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August 01, 2008

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Not only am I convinced that schools are not aware of this bias, but I would be willing bet that even presented with the evidence of such a bias, they would refuse to correct for it.

Too bad for you that your result didn't come out the other way. If you had come up with a statistically valid justification for subtracting off from high scoring male SAT scores then you would have been declared one of the most important academics of our time by many politically correct professors. And I bet you could have made millions consulting for elite college admissions offices.

Not quite the same issue, but Oxford was in the news a couple of years back when it emerged that some admissions fellows were discounting female students' grades on the basis that they were more likely to reflect conscientiousness than talent.

What would we expect about the degree to which score variance is driven by ability vs. errors? I know there have been suggestions that females are more risk averse test-takers, which would presumably lower their error variance relative to males. More general concerns about stereotype threat etc. could also suggest that the expected error in female scores is negative --- though I'm unsure how much weight to put on such factors.

More generally, I would think that there must work out there on the predictive ability of tests scores in determining some set of relevant outcomes (pass rates, job placements, etc.) for males and females, which might be more directly relevant to the question?

I love this post. Narcissistically, cause I've thought of it too :)

Another way of thinking about the core issue: Information/evidence (here, about the college candidates: sex or race or SES or library rental history) ALMOST ALWAYS continues to MATTER; better, newer information just reduces its importance. New information (an SAT score, a single day's results from random questions) hardly ever completely subsumes (obviates) all the earlier evidence.

Call all the Ashkenazim who score 700V "Judd." Call all the gentiles who score 700V "Bob." I would make so much money if anyone wanted to bet on the Bobs' versus the Judds' retests!!

Matt Huang is completely right, and dryly understates it. Schools actually "correct" in the OPPOSITE directions by being overly impressed with unusual results, both as a matter of public policy ("diversity") and by going gaga over dramatic miracle-type stories (doing poorly in the past is "hardship"; the more it contrasts with a current test result, the more is "overcome").

@James Miller: Indeed.

P.S @James Miller: it's easy enough to get the result to come out the other way. Just fix the ability variance A equal for males and females, instead of the error variance E. Then the discount ratio for females to males is just R.

From what I could find, approximate mean and standard deviation for SAT math scores are 500 and 100, so variance is about 10,000. Robin's exemplar ratios N of measurement error variance to score variance, 0.1, 0.2 and 0.4, would therefore correspond to measurement error standard deviations of 32, 45 and 63 respectively.

Juniors who re-take the SAT as seniors gain an average of 13 points while 1 in 25 gain 100 points or more. If we assume that the 13 points is due to greater experience and intelligent, this suggests that 4% of students score 87 points above expectation due to measurement error. That would imply a measurement error SD of about 50, roughly in the middle of Robin's sample range, close to the N=0.2 value.

GNXP on the issue here. With a picture for those who don't feel like reading.

No doubt there are many other factors to consider in comparing male and female candidates

No doubt. Correcting the scores by 1, 5, or even 20 points doesn't make much of a difference in the overall quality of an admissions application. So you have to consider the cost-benefit ratio of implementing the corrections. It's probably not worth doing.

I know a guy who got into Princeton with a 1200 on the SATs. Not bad, but not stellar either. He also wrote articles for the local (not school) newspaper while in high school and was an accomplished violin player. Just goes to show that you can make up for hundreds of SAT points with other skills, talents, and accomplishments.

Hal, retest variance gives a lower bound on error variance; we also expect a systematic error, which is the same for each retest. If a systematic variance was about the same size as an unsystematic variance of N=0.2, then we'd really have near N=0.4.

MZ, so you'd predict that if some school did make this change people wouldn't bother to lobby to change it back?

the berkeley math dept. is sexist!

also, your blog makes me want to throw up :-)

Right, as was first pointed out to me in 1999, the black guy who got a perfect score on the SAT is less likely to retest at that level than the Ashkenazi guy with the same high score. So, a Bayesian approach to college admissions would knock some points off black high scorers on the grounds that their impressive scores are more likely to be flukes.

But no college does that, and it's easy to see why. Colleges have a hard enough time defending the use of the politically incorrect SAT without doing something so prima facie unfair that it can only be justified by an insight so sophisticated that it eluded even the Overcoming Bias boys for years.

Overall, it's better just to rely upon the relatively high consistency of the SAT and ACT in a colorblind manner than to play Bayesian games, as interesting as they are to contemplate.

Steve, shouldn't we recommend all bias corrections we can think of, regardless of how politically popular they are? If we only suggest politically popular corrections, how sure can we be that will add up to a net improvement?

Is there any reason to suppose that the error would be additive, as opposed to multiplicative? What if S = A * E?

Robin,
shouldn't we recommend all bias corrections we can think of, regardless of how politically popular they are? If we only suggest politically popular corrections, how sure can we be that will add up to a net improvement?

If any of us suggested ALL the corrections we understand (as obsessive expert generalists)...we'd be ignored (as out-of-touch) before having a chance to be appreciated.

Relatedly, Eli is right to build some complicated ideas up, piece by piece.

Your personal list of "absurd" beliefs is awesome (to me) and laughably outlandish (to most, including essentially everyone whose powers we hope to convince). For a relatively uncurious (overconfident) audience, would you want to LEAD with that list?

Sadly, we can't be much SURE that ANYthing we advocate will lead to good (due to likely nonreactions, bad reactions, misapplications, unintended consequences...). In various contexts, we should just "pick our spots" and work/hope for the best.

There is TONS that is simply "off the table" nowadays, or any existing polis. Steve Sailer, ironically (because you're asking HIM if "we" should be more extremely whole-truth-and-nothing-but), addresses more politically infeasible topics more in-the-trenches persuasively than anyone I know...and his underpopularity speaks volumes.

How can we tell whether males have larger error variance or ability variance? If it's per-test error, it's easy to measure, but is the data available?

This has been discussed at Gene Expression in years past. I agree that the prima facie unfairness is too great to transparently implement an algorithm that made use of this information. Even if it were understood to improve accuracy, advocates would raise a danger of a positive feedback effect, where members of groups to which a statistical penalty is attributed feel the system is rigged against them and disidentify with it, further reducing their performance and increasing the penalty.

A related phenomenon is that when your measure of something that varies between individuals (wealth, intelligence, etc) is noisy, you will tend to underestimate the degree to which it is influenced by any particular factor (genes, in utero environment, schooling, etc). Psychometricians do know about the effects of imperfectly reliable tests and at least sometimes adjust for them.

Blech- you all need to get over test-scores and priors an just look at what people *do*. The truly talented will make their way whether or not they get into princeton... Tests seem to grossly undervalue real-world ability... navigating bureaucracies, dominance hierarchies, basically getting things done... What's truly annoying is when someone who is *clearly* talented by all real-world measures is ignored for not having the proper intellectual pedigree... No Ivy-league degree? Spent 5 years as a real-estate agent in Bejing? 1 and 1/3 kids? Blah! Away with you heathen! Loser fuck-up!

What is the point of social science, or indeed any analysis, if we only ever use it to offer support for beliefs people already have? My argument above is very simple and solid, and took only four paragraphs and two simple algebraic equations. Why ever bother with a twenty page journal article, or a three hundred page book, full of math if even this short post is too complex to persuade anyone who doesn't already believe the conclusion? Shall we just wait until society decides, for example, that we need affirmative action and then go searching for data and models to support that conclusion, and then wait for society's next political fashion so we can go find support for that?

Robin wrote "What is the point of social science, or indeed any analysis, if we only ever use it to offer support for beliefs people already have? My argument above is very simple and solid... Why ever bother with a twenty page journal article, or a three hundred page book, full of math if even this short post is too complex to persuade anyone who doesn't already believe the conclusion?"

Colleges do a horrible job at rationally considering issues of race and gender. Thus, the fact that the argument you present here won't persuade anyone to change their position doesn't imply that on most issues college professors are unwilling to be persuaded by logical arguments.


if even this short post is too complex to persuade anyone who doesn't already believe the conclusion?

Nah. It's merely too complex to persuade 99%+ of those who weren't already there ;)

(To be persuaded, someone must both understand your analysis on its face + be confident that there's nothing for another side that's unspoken and more important.)

All good arguments matter to SOME people (however few)...and that has to be enough to justify publishing. I'm reminded of the "Elite For The Elite" Demotivator a few weeks back.

Robin, your broadcasting niche is well-established to include WEIRD. So, don't hold back! (...much, if any...)

...though I'd be curious to see the different reaction (even here at OB) if you posted the obvious even stronger and more striking version of your 4 simple paragraphs for racial differences (...with their differing means as well as variances) instead of gender. Bet on the amount of firestorm (now or later)? On impact to your career? (especially any govt-funded advisory role you might accept...)

I wish each truly great WEIRD idea (like Idea Futures) could attract several popular advocates with otherwise perfectly mainstream reputations...

Shall we just wait until society decides, for example, that we need affirmative action and then go searching for data and models to support that conclusion, and then wait for society's next political fashion so we can go find support for that?

Whether we're the ones to do it or not...surely you've noticed that the marketplace most rewards EXACTLY THAT! So, it will get done (however tortuously -- e.g., Rawls) and it will break big.

Perhaps the most effective subversion (of a rising dominant trend you actually oppose) would be to become its best spokesperson (and therefore gain chances to limit its scope!!)...if only one were psychologically capable of such.

Robin wrote:

"What is the point of social science, or indeed any analysis, if we only ever use it to offer support for beliefs people already have? My argument above is very simple and solid, and took only four paragraphs and two simple algebraic equations. Why ever bother with a twenty page journal article, or a three hundred page book, full of math if even this short post is too complex to persuade anyone who doesn't already believe the conclusion?"

The post is not too complex to convince many quantitatively sophisticated academics that the bias discussed exists, but it is too complex for many other academics, not to mention unsophisticated, rationally ignorant, rationally irrational voters, particularly if it is presented inaccurately by the media or political advocates. The people to contact on this would be the Educational Testing Service's research division or the College Board, organizations of psychometricians that produce score interpretation recommendations for their tests. I expect that they will understand the point of the post but will also explain that the political pressures they operate under prevent them from drawing attention to it.

Robin- Nothing in your post proposes what to do about said bias, or even suggests that anything should be done... It is an exercise in simple mathematical equations and that is *all.* What did you think we would say, "Oh dear! Women with high math scores *still* are likely to suck at math... better not hire all those female grad-students just because they got high GREs!" I mean, God, what *was* the point of your post? "Nah nah nah nah nah nah- you actually are different in spite of equally test scores! I'm smarter than you are 'cause I gotta penis! And *that* prior is the most reliable predictor of success *anyone* has to go on! I mean, how better to cut down half of the work of selection, then automatically taking a man over a woman! I mean, look at history: Napoleon, Mussoulini, Hitler, Churchill, General George Washington, every US president for that matter! Women... what's the point... Oh yeah, sex! Damn bitches seem to want insane criminal types for some reason...Gee... Stupid hoes... You know, prisoners do really appreciate it more than I ever could... Maybe I should spend 5 years in the slammer 'cuz then I'll actually get laid! Maybe they could teach me how to keep my hoes in the rows..."

Really.

And you wonder why women can't stand the things you post?

There are few enough people willing and capable of doing what is necessary without excluding them prima-fascia for not fitting your priors. Stop trying to divide people by color, gender, sexuality, whathaveyou and come up with some real-world, actually useable solutions already.

Lara, if it was such an "exercise in simple mathematical equations", why are you getting so worked up over it? I wonder how the world will progress when a man is not allowed to post simple mathematical equations without accusations of being sexist.

And why can't a person post a problem without proposing a solution? I don't see any precedent for this attitude in the world of science. That's like telling Young, "Oh, you performed that double slit experiment! You've just shown Newtonian Physics is wrong. Whats the point? I don't see your solution to the problem. Go work on a solution instead of coming up with problems. Idiot."

conchis: Not quite the same issue, but Oxford was in the news a couple of years back when it emerged that some admissions fellows were discounting female students' grades on the basis that they were more likely to reflect conscientiousness than talent.

Interestingly, in the cited discussion, some people argued that men excel disproportionately in various fields because of their greater "obsessiveness"; and this intended not to diminish their achievements but merely to explain why they achieve. But it seems that in the eyes of some Oxford admissions fellows, when women work hard, it's mere "conscientiousness".

Perhaps that is specific to the culture of Oxford University. I have heard someone from Oxford denigrate someone as having merely a "hard work first [class honours degree]". Is that a concept that would even occur to anyone at an American university?

Knowing what a Bayesian approach tells you about the expected performance of various candidates is useful information even when you want to trade off the feature being measured for the promotion of social integration, or to reduce the salience of such unchosen features: it helps to measure the extent of discrimination and to make the tradeoffs clear. The political conflict over such analysis comes because people don't want others with different values or biases to be bolstered in views that they oppose.

Lara, Robin has identified an existing bias in favor of women, not proposed a policy in favor of men.

The GNXP references given so far do not appear to mention the topic under discussion here. Rather they simply discuss the well-known male/female variance difference.

I thought over this issue a while back, and concluded that the argument against such "corrections" is not so much an argument against the principle of regression to the mean, but rather, an argument that people's social reputation should be construed in such fashion as to depend only on proximal factors under their volitional control.

The point of good reputation / bad reputation is not just to provide information about people, but to encourage people to behave well in order to acquire good reputations - this requires reputations to depend on behavior. Consider the effect on the Tit for Tat equilibrium, of some agents starting off with a string of Defections that they never actually made, listed against them.

If you want to distinguish gang members and, say, not offer them taxi rides, then discriminate against a certain style of clothing, not against people colored black. Clothing is volitional, color is not.

Eliezer, that sounds to me like an argument to make corrections for both non-volitional and volitional factors, and then make some *extra* corrections for volitional factors where appropriate. It also seems important not to let game-theoretical factors like this influence your actual beliefs (e.g. your estimate of someone's intelligence) as opposed to your actions.

I mean, what you're saying is that, when two people score the same on the SAT but prior knowledge says one is probably a bit smarter than the other, we should discount this prior knowledge because doing so rewards the stupider one for doing better on the SAT than reflects his intelligence, even though doing better on the SAT than reflects your intelligence doesn't benefit others in any obvious way.

All, I find it curious that my argument is seen as being against women, when it is only hurts high scoring women - it helps low scoring women. Why is it that when we discuss wages the usual presumption is that it is better to help low wage folks, even if that hurts high wage folks, but the opposite presumption holds for female test scores?

Eliezer, with steven I don't follow you - we are usually comfortable with social reputations that depend on a great many things not under volitional control. People have social reputations for being pretty, witty, having popular friends, having successful or failed businesses, and so on. And the adjust score I propose would still be under your volitional control.

Lara, I thought I was clear about my proposed solution: adjust the scores when ranking candidates.

Vulcan- That was posted in response to Robin's comment: "Why ever bother with a twenty page journal article, or a three hundred page book, full of math if even this short post is too complex to persuade anyone who doesn't already believe the conclusion? Shall we just wait until society decides, for example, that we need affirmative action and then go searching for data and models to support that conclusion, and then wait for society's next political fashion so we can go find support for that?"

Which I interpreted as suggesting the need that we actually *do* something about women's test scores not meaning the same thing as men's, and even women's achievements having more to do with luck than skill just based on variance and priors... Indeed, he just said: "Lara, I thought I was clear about my proposed solution: adjust the scores when ranking candidates."

Which is absurd and dangerous! It doesn't matter if he's right about the math- when you start introducing systematic discriminatory practices based on priors beyond people's control, you get totalitarianism... It's just... DISGUSTING. And from a *libertarian*??? INCOMPREHENSIBLE. I mean, lets make it so black people *can't* vote again, since statistically they are less likely to be up-to-date on the news... I'm disturbed that so many people are flocking to Robin's aid on this... I guess I can start to see how Hitler was able to convince the Germans that Jews were sub-human in spite of having any seeming humanity...

Robin, defending this disgusting prejudgement: "People have social reputations for being pretty, witty, having popular friends, having successful or failed businesses, and so on. And the adjust score I propose would still be under your volitional control."

BAD EXAMPLES!!! Pretty: Lose the fucking weight, dress better, get nicer beauty care products... trust me, unless you're quasimodo, this makes a *temendous* difference in how you will be perceived by others. You don't need to be blessed with 'natural' beauty, whatever that's supposed to mean anyway.

Witty: Well, that's just obvious. Go read an Oscar Wilde Play and go to a few bars to practice... Demosthenes? Anyone?

Having Popular Friends: Now this is a moral choice... Do I *use* that person? I don't like to, and so opt out, but yeah, this would not be a problem if one was determined just to look good for the judges... Maybe I should change my name to Larry and start dressing up as an orthodox Jew to please the great and powerful Hanson...

Having failed or successful businesses: Well, there is an aspect of luck here, but there is also a tremendous aspect of skill and foresight involved. A friend of mine made it semi-big in the dot-com bubble, didn't sell his company, and lost it all... Learning experience maybe?

But don't listen to me... I'm just a woman... My priors tell you that in all odds anything I have done is more likely luck... They asked me if I cheated on my CTBS memory test with the word associations when I was in 3rd grade, cuz I got them all right... Oh, lucky me...

Lara, there is difference between having some control over a reputation factor and having complete control. You would still have some control over adjusted scores, just as you have some but far from complete control over being pretty.

Robin- you just don't get it. How big of an effect do you really think this variance has to be justifying deliberately discounting it? You, of all people, should know that human beings are extraordinarily bad at appropriate mental calculations... let's just say it's a 2% discount.. well, you tell the judges that and in their minds there's this difference that they know they have to account for, but it's in what in total is a subjective decision... so Jane is slightly more qualified than Jim, but we know Jane's a woman, and we *should* discount for that, so take Jim... Now Moira is more than slightly more qualified than Marco, but this bias is already in the judges head, so they *overcompensate* and take Marco, reflexively... This is to a large extent *already* how it is without giving the judges more ammunition for automatically taking the man over the woman or the white over the black... Do you really think women are overvalued in science? I personally don't like affirmative action, because it casts a shadow on the abilities of the people who are there legitimately... 'Black woman? Oh, affirmative action case, obviously.' Then again, would the judges ever let her in without it?

Why is it that when we discuss wages the usual presumption is that it is better to help low wage folks, even if that hurts high wage folks, but the opposite presumption holds for female test scores?

Excessive taxing of the rich and giving to the poor would be bad - if it caused all your talented individuals to emigrate to where their efforts would be better appreciated.

However, the fact that the rich tend to set the laws, control the police, run the country, and control the voters like sheep may mean that the extent of wealth inequalities is larger than is necessary for efficient operation - at least in some countries.

I don't see an analogous effect for female test scores.

Also, the distribution of wealth is a bit different from the distribution of intelligence - the tail of the distribution of wealth extends mosly in one direction - so there are many poor folk and only a few rich ones.

The taxi driver replies:

Look, I'm trying to stop my taxi getting trashed, and make sure my fares pay. I prefer to leave the job of self-sacrifice aimed at reforming society up to other people. Also, isn't much of the point of reputations that they are difficult to forge? If you could buy a reputation as easily as you can buy a suit, all my fares would be wearing them.

Robin: *My argument above is very simple and solid, and took only four paragraphs and two simple algebraic equations. Why ever bother with a twenty page journal article,...*

I'm persuaded by your math, but I'm not the average person, or even the average academic. There is value in a little more exposition, especially for a fact of such general interest.

My suggestion: write a somewhat longer explanation for people less familiar with statistics. A short explanation of the statistics you use (I know it's very basic), an analogy to something non-controversial (e.g., pixel values in an image) showing that this is just a general result of noisy measurements, followed by your conclusion.

You might also look for a place in academia where female test scores overpredict performance, and see if the ratios there agree with your data.

I'd also suggest leaving out the normative conclusion that we *should* correct for this bias. Stick to the facts, it makes irrational criticism more difficult.

What I'm suggesting is that "reputation" is a construct that glues society together. It is not the same as your expected performance. "Reputation" is something you're supposed to earn only by actual performance, because part of the point of keeping track of reputations, and treating people solely based on their reputations, is to encourage performance. If you allow anything into the reputation-construct that isn't performance, even the smallest shred of it, human beings react to this as tremendously unjust and they will cease to respect the system. (At least if they have the prior sense that the metric was supposed to be performance-based. This is one of those systems with a selective off-switch - i.e., all the fantasy novels about the heir to the throne with royal blood, and so on.)

If you find yourself saying something like, "People shouldn't be subject to systemic penalties they can't avoid, for something they never did, that they can't possibly dig themselves out from under no matter what they do," then this is the moral intuition I'm defending and trying to formalize. Deploying any possible test you like, is one matter - there's at least the theoretical possibility of trying harder and scoring better. If you penalize someone for sex or color, they can't change their sex or color by trying harder.

Now, yes, there are kids out there who won't score 1600 on the SAT no matter how hard they try. But that's a different category of injustice, that will take a higher order of technology to remedy.

If you can't see the difference between these types of injustice, then I suspect you of trying too hard to ignore it.

And yes, the second kind of injustice is actually much worse - but the first kind of injustice is a human injustice, not an injustice of Nature; so it is easier to remedy; and therefore, easier to acknowledge as a problem.

I agree we should be careful to separate data from inference about data, and I agree we are often irate that others draw what we consider to be inaccurate inferences about us from data about other people. But most uses of the word "reputation" surely do depend not only on what people did, but also on factors outside their control, many of which can be greatly influenced by the rest of us. For example, whether you get accepted to a school depends now on your nation and state of residence, whether an admission officer just likes your style, whether other students voted for you for class president, whether a teacher liked you enough to write a nice letter of recommendation, whether you were pretty enough to be selected for the lead in a play, and whether an school paper editor thought your articles popular enough with readers. Even your actual SAT score depends on how many times you can afford, in time and money, to retake the test (most schools just use your max score).

It's hard enough for me to calculate my actual beliefs, I'm not sure I also want to have to calculate the beliefs I would have had if I knew some things but not others.

...and worse, people might confuse those two things.

Robin- And you want to add more unjust appraisals to this list, why?

The results of ONE test are highly unreliable. It does not filter out a certain percentage of lucky fellows, those wunderkinder who later, unavoidably fail. It could be easily avoided by retesting the outliers.

Lara, I agree with some of what you say, especially this:

so Jane is slightly more qualified than Jim, but we know Jane's a woman, and we *should* discount for that, so take Jim... Now Moira is more than slightly more qualified than Marco, but this bias is already in the judges head, so they *overcompensate* and take Marco, reflexively

However, I'm not sure if I agree with "when you start introducing systematic discriminatory practices based on priors beyond people's control, you get totalitarianism". We systematically discriminate against kids, by not allowing them to drive until they're 16/18 because statistics show that kids really suck at driving cars.

Also the Hitler comparison was unnecessary. (Godwin's Law)

It's interesting to imagine actually applying this policy, and the resulting implications.

It would be a reversal of affirmative action - women and under-represented minorities would be penalized on their test scores, while those minorities who do better than average, perhaps Asians and Jews, would get a bonus (here we are focusing on high end scores). There might be an effort for members of certain minorities to identify sub-groups which have scored exceptionally well. Perhaps females do somewhat worse than males, but females from Massachusetts (I'm just making this up) do better, so they deserve a bonus rather than a penalty. Meanwhile members of below-average minorities would try to hide their minority status in order to avoid the penalty.

So from one direction there would be constant pressure to discriminate more finely in order to benefit groups whose priors are high, and in the other direction there would be attempts to avoid being caught by fine-tuned discrimination that would identify low-prior groups. At the same time people might try to lie about characteristics in order to gain the benefits of being associated with a high-scoring group.

All, I find it curious that my argument is seen as being against women, when it is only hurts high scoring women - it helps low scoring women.

Since people are probably imagining an admissions officer with high standards doing an evaluation based partly on SAT score, the general view is not inaccurate. If there's a fixed Pass/Fail level, then your argument either hurts high-scoring women or helps low scoring women, but not both. (Which possibility occurs depends on whether the bar is set above or below the mean.)

When discussing policy adjustments, I'm surprised no one has mentioned trying to decrease error, which could be done fairly simply by increasing the number/length of tests.

Robin wrote: "All, I find it curious that my argument is seen as being against women, when it is only hurts high scoring women - it helps low scoring women."

Colleges with high average student SAT scores reject many students for admissions, but colleges with low average student SAT scores often reject few students or have open admissions. Consequently, a woman with a SAT score well below the mean would get very little if this score was slightly increased whereas a woman with a score much above the mean would have a high expected loss from having her SAT score slightly decreased. Thus Robin's plan would harm high scoring women but not much help low scoring ones. Still, this is not a valid reason for opposing it.

Razib who wrote "your blog makes me want to throw up" cites an article saying that the ratio of men to women who have high IQs is much lower than the ratio of men to women at Berkeley's math department. Razib thinks this means that Berkeley's math department is sexist. But before you label Berkeley's math department sexist you need to prove that men with high IQs don't like math more than women with high IQs do.

I'm hesitant to enter this discussion, and I especially don't want to say something absurd like "Even if your maths is right, you're still wrong." However:

All, I find it curious that my argument is seen as being against women, when it is only hurts high scoring women - it helps low scoring women.

I suspect that, as a matter of psychological fact, the annoyance of being told "you will never have members in the top 1%" is greater than the comfort of "you will never have members in the bottom 1%".

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