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.
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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.
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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.
Maybe, Sbard, but I’ll bet that Graciela Chichilnisky would disagree.
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.
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
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.
“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.
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.
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.
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
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”.
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?
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)
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).
I like this site, at the dawn of Christmas, like all the sites frequented by friends sent holiday wishes ahead of time, a happy holiday.
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.Â
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. Â
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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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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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?
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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