## Saturday, June 27, 2009

### More on the Male-Female SAT Math Test Gap

It's well known that for the SAT mathematics test, a) male high school students in the U.S. have higher scores on average than females, b) the gap is large and statistically significant (+30 points), and c) the male-female math test score gap has persisted over time, since at least 1971, and probably much longer (see chart above, data here from the Dept. of Education).

One explantion for the female-male math test score gap is summarized here by Janet Hyde et al.:

In 2007 the SAT was taken by 798,030 females but only 690,500 males, a gap of more than 100,000 people. Assuming that SAT takers represent the top portion of the performance distribution, this surplus of females taking the SAT means that the female group dips farther down into the performance distribution than does the male group. It is therefore not surprising that females, on average, score somewhat lower than males. The gender gap is likely in large part a sampling artifact.

MP: In other words, it is only because more females than male take the SAT exam that males score higher on average than females, and if the sample sizes were more equal, the difference in mean math test scores would disappear.

Consistent with this explanation of the difference in mean math test scores would be the following assumption:

Ceteris paribus, if the number of females taking the math SAT exam relative to males (and female percentage of total) increases over time, the male-female math test score gap should INCREASE over time, since an increasing number of females (and increasing percent of total) taking the SAT should lower female mean math test scores over time relative to male math test scores. Reason? The increasing number of females taking the SAT will "dip further down into the performance distribution" over time.

Using
Census Bureau data, the chart below shows that females taking the SAT exam as a percent of the total increased from 50% in 1975 to 53.6%, as the male percentage has decreased from 50% to 46.4% over that period (see chart below).

According to the reasoning above, as the number of females taking the SAT exam increased over time (along with the percent of total) relative to males, the mean female score should have decreased relative to the male mean score, and the male-female gap should be INCREASING over time, theoretically.

But that is exactly the opposite of what has actually been happening. The chart below shows that the male-female gap has actually been decreasing over time, even as more females took the test relative to males, from a high of 46 points in 1977 to a gap of 33 points in 2008.

Bottom Line: The gender gap appears to be more than just a sampling artifact, since the decreasing male-female math test score gap is exactly the opposite of what the Hyde et al. hypothesis would predict.

Update: Additionally, if the number of females taking the test increases over time, the Hyde hypothesis would also predict a falling mean female math test score over time, when in fact we see the opposite: a rising female mean SAT math test score.

Comments welcome.

#### 26 Comments:

At 6/28/2009 12:44 AM,  Anonymous said...

Your logic is impeccable.

At 6/28/2009 6:13 AM,  Anonymous said...

If the mean and var are the same for the pop., shouldn't the female scores show more var than the male scores in the sample due to the larger female count?

At 6/28/2009 6:46 AM,  Steponic's Economics said...

Does the College Board offer an explanation? It seems to me they would have a vested interest in removing gender and race bias from their test.

At 6/28/2009 7:48 AM,  Anonymous said...

"It seems to me they would have a vested interest in removing gender and race bias from their test."

Why do you make an unsubstantiated assertion that there is bias in the test?

At 6/28/2009 9:15 AM,  Anonymous said...

It seems to me they would have a vested interest in removing gender and race bias

Not if the bias is for girls. The current 2400-scale scoring table proves it.

At 6/28/2009 10:21 AM,  PotatoChef said...

How can a math test contain a gender or race bias?

That is such an "easy out" that it is almost laughable.

At 6/28/2009 5:37 PM,  Anonymous said...

Both your conclusions and Ms. Haley's conclusions are nothing but hot air. You're both right when operating under your own choice of faulty assumptions:

Normal (or symmetric) distributions
Distributions which do not change over time
Selection bias
Mesokurtic distributions
Equal variances of the two distributions
Independence

Change your model of the distributions and you can justify the hypothesis of equal (or unequal) means with nearly any test results. Until one rigorously analyzes the distributions, all the hypothesis tests are invalid. The discussion is interesting and rigorous analysis should be done, but reaching any conclusions at this point is ridiculous.

At 6/28/2009 5:56 PM,  Anonymous said...

@Anonymous 0613:

No, if the underlying distributions of males and females are identical or have the same variance, then the larger sample size will result in a smaller variance in the sampling distribution. The sample size is in the denominator. Therefore the sample mean is more likely to be closer to the population mean for women than men. The sample sizes are so large for both populations that the standard error is near zero.

Hyde's argument might be true if math ability for women is skewed to the left. Then as sample size increases you'll sample more from the left tail than the right even though the male and female means are identical. There are tests for skew and kurtosis that can be done.

Some researchers think that black test scores are declining because more low-ability blacks are being urged to take the test by counselors. This results in selection bias. This is, ironically, the result of affirmative action which gives blacks preferential admission.

At 6/28/2009 6:22 PM,  Anonymous said...

Thank you A 5:56, I was unclear.

Basically, I was thinking along the lines of
1) assume that both male and female math talent is ~ N(0,1)
2) SAT math test measurese this talent accuarately
3) The group taking the SAT is not a random sample from the overall population but is skewed to the top range. For simplicity, assume that the female SAT sample represents the top 40% of all females. While the male SAT sample represents the top 35% of all males.

Doing the math:
E(SAT-Math|F) = .966
and
E(SAT-Math|M) = 1.058

Var(SAT-Math|F) = .312
and
Var(SAT-Math|M) = .288

Hence the variance should be larger for the female SAT sample than for the male. I believe that as long as the female sample is larger than the male sample and both represent less than 0.5 of the population this will hold. I'll work on showing this to be true or not.

At 6/28/2009 6:42 PM,  Robert Miller said...

This comment has been removed by the author.

At 6/28/2009 10:06 PM,  Anonymous said...

If there is a gender bias, it is in the life long preparation - not the test itself.

At 6/28/2009 11:22 PM,  Robert Miller said...

This comment has been removed by the author.

At 6/28/2009 11:29 PM,  Jim Egnor said...

On a more rudimentary level, what's the point here? Ok...there's a gap. Has been seen for years. And? Are you arguing for greater monies to study this phenomena? Are you wanting to see more efforts to increase math aptitude among women in this country? In the grand scheme, realpolitik POV---what are your postings regarding male/female inequities attempting to show?

At 6/29/2009 12:36 AM,  OBloodyHell said...

> On a more rudimentary level, what's the point here? Ok...there's a gap. Has been seen for years.

They've been claiming for some two decades or more that this is a sign of some cultural gender bias against women learning math and the "hard" sciences (they, strangely enough, don't seem to feel there's any substantial problem with the pro-female/male disparity in reading scores).

In actuality, the fact that it runs across all cultures, as shown recently in this blog with stats from the PISA study.

If there were not probably some deeper, internal-mechanical reason for the difference in innate math ability (and, of course, on vocab/reading comprehension on the other side), then some culture should show a tendency towards females excelling at math. None of them do.

Feminists cannot abide by the idea that there are, in fact, innate differences between the genders -- it is anathema to their entire raison d'etre, and it just ticks them off.

The fact is, there are, indeed, some jobs which men tend to be more likely to be better at, and others women tend to be more likely better at.

I've never had a problem with this idea. It doesn't say anything about what an individual is capable of, and as long as that is kept in mind, it doesn't matter that guys run better at math or worse at vocabulary/reading.

But say that around a feminist and your head will not remain on your shoulders for much greater than a fraction of a picosecond.

At 6/29/2009 8:00 AM,  Jean-Paul said...

None of them do.
Well, in fact, one of them does. Qatar.

You can check at:
http://www.oecd.org/dataoecd/30/17/39703267.pdf
on page 320.

But for Austria, Chile, Colombia, Germany, UK, Italy, Luxembourg, Portugal, Australia, Slovak Republic, Canada, Switzerland, Netherland, Finland and Brazil, males significantly outperform females.

At 6/29/2009 8:56 AM,  Junkyard_hawg1985 said...

Mark,

Great post. I said earlier that the SAT data did not in itself prove boys are better than girls in math. My concern was the selective nature of of people taking the test. The plots you showed of the % of SAT takers by sexes and difference in math scores over time have fully answered my concerns on the data. Great post!

At 6/29/2009 9:09 AM,  misterjosh said...

I agree with your reasoning, but I would note that SAT scores from year to year aren't really comparable. The gap is comparable obviously, but not the actual scores themselves.

At 6/29/2009 10:28 AM,  Anonymous said...

Misterjosh:

You say that the test scores are incomparable over time. I assume because you think the tests change, the level of preparation changes, and the characteristics of the test takers change over time. In that we are in agreement.

However you say that the gap is still relevant. As the test, preparation and characteristics change over time, the variance may grow or decline for men and women so the gap must be tested for significance within each year and the magnitude of the gaps from year to year will be comparable by the number of standard deviations from the mean not by the absolute size of the gap.

The suggestion someone made that this is a census rather than a sample is intriguing. This implies there is no inferences being made about innate verbal and math ability, only the fact that males who take the test do much better than females who take the test. No hypothesis tests need to be done because its a population and makes no external inferences. If that leaves an unanswered question about why genders perform differently and what if anything to do about it then that's a problem. Remember that the purpose of the test is to determine the likelihood of success in different majors in college and allocate admissions efficiently. The test is not an IQ test for any of the subjects. The math questions are all basic algebra and geometry which must be answered in a short timeframe.

At 6/29/2009 11:10 AM,  jonathan said...

I question whether this issue can be addressed within the confines of this data. Example: I have two daughters, one who was AP math - A student in BC track calculus - and another solid in honors math. Both feel that cultural typing and male aggression play a role in how girls view math. The elder had to fight it out to be heard in a room full of Russian and Asian boys who competed to get the answer first without working together. (She would come home crowing about beating them, but fuming at the same time.) The younger tells me how boys dominate class discussion in math and how the girls are made to feel that math is what the boys do, even if a particular girl is better.

There is evidence - see Steele - that expectations matter in testing, particularly in standardized tests. I can't make any firm statements about this, but 30 points may be at least partially explained by an expectations gap that has been internalized by girls - and maybe by boys in a positive manner.

I don't see any meaningful explanation in the score data by itself so one needs to find other data that bears. My daughters anecdotal evidence points toward expectations.

At 6/29/2009 11:18 AM,  KJ said...

I can tell you exactly what's behind this from seeing first hand the nonsense that goes on in public schools. Many teachers today overtly and I mean overtly favor girls over boys in their teaching and attitude. They are doing this in their view to make amends for prior wrongs.

At 6/29/2009 12:08 PM,  Robert Miller said...

This comment has been removed by the author.

At 6/29/2009 1:30 PM,  Jeffrey Ellis said...

Hyde's explanation is pure crap.

"[T]he surplus of females taking the SAT means that the female group dips farther down into the performance distribution..."

Similarly, it means it dips higher up into the performance distribution as well. Duh. Net effect: the average female performance remains unchanged.

If other females are as smart about math and statistics as Hyde I think we can call this mystery solved. ;-)

At 6/29/2009 5:10 PM,  Anonymous said...

Jesus, what do you expect! She's a professor of Psychology and Women's Studies. Her knowledge of Statistics is novice, at best, and her ideological bias is obvious.

She's not a scientist. Her hypothesis is a preconceived conclusion in search of supporting evidence.

At 6/30/2009 1:53 AM,  OBloodyHell said...

> None of them do.
Well, in fact, one of them does. Qatar.

Conceded. Missed that one.

It does not disprove the point, however.

At 6/30/2009 2:05 AM,  OBloodyHell said...

> The elder had to fight it out to be heard in a room full of Russian and Asian boys who competed to get the answer first without working together. (She would come home crowing about beating them, but fuming at the same time.)

And how is this relevant, please? Are the boys colluding against her? Or are the competing just as much with one another as against her?

If she can't handle the competition, then she needs to work on that, not to interfere, argue with, or complain about male competitive instincts.

This is hardly a trivial matter.

I had a friend whose fiance (now wife) was the most experienced teller at a bank, with 3-5 years worth of experience there.

When a head teller position came open, the bank hired a guy with only about 6 months experience for the position.

She was going to put up with it.

By contrast, a guy in that situation would have
a) Verified other employment, possibly taken it immediately.
b) Having done "a", they would have gone to their superiors at the bank and made it clear in no uncertain terms that they would not tolerate being passed over like that.

If there is a reason why it is that men actually go up the ladder faster than women, as feminists allege (debatable when you actually look at stats -- but let's take it as a given), then part of it is certainly this sort of thing -- the guy in question almost certainly went after that HT position aggressively, whereas she just "expected" to get it.

The phrase, "the squeaky wheel gets the grease" comes to mind.

Properly controlled aggression pays off in the work environment, both internally and externally to the company in question.

There's a reason for that. Fiddle with that principle at your own risk.

At 6/30/2009 2:07 AM,  OBloodyHell said...

> (God, I wish people would at least pick a phony name in the Name/URL block for easy reference)

Moved, seconded. There's no excuse for being this lazy, sorry.