritize family over work, as you say. However, all of these other explanations have been mentioned in this discussion, and they were mentioned by Summers, too, so your accusations are baseless.

41 professor July 29, 2008 at 10:35 am

As Heckman explains, ‘innate ability’ doesn’t exist. You are born with an initial ability to acquire skills. From there your human capital begins to be built. As you acquire more abilities, it becomes easier to acquire even more skills and knowledge. If parents foster girl’s mathematical side with slightly greater variance over time, or if schools/teachers/peers are slightly less consistent in their encouragement of girls in science, across the cross section, we would expect the variance in math ability of women to grow over the course of childhood, relative to men. This would also cause the quantitative test mean of women to fall relative to men over time.

If on the other hand, the variance ratio remains constant at all ages of development, then we would be led to believe that the cause of the disparity is not due to inconsistent encouragement across the cross-section, but instead, may be driven by a biological initial condition at birth.

The other possibility, is that math tests are a noisier measure of female quantitative ability. This would cause attenuation bias; a lower mean and higher variance for women rather than men.

Neither the noise measure theory nor the encouragement theories would most likely hold, since they imply greater variance on math tests for women…not men. Furthermore, the means between women and men have converged.

We are left with two possibilties:
1) women are, at birth, endowed with a tighter variance of the ability to acquire skills. The variance will grow over time as in human capital theory, however, so will the variance for men. The ratio of men/women variances, however, will remain constant and greater than 1 as long as boys and girls are treated the same way. If encouragement is less consistent for women over time, the variance ratio will fall closer to 1. Since the mean for girls doesn’t fall compared to boys over time, we are led to believe that encouragement isn’t the driving force.

2)Math tests are a noisy measure of mens abilities. This would explain the higher variance of men, but would imply that the true ability of men is higher than reported by exams. This is far less believable since it implies that a male dominated quantitative profession is less precise at their evaluation of male quantitative abilities than that of female.

-an academic worried about publishing this kind of work and being black-balled as a sexist

42 lb July 29, 2008 at 11:12 am

@JL: i think you need to cool down, since i did not accuse Summers of anything.

Some of the posters above did recognize the value of career choice and social pressures in becoming an academic in maths, others did not, but went deeper into baseless computations. Personally, I don’t believe that looking at these tests is going to give you that much information about the future tenured and mathematicians and Fields laureates.

Just as my colleagues from the start of the PhD in applied maths would not. From 6 people in the beginning, 4 were females, and from these, 2 dropped out of academia after 2 years of graduation, despite a good start with publications and obvious talent. From talking to other people in applied maths, it seems to me that, after a certain level, innate abilities have little to do with one’s career. It is not so much about potential as much as the will to stick to it.

of course, you can continue with the inflammatory rhetoric.

43 steve Hsu July 29, 2008 at 11:42 am

lb,

There is actually pretty good evidence for positive returns to high ability in science. That is, although motivation, drive, luck, etc. all play a role, the set of most successful scientists seem to also have significantly higher *ability* levels than the average PhD. This suggests that relative populations in the tail are somewhat predictive of the distribution of Nobel prizes and Fields medals.

See this study of 64 randomly selected eminent scientists. Virtually all the people in the sample were +(3-4)SD or so.

http://infoproc.blogspot.com/2008/07/annals-of-psychometry-iqs-of-eminent.html

As I said before, I agree with you that there are many factors other than ability which disadvantage women in science, including outright bias. But let’s at least get the ability story right.

44 Jim Hu July 29, 2008 at 12:02 pm

I didn’t see your linked post while composing my reply. I see that you address the threshold argument.

45 loki on the run July 29, 2008 at 1:55 pm

Steve Hsu says:

(Did you go to Gunn? :-)

Well, either Gunn or Paly, but I have heard that there are more Chinese at Gunn.

I have also heard that many of the early Chinese in the Bay Area, and even San Fran, were 四邑 speakers …

46 Renee July 29, 2008 at 2:18 pm

(My point, by the way, is that social context really does drive a lot more things than many people want to admit. Many people love to think “I got here all by myself” but, well, no, you didn’t. A lot of things happened along the way. I am skeptical about this study’s ability to completely filter out all social effects, because they’re so bloody all-pervasive. Even still, though, there is SO MUCH room for improvement in womens’ abilities before we hit any topping out that it is simply silly to focus on innateness until we get a lot closer to anything even remotely resembling equality.)

47 lb July 29, 2008 at 3:13 pm

@hsu: thank you for the links. I am not saying that having quantitative skills does not have a good payoff (at least by becoming a quant),but i think few people study maths for monetary reasons. There are easier ways to make money. Besides, if you do pure maths, it is quite a different story.
Also, like Renee, I doubt one can separate innate abilities from the other factors, or that we can measure potential of excellence in skills one never acquires.

@Renee: thank you for making the same point in a much more articulate manner.

48 steve Hsu July 29, 2008 at 3:59 pm

Lanny,

I agree with your take, although many quants who have decent mgmt/sales/leadership/trading skills can easily move out of research into more lucrative positions. Also, $250k is pretty low — that’s a starting salary in a lot of places. If you manage a group of, say, 5 quants at a big bank you would make many times that.

This discussion is not about how great life is for +4SD math people. It’s about whether the unbalanced gender ratio in, e.g., physics departments (or the stat arb group at DE Shaw) is *by itself* evidence for strong gender discrimination. I doubt that anyone who understands the realities would really make that case, although many are trying to.

There are scary proposals to force NIH and NSF to *require* equal gender representation among their grantees (like Title IX in college sports). Think for a bit about the consequences of that, and perhaps you’ll see why I am wasting my time exploring this issue in a systematic way.

See here:
http://tierneylab.blogs.nytimes.com/2008/07/23/sex-bias-and-data/

49 daumier July 29, 2008 at 4:05 pm

@Hopefully Anonymous: When the president of Harvard gets drummed out of his job for having the temerity to point out facts well backed by the scientific process, and then two years later having the liberal mainstream media continue to misrepresent those facts, shouldn’t the topic have “psychological salience” for _anyone_ who cares about the scientific process, and the truth in general?

50 Pat July 29, 2008 at 4:55 pm

londenio, How do you explain the recent trend of more and more women being accepted into colleges. Top universities are finding it ever more difficult to find qualified male students. From what I have experienced, and colleges have realized, the far more obsessive sex (academically) is female.

51 The Epicurean Dealmaker July 29, 2008 at 5:59 pm

Alex — What you did not mention and I find slightly disturbing is that the authors of the most recent paper are all women. Not having access to the paper, I have not read their presentation, but I find it potentially worrisome that these authors (in your words) “downplayed” the variance angle. Perhaps I am overinterpreting this, but I would hope that science would trump politics in its own domain, at least.

52 Renee July 29, 2008 at 7:38 pm

Maybe, Sbard, but I’ll bet that Graciela Chichilnisky would disagree.

53 joel July 29, 2008 at 9:49 pm

This whole argument is so lame.

If a woman wants to do math/physics/engineering etc, there is nothing holding her back.

That these careers stink in many ways compared to law, medicine, and finance likely influences practical people to avoid them. Woman are certainly more practical than men.

But, why should anybody think there “should” be more women in certain careers. Don’t they think women can make up their own minds?

What right does anybody have to pressure some young person into a very demanding, and in the end, not very rewarding career. BTW, anybody with a family knows family is way more important than some silly research job.

54 steve hsu July 29, 2008 at 10:16 pm

Jim wrote “I will assuming, in turn, that you are not making the claim that variance explains almost all of the status quo, so there is NO significant contribution of gender discrimination to the observed distribution.”

Nope, I am definitely not claiming that. I would only make the limited claim that the variance difference does have an effect on outcomes, which is stronger in some (math intensive) subfields than others. Check out the Tierney blog post (NYTimes) I linked to previously for some interesting statistics on fraction of female PhDs vs professors by discipline. Before you look at the numbers, guess what the ratios will be like — is the pipeline more/less leaky than you thought?

http://tierneylab.blogs.nytimes.com/2008/07/23/sex-bias-and-data/

Regarding journalists, this Penn linguist claims that less than 1% of the population understands “distribution talk” — he makes an analogy to certain Amazonian tribes that have no word for numbers like “ten” :-)

http://infoproc.blogspot.com/2007/10/bounded-cognition.html

55 econgirl July 30, 2008 at 2:11 am

Alex- very well said, in my opinion. I even used your summary as a jumping off point for my own blog entry (www.economistsdoitwithmodels.com). Thanks!

P.S. I am female and find the desire to prove equality in this regard ludicrous and even unproductive.

56 Hughes July 30, 2008 at 2:45 pm

“Note that we are assuming that mathematical ability is normally distributed – we know the data fit this distribution around the mean but we don’t know much about what happens at the very top.”

How exactly do we know the normal fits the data around the mean? Was this tested or is this supposition? You have to remember that nothing in nature is normally distributed. The study authors should have bootstrapped.

57 steve Hsu July 30, 2008 at 3:46 pm

The study’s Asian-American data is from a tiny population in Minnesota (Hmong?). The data are anomalous — for example, there appear to be fewer Asians at 99th percentile than whites, which is atypical for the US or the world. See the PISA data linked to above for worldwide data which doesn’t show the M/F reversal that the study authors mention. Only a desperate ideologue would try to base their conclusions on that sample / data.

Also, this whole thread has been about whether 99th percentile is adequte to characterize the professoriate in math intensive fields, and I feel it is not. If you go out beyond 99th percentile you *can* account for observed ratios using variance alone, although i don’t feel it is the whole story by any means — and neither did Summers, if you actually read his remarks.

58 The Epicurean Dealmaker July 30, 2008 at 4:18 pm

DT — Have you actually ever read what Summers said?

Here it is: http://www.president.harvard.edu/speeches/2005/nber.html

If you take the time to do so, you will discover that Summers proposed no single factor as an explanatory hypothesis for the data. He proposed three principal hypotheses, which he ranked in probable order of importance as contributing factors to the observed “achievement gap,” as you put it. Innate aptitude was second. (Preference was first, and socialization/discrimination was third.) He did not attempt to assign weights to each of these factors, other than ordinal ones, as I have mentioned.

The speech is a careful, balanced, and reasonable exposition of a highly charged subject. Which is far more than I can say for many of the Summers-bashing comments I have read here and elsewhere.

The man is no angel, but he deserved better than the public lynching he got.

59 the buggy professor July 30, 2008 at 4:57 pm

The following commentary was left at Mark Perry’s admirable web site in a similar thread . . . Mark a former Ph.D. student of Tyler Cowen. It deals with the misconceptions of a fair number of posters at both Carpe Diem (Perry’s site) and in this thread about why Summers got the boot in the summer of 2005.

You will note some links to the Harvard hullabaloo and pc-prodded witch-hunting in which Summers, a woman Harvard scholar, and some exchanges on her Commentary article (written back in 2005). The commentary should also illuminate how Summers, long before his January NBER talk, had antagonized more and more of the pc pulpit-pounding faculty members . . . aghast at any challenges to their ideological shibboleths, nostrums, and sacred dogmas. And, predictably, lifting themselves into frenzies of below-the-belt, totalitarian tactics to silence any critics of their fervently held high-octane credos.

……………..

I think whether he should have been fired or not depends on what he was hired to do as the president of Harvard. Was it his job to use his academic brilliance as a speaker and researcher, or was it to use his judgment and tact to fulfill Harvard’s mission as a world-renowned institution of higher learning? If it was the former, he should not have been fired, if it was the latter, he probably should have been fired. — Walt g.

Walt:

As usual, I appreciate your thoughtful and always civil-like comments. In the Summers case, however, they reflect a misunderstanding on your part.

You see, those alternative hypotheses offered by Summers on women in the sciences were set out and discussed by Summers at a conference sponsored by the NBER — the National Bureau of Economic Research — on women’s roles and problems in those disciplines. Summers was specifically invited as the keynote speaker.

Click here for his own views of what happened: http://www.president.harvard.edu/speeches/2005/womensci.html

……

So, you see, he wasn’t just ad-libbing or misusing his influential Harvard post or even reducing the problems of women in the sciences to just biologically-influenced IQ levels at the upper end of the high-IQ tail.

Whether he was fired a few months later for those unpolitically correct views only is another matter.

…..

As it happened, he had alienated ever larger numbers of Harvard faculty — mainly in the humanities and social sciences (some of them anyway) — because he had done such fascist things as reintroduce ROTC at Harvard . . . cancelled in the late 1960s. As well as defending Israel openly in its fight against brutal suicide terrorism, and supporting the US war on terrorism.

He had also alienated a leading black studies scholar, a philosopher, who Harvard had given the distinction of University Professorship (or something like that) . . . an award usually given to Nobel prize winners and the greatest scholars in other disciplines — anyway, no more than 15 or so in any one decade or so.

That scholar, Cornel West, who deserved that status as about as much as you or me — he has no influence of any note in contemporary philosophy —- had done such notable scholarly work as help run Al Sharpton’s presidential campaign and additionally a book on rap-music (which West modestly described on his web site as the most important work ever written on music anywhere, any time).

So what did Summers do?

As the new president, he tried to meet every one of the faculty’s members and discuss their work with them . . . remember, Summers himself was the youngest full-professor in Harvard history. When he met West, he urged him to do some “serious work” that justified his rarified status.

Affronted — how dare someone suggest that Cornel West do what he was supposed to! — West huffed out, called a press conference, and soon departed back to Princeton.

…….

Whether or not Summers was a good president of Harvard, then, was a secondary issue in his departure under fire. Not the most politic of men, he nonetheless had a very large minority — mainly scientists on the Harvard faculty — who supported most or all of his three or four years in office.

He was let loose, overwhelmingly, for colliding with the politically correct brethren on the faculty, with the talk at the NBER conference only the last affront to pc intellectual idolatries.

……

Ruth Wisse, a tenured Harvard scholar, wrote a remarkably perceptive article on all this in Commentary Magazine, one of the two or three most influential conservative publications.

(Clarifying remarks: A former liberal magazine, its chief editor, Norman Podhoretz had been appalled by the anarchy and semi-totalitarian behavior of the “New Left” radicals of the late 1960s and early 1970s . . . exactly in the same way the editorial heads of the Public Interest (by far the most influential public policy journal in American history) were. Nathan Glazer and Irving Kristol, two prominent scholars, moved the new journal quickly in the early 1970s toward a conservative, scholarly based journal assessing public policy and cultural matters and — along with Commentary — were the intellectual influences that most mattered in the Reagan era of the 1980s and into the 1990s.)

Though Wisse’s article is gated, the follow-up exchanges with other women at Harvard in the letters column of Commentary can be found here:

http://www.commentarymagazine.com/viewarticle.cfm/women-s-choices-11116?search=1

…….

More generally now: there have been and remain totalitarian-tendencies among the New Left tenured radicals at US universities that have sought to stifle free-wheeling scholarly discussion of any controversial issue that infringes on their politically correct attack-dog roles . . . all self-anointed.

Take the case of Richard Rorty, the most prominent public philosopher in the last 50 years . . . a gadfly who started on the left and became about the only gifted philosopher in analytical and post-analytical philosophy to be embraced for a couple of decades by these arrogant, aggrandizing New Left types.

By the mid-1990s, Rorty had become disgusted with his supporters. He openly denounced them in a distinguished book — Reading Rorty, where virtually every prominent philsopher in the US and Europe engaged him in open exchanges. In particular, as he admitted to one of France’s leading philosophers who criticized him for abetting the growth of irrational philosophy in France (and elsewhere), he was chastened and regretful.

And he specifically attacked the New Left as semi-literate, arrogantly self-righteous, uninformed, and politically useless. He went further. In that scholarly book aimed at professional professors, Rorty — who two or three years earlier had also upbraided his former New Left idolators for their anti-Americanism and lack of patriotism — called them outright “creeps.”

…….

Hope this clarifies matters . . . including my own battles with the radical left — I a moderate independent who cares about free speech and open discourse in our universities and in wider public life.

Fortunately, these tenured radicals are close to retirement now, and many have already gone into the oblivion they so richly deserve.

But note. Though the newer generation of scholars replacing them haven’t the same alas, the cultural combats are still afflicting us and require diligent battle-ready men and women to engage in open combat with them when they resort to their rag-bag of totalitarian-like tactics:

* Never discuss alternative points of view in their indoctrinating courses. And never have alternative readings in your class syllabi.

* Ridicule students who contest your views, even when the students are polite and informed.

* Support the efforts to drive off campus any invited speaker to the right of Al Gore.

* Send storm-trooper students (and hang-on scum from off campus) into classrooms like mine if I dare not toe the canon-designated line.

* And if none of this works, set up witch-hunting tribunals where the prosecutors, judges, and juries are one and the same (carried out in secrecy a la Stalinist purges of the 1930s). These kangaroo courts, observe quickly, will likely invoke some campus rules about hate-speech — invariably found to be unconstitutional if challenged in our legal courts — while allowing venom and bile by the boxcar load to be dumped on the heads of any vicious, fascist-like pc-critics. (I always tell my former students on faculties elsewhere, if attacked in this latter way, to bring a lawyer to those ideological purges — which are presumably intended to be an auto-de-fe of the sort the Spanish Inquisition preferred, followed by mass burnings at the stake; and if the purging Inquisitors refuse to allow the lawyer to be with them, to have the lawyer say . . . “Fine. I have already called the press. I will denounce these Stalinist-like tactics and disclose the civil suit I have already filed in court against each of you and the university.” Note that there are several organizations in the country that will offer free legal aid of this sort.)

……

Wait though!

We haven’t completed the list of crackling totalitarian-inspired silence-the-critics chincaneries.

Namely. If none of the above ploys suffice to black-out any and all criticisms, our politically correct, post-modern, identity-obsessed brethren can do what some of the women scholars did at Summers’ talk: rush out, cry you felt sick (this happened: one MIT woman scientist said she was about to vomit in the auditorium, the poor girl!), call the press, and get semi-informed, pro-pc ideological journalists to pound the tom-tom drum for you far and wide.

……

Michael Gordon, AKA, the buggy professor. http://www.thebuggyprofessor.org

60 htb July 31, 2008 at 3:02 pm

Several posters have wondered about whether the correct measure for a truly gifted mathematician is 99% or 99.99% or something else. I’d just like to say that during school I consistently tested between 98.8% and 99.4% in mathematics achievement and capability.

I don’t have the ability to be a world-class mathematician.

With sufficient effort, I could have gotten an undergraduate degree at a good college in mathematics, but probably not a masters *at a good school*.

On the other hand, at the local not-very-good state university, I might easily have been the best math student they’d seen in a decade, even in their masters program (which is heavily tilted towards teaching math to young children). This may illustrate the difference between “getting a degree” or even “becoming a professor” and “being a world-class mathematician”.

61 Michael F. Martin July 31, 2008 at 4:40 pm

Wouldn’t China’s one-child policy provide a pretty good instrumental variable for statistically testing the cross-correlation between differences in variance and sex-ratio at birth? Do we have the data for both?

62 Melykin August 2, 2008 at 6:37 pm

The results of the prestigious Putnam Mathematical competition, which is written by undergrad students in North America, would be a good place to get data regarding this matter.

http://www.maa.org/awards/putnam.html

Every year the five top scoring students are given the designation “Putnam Fellows”, and these people are surely in the very extreme fringe of mathematical ability. Looking at the data from 1992 – 2007, the ratio of M:F in the Putnam Fellow population is about 19:1.

Since this is an undergraduate competition, this population wouldn’t likely be biased by women dropping out of academic life to have babies (the way studies involving grad students or professors might be)

63 Ian Stuart November 26, 2008 at 9:18 am

As the article pointed out, when you discuss professors at the top univerisities you ARE talking about the top one percent or less. Also Summers did NOT indicate that he would not pursue policies to try to increase the number of women in tenured positions in the sciences. What he said was that DESPITE such efforts the numbers had not significantly increased and that PERHAPS we should look at whether there were some gender specific differences (and not just in variance).

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65 Jacob Israel December 11, 2008 at 12:26 am

Why is there no reference to TIMSS which shows that at the 12th grade level, whose scores are very different from the 8th grade level in both directions (up for most countries, VERY much down for the US), Norwegian boys scored 2 standard deviations higher than Swiss boys (589 vs. 519)? Or that Swiss boys scored 2 standard deviations higher than Swiss girls (519 vs. 444). And Swiss girls scored another standard deviation higher than American girls (444 vs. 393), for a total of 5 standard deviations of separation between American girls and Norwegian boys?

SAT scores for 12th graders show that boys in Catholic states score almost two standard deviations lower than boys in Protestant states. And girls in Catholic states score another two standard deviations lower than boys in Catholic states, for a total of 4 standard deviations of separation between Protestant boys and Catholic girls. They also show that two thirds of those who score over 600 in SAT math are boys and only one third are girls. 

NAEP confirms the phenomena, plus provides the additional insight that blacks score another 5-9 standard deviations lower than Whites, and that blacks in the District of Columbia have an IQ which is 4 IQ points lover than the average for American blacks, another half of a standard deviation.  

While not every step along the way is necessarily cumulative, it’s not impossible that the total number of standard deviations of separation between American black females in DC and boys in Norway is a total of 14 to 18.5.

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69 Tom Grey February 6, 2010 at 10:23 pm

not safe to assume that larger variance means more men in the upper tail.
It’s not really an assumption — if the data shows it.
Last year, how many high school students took SAT tests?
How many men scored top possible 800?
How many women?

The data, not assumptions, show men dominate the top.
Also true in chess playing (great ‘relations’ foto, tho I was wondering if the Gibraltar apes were the siblings of humans).

The domination of variance over average ability at the very top is one likely reason.

no possible question that women have been historically discriminated against in academia.

But there is a question about how much discrimination against women in top math / physics groups in the last 20 years. In fact, in all academia over the last 20 years.

Does the NBA discriminate against Hispanics? I don’t think so, but I think it’s a possible question.

Oh wait, the SAT scores since 1996 have been modified so that a prior 780 would now be 800.
http://www.greenes.com/html/convert.htm
Maybe the “top” can be increased enough to hide the differences in the very top.

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72 Jay J October 13, 2010 at 3:38 pm

I wanna stick up for Tom West’s point a bit:

If you’re the President of an institution like Harvard, the institution can rightly expect you to exhibit a certain about of decorum when it comes to socially controversial topics. Sure Summers was in an administrative position, but a position like has overlap with something like a real estate agent, lawyer, PR person, etc. Part of the job description is to show some delicacy in public. It’s also not unlike meeting your significant other’s parents at Thanksgiving; there may be all sorts of ways your significant other’s family deludes themselves, but it’s not your job to disabuse them of their notions, no, your job requires much more diplomacy than that.

On the other hand, we are in a culture where taboo is much more active a social enforcer than people seem to realize, and our ostensible principles are often simply masked passions. Many segments of left take pride in relying on open discussion and the dictates of reason, but even there, those principles often give way to taboo.

We have a pop culture of educated laypeople and academics. These are the very people (particularly the latter) we ought to be able to expect to be consistent (which is a sign that reason and principle really guiding the way, rather than tribal taboo).

So, for one thing, what Larry Summers said caused a reaction in our intellectual pop culture that is discouraging. But, being the President of Harvard entails realizing that as discouraging as it is, our intellectual pop culture just can’t handle certain things, and you can’t just publicly speculate on sensitive topics as if you’re at a card game or in the duck blind with your buddies, because this kind of behavior will do no good, and it will cause particular trouble for the institution you represent (this goes for heads of institutions, but not for academics whose job it is to find the truth).

What I’ve said so far may imply that I believe Summers is exactly right, but I’m not so sure. I only think the way many people reacted is very disappointing… I do believe it’s a multifaceted topic (biological, cultural, historical) and a modicum of intellectual modesty should be exhibited when speculating on such topics, (even in a duck blind), especially if one is is such a position of academic authority. Having said that, I think those who make more legitimate arguments in favor of holding Summers accountable (like Tom West, or Skorri) for what he said should also acknowledge that it’s unfortunate that Summers would have received much the same reaction if he had calibrated his statement perfectly; so long as it had the part about innate biological differences between the genders playing a possible role, it would have been very controversial.

And, since I started this post with a tip of the hat to the “left” side of this disagreement, l want to end posing a question to that side, because even after all this time has passed from Summers infamous statement, I’m still wondering: Are we arguing about whether the content of Summers statement was beyond the pale because he said that there are innate biological differences between the genders, or, whether, even if the data support Summers’ speculation, this data doesn’t explain the width of the gender gap?

73 namae nanka November 25, 2010 at 7:24 pm

The simplest way to settle the debate and remove the discrimination against women is to have sex-segregated colleges for men and women.

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