Disparate variances can result in disparate performance, so Hyde et al. addressed the issue:

All [variance ratios], by state and grade, are > 1.0 [range 1.11 to 1.21...]. Thus, our analyses show greater male variability, although the discrepancy in variances is not large.

But what does "not large" mean? Since psychologists don't do Fermi problems, I will lend a hand. Hyde et al. set the context for the meaning of "not large" in their opening sentence:

Gender differences in mathematics performance and ability remain a concern as scientists seek to address the underrepresentation of women at the highest levels of mathematics, the physical sciences, and engineering.

And again in their conclusion:

There is evidence of slightly greater male variability in scores, although the causes remain unexplained. Gender differences in math performance, even among high scorers, are insufficient to explain lopsided gender patterns in participation in some STEM [science, technology, engineering, and mathematics] fields.

"Not large," then, means that the male-female variance difference exerts negligible influence on the number of women with sufficient cognitive resources to function "at the highest levels of mathematics, the physical sciences, and engineering." We ask then: What numerical ceiling does a variance ratio of 1.15 impose upon women in these fields?

For definiteness, consider tenured faculty in math-intensive departments of research universities. Approximately 14,000 doctorates are awarded in these fields each year. (The precise number in not important.) If we assume an average of one new Ph.D. per year per tenured faculty member, the number of doctorates granted roughly equals the number of faculty positions to fill. The pool of US residents from which these faculty are drawn is, by and large, that segment of the population between the ages of 25 and 65. Of these, about 81 million are women and 79 million men.

Consider 2 cases:

Case I. All 160 million men and women compete for the 14,000 tenured positions, the seats being filled in rank order of ability. The ability distributions of men and women in the 25 to 65 year-old cohort will sort out who gets tenure.

Of course, this case will not obtain. Some (perhaps many) able men and women will find rewarding work elsewhere. So, consider Case II.

Case II. One in ten with the required talent chooses to compete in this segment of the marketplace. (In the rank order calculation this is equivalent to expanding the number of tenured positions 10-fold to 140,000 (still in the top 0.1% of the adult population).

Somewhere between Cases I and II lies the truth. The two cases give us bounds on the ceiling imposed by the variance ratio.

Finding the bounds is straightforward.⁵ With a male-to-female variance ratio of 1.15 and zero gap in the means, filling tenured faculty slots in rank order of ability imposes the following bounds:

Case I: No more than 26% of faculty positions will be occupied by women.
Case II: No more than 33% of faculty positions will be occupied by women.

Thus, "the discrepancy in variances is not large" means that it only restricts female participation in this segment of the marketplace to between 26% and 33%. And we have not considered the sex gap in the means. Accounting for it, the bounds drop to between 16% and 22%.

Gender differences in performance on the SAT Mathematics test are widely publicized and contribute to the public's view that males excel in mathematics, compared with females. In 2007, males scored an average of 533 ± 114 (mean ± SD = 114) on the Mathematics portion of the SAT, compared with an average of 499 ± 111 for girls. For many reasons, these data tell us nothing about gender differences in mathematics performance. Chief among these reasons is sampling. The SAT is taken almost exclusively by college-bound students, and even then, some college-bound students do not take it because their intended college requires some other test such as the ACT. Therefore, there is no well-defined sampling frame that would permit broader generalization. Perhaps more important is the fact that, coupled with the current trend for more females than males to attend college, the SAT is taken by more females than males. 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.

All of the above is true, even the conclusion that "the gender gap is in large part a sampling artifact." So what if anything is wrong with this analysis?

In 2007, 23,281 boys scored 750 or better on the math SAT compared with 11,852 girls. At this, the highest level of achievement reported, boys outnumbered girls 2 to 1. It is true that adding girls to the low end will lower the girls' average. It is also true that many if not most college bound students in ACT states do not take SATs. And yes, it is true all this introduces significant sampling problems. But none of this has anything to do with the 2 to 1 ratio of boys to girls in 750 plus territory. That has to do mostly with male variability. Happily, Hyde et al.'s paragraph can be repaired easily by replacing "these data tell us nothing about gender differences in mathematics performance" with "the full SAT data confirm the male advantage in mathematical ability."

Hyde et al. continue:

This conclusion [that the gender gap is in large part a sampling artifact] is verified by results from a study of the ACT. It, too, is taken by a selective group of college-bound students. Traditionally, males have had a slight advantage of 0.2-0.3 points on the composite score. In 2002, two states, Colorado and Illinois, mandated the administration of the ACT to all high school students in those states. ... The gender gap in scores disappeared when the test was administered to all students and, in fact, a slight gap favoring females emerged. These findings support the conclusion that the male advantage on the SAT mathematics test is largely an artifact of sampling.

The ACT composite score is a weighted average of English, math, reading and science scores. In Colorado and Illinois, where there are no sampling problems, as well as nationally where there are, girls outscore boys in English and reading, while boys outscore girls in math and science. And, as in SATs, boys outnumber girls 2 to 1 in the highest ACT levels of mathematics achievement.

By making use of the Colorado and Illinois ACT data we can estimate the effect of sampling error. In 2007, the gender gap nationally on the math ACT was 0.21 SD. The same year in Colorado and Illinois, absent sampling error, the gaps were 0.13 SD and 0.16 SD, respectively. The difference between the national and full-cohort gaps is the contribution of sampling error to the math gap. Thus, "largely an artifact of sampling" means that sampling errors caused the gender gap to be overestimated by approximately 0.05 to 0.08 SD, leaving behind a generous gap of between 0.13 and 0.16 SD, remarkably close to those we shall presently reveal. Sampling error not withstanding, the math gap proves durable.

The Guiso research team analyzed test scores from standardized tests offered to teenage students in many countries. Because the degree of women's emancipation generally differs from country to country, the opportunity to find relationships between mathematical proficiency and gender neutrality presented itself.

Guiso et al. looked test scores from the Program for International Student Assessment (PISA). These tests are designed to assess, cross-nationally, student competencies in science, math and reading. They are given triennially to representative samples of adolescent students ranging in age from 15 years 3 months to16 years 2 months. In 2003, approximately 300,000 students from 41 countries took the tests, increasing to 400,000 students from 57 countries in 2006. Guiso et al. looked at the 2003 data.

To carry out the analysis, the researchers needed a gauge of national gender neutrality. They found four highly correlated measures. Of these, the Gender Gap Index (GGI), an instrument developed by the World Economic Forum in 2006, produced the largest correlations with sex differences in math performance.

The GGI employs four measures of women's emancipation: economic participation and opportunity, educational attainment, political empowerment, and health and survival. It ranges in value between 0 and 1. A value of 1 corresponds to complete gender neutrality; zero to macho heaven. According to the GGI, Northern European countries are the most gender neutral. Latin American and Muslim countries dominate the bottom end of the scale. Both in 2006 and 2007 Sweden lead all nations with GGI scores of 0.8133 and 0.8146, respectively. Norway, Finland and Iceland followed closely behind. Yemen, with scores of 0.4510 and 0.4595, bottomed out the list.

Utilizing data from PISA 2003, Guiso et al. ran linear regressions using gender neutrality measures like the GGI as independent variable. Girl-to-boy ratios in the 95th and 99th percentiles, as well as gender gaps in the mean were the dependent variables. Correlations between sex differences in performance and gender neutrality emerged, from which the authors concluded that gender-neutral environments result in gender-equal math performance.

1. Innate mathematical ability is distributed differently in men and women.

2. Native ability distributions (not test-score distributions) of both men and women are the product of many alleles, and therefore are well approximated by Gaussians (central limit theorem).

3. The variance ratio and mean difference between male and female ability distributions are biological constants, characteristic of the sexes and independent of race, geography and national culture.

4. The influence of gender neutrality on test performance may be treated as a perturbation.

The consequences of the theory of Everyone are readily revealed in diversity space, a construct first introduced in Intelligence, Gender and Race. Let's do a quick review.

Points in diversity space specify the proportions of each of two groups that attain various thresholds of achievement. The thresholds, themselves, are not specified, only the proportions of each group that reach them. Underlying attainment is ability, which in general will be distributed differently between- the two groups.

Figure 1 shows a diversity space of boys and girls. The point illustrated corresponds to a threshold reached by 60% of girls and 40% of boys. It might, for example, represent the proportions of girls and boys that pass a reading comprehension test. The passing grade would then be the threshold. In the illustration, girls have more of whatever it takes to reach the threshold.

Figure 1. Diversity space of girls and boys. The point shown represents a threshold reached by 40% of boys and 60% of girls.

The utility of diversity space lies in the fact that not all points are allowed. Those that are, are determined by underlying ability distributions. If we know the distributions, we can generate the loci of points, but more often we observe the points and from them deduce the ability distributions.

Consider now the diversity space of girls and boys populated with points obtained from PISA test scores. In previous applications of diversity space, I employed multiple thresholds. Here, I use PISA math level 5 as the single threshold. (PISA defines six levels of proficiency, one to six, six being the highest. Level 5 corresponds to approximately the 85th percentile.) To populate diversity space, divide boys and girls into subgroups corresponding to the boys and girls from each participating country. In this partition, each country contributes one point to diversity space for each year it takes part in PISA testing.

The theory of Everyone asserts that mathematical ability in both men and women are Gaussian distributed. Figure 2 displays in diversity space several curves from the family of curves that can be generated from Gaussian distributions. The curves in the figure were generated using arbitrary choices of the distribution parameters. Of the infinity of such curves, one will best fit the data. Our job is to find that curve and the ability distributions that generate it.

Figure 2. Several of the infinity of curves representing possible loci of points in diversity space. The curves were generated using arbitrary values for the variance ratio and mean difference in the ability distributions of boys and girls.

In both PISA 2003 and PISA 2006, the percentage of students that reached level 5 or higher varied wildly from country to country. For example, in 2006, approximately 0.04% of Kyrgyzstani girls and 0.08% of Kyrgyzstani boys reached level 5 or higher, while that same year 30% of Korean boys and 24% of Korean girls achieved at this level. Between these extremes lay many countries, resulting in a good spread of points in diversity space.

A 1/N weighted least squares fit in diversity space of PISA 2003 and 2006 data revealed the underlying ability distributions. Table 1 shows the resulting distribution parameters. Included for comparison are parameters from Project Talent, a 1960 survey of 73,425 15 year-olds representative of the entire population of 15 year-olds in the U.S. -- students and nonstudents alike. After almost half a century Project Talent remains one of the best assessments of cognitive sex differences ever made.

Allowing for statistical fluctuation, it is evident from Table 1 that: 

MATH-ABILITY DISTRIBUTIONS OF MEN AND WOMEN HAVE NOT CHANGED SIGNIFICANTLY IN AT LEAST HALF A CENTURY.

Figure 3 illustrates the fit in diversity space of theory to PISA data. In the three years between PISA 2003 and PISA 2006, Iceland, Korea, Macao-China and the Netherlands, outliers in 2003, migrated sharply back into the mainstream of performance toward the theory of Everyone prediction. The Czech Republic moved in the other direction away from prediction. None of the swings was accompanied by a corresponding change in gender neutrality. They are statistical fluctuations whose size warns against overinterpreting data from a single PISA year.

Figure 3. Wild swings in diversity space were observed between 2003 and 2006. The Netherlands, Iceland, Korea and Macao-China, outliers in 2003, moved sharply toward the predicted locus of points, while the Czech Republic moved oppositely away from prediction.

THE GIRL-TO-BOY RATIO AT ALL LEVELS OF ACHIEVEMENT IN MATHEMATICS INCREASES MONOTONICALLY WITH THE GENERAL INTELLIGENCE OF A POPULATION.

That is, in smart countries, girls will perform more like boys than they will in dumb countries.

Figure 5 illustrates the male and female math-ability distributions. The minimum ability, λ₀, required to reach a particular but unspecified threshold of performance is marked on the ability axis. The ratio of girls to boys, R, at and beyond the threshold is given by:

To evaluate the girl-to-boy ratio, R, for a country, it is convenient to choose the origin as the mean male ability, marked as μ in Figure 5. Accordingly, we replace λ₀ by λ₀ - μ in (1), and after some arithmetic obtain:

In (2), Δ is the mean ability gap (male - female); ρ is the variance ratio (male/female); and both μ and Δ are in units of the standard deviation of the male distribution.

The expression (2) tell us how the girl-to boy ratio, R, varies with mean male mathematical ability, μ. It is plotted in Figure 6. Arising from the theory of Everyone, (2) is quite general. It applies at all levels of proficiency. The ratio, R, as given by (2) is non-negative for all values of μ and increases monotonically with μ.

One story told by (2) and illustrated in Figure 6 is that girl-to-boy ratios at all levels of achievement are greatest in countries with high mathematical ability -- more precisely, high male ability. But the distinction is without a difference. National mathematical ability and national ability of men correlate at r = 1.00. Both are excellent proxies for national intelligence as well. PISA means correlate at r = 0.85 with the Lynn and Vanhanen compilation⁹ of national IQ; PISA means of males correlate at r = 0.87. One could even argue that PISA means, whether national or male, better assess national intelligence than do the IQs derived by Lynn and Vanhanen from often sketchy data and dubious assumptions based on the Flynn effect. (Alternatively, one could take these correlations as evidence in support of the L and V compilation.)

Figure 6. The theory of Everyone predicts how the girl-to-boy ratio, R, varies with male mean ability, μ, a proxy for national IQ.

Figure 7 compares predicted with observed girl-to-boy ratios in PISA levels 5. Though the ratio increases with national intelligence (or the proxy PISA mean score), it would take a nation of cognitive giants to maintain a girl-to-boy ratio near unity. Mean scores of the current highest-performing nations would have to increase by almost two standard deviations.

Figure 7.  PISA data viewed against the theory of Everyone prediction.

Table 2 lists the average national IQ and GGI of the bottom and top performing PISA countries. Between the highest and lowest performing countries is an IQ gap of 0.7 standard deviations. There is also a GGI gap of 1.2 SD in the same direction. Thus, the lowest-performing countries, dominated by Muslim and Latin cultures, are not only cognitively limited, but from a feminist perspective culturally needy.

F = F₀(Δ, ρ) +  α (1-GGI) + β (1-GGI)²  + γ (1-GGI)³ + O([1-GGI]⁴)                 (3)

In (3), Δ and ρ are the mean difference and variance ratio, respectively, of the ability distributions of men and women. With the threshold set at PISA level 5, a least squares fit of (3) in diversity space produced the distribution parameters listed in Table 3. Included for comparison are the results of the original 2-parameter fit ignoring cultural contributions.

The inclusion of perturbative terms increases the gap between the boy and girl distribution means by 0.02 to 0.03 SD; the variance ratio remains essentially unchanged. The previously estimated numerical ceilings on women in the math-intensive marketplace remain intact. If anything, they are a bit lower.

Figure 8 graphically represents the relative contributions, determined by (3), of nature, F₀(Δ, ρ), and nurture (the perturbative terms) to the percentage of girls reaching PISA math level 5. Data from all countries that took part in either PISA 2003 or PISA 2006 are included in the figure. Female emancipation not withstanding, there is a formidable biological barrier to overcome before the math gender gap can be closed.

Figure 8. Relative contributions of nature and nurture to the percentage of girls reaching PISA math level 5 or above. Data from all nations participating in PISA 2003 and 2006 are included.

In brief, we have seen tonight that the gender gap in mathematics has been stable for at least half a century; that sex differences in ability-distribution means and variance ratio are independent of race, culture and geography; that female math performance is closest to that of males in high-IQ countries; that culture plays a role in math performance, albeit small; and that the theory of Everyone accounts for all of the above. If these results are unsettling, take comfort knowing that no presentation of fact, regardless how compelling, will keep the gap buster from her noble calling. Thank you for listening to a few of the facts.

[2] JS. Hyde, S. M. Lindberg, M. C. Linn, A. B. Ellis, C. C. Williams, "Gender similarities characterize math performance" Science 321, 494 (2008)

[3] J. S. Hyde, E. Fennema, S. Lamon, "Gender differences in mathematics performance: A meta-analysis" Psychol. Bull. 107, 139 (1990).

[4] Among social scientists the term "effect size" can mean different things. As used by Hyde et al. and many others, it is the difference between two group means (in laboratory units) divided by the population weighted root mean square standard deviation of the two distributions.

[5] Let N_W and N_M be the numbers of women and men, respectively, that compete for N_S tenured-faculty slots to be filled in rank order of ability. The quantities P_W and P_M are the ability distributions of women and men respectively, and a is the ability of the dimmest bulb to obtain tenure. Then the following relation must be satisfied

The two terms on the LHS of (1) represent the numbers of women and men, respectively, who succeed in gaining tenure. Numerical solution of (1) gives us a value for a from which we can evaluate each term individually.

[6] C. P. Benbow, D Lubinski, D. L. Shea, and H. Eftekhari-Sanjani "Sex Differences in Mathematical Reasoning Ability at Age 13: Their Status 20 Years Later" Psychological Science 11, No. 6, 2000.

[7] The difference between the male and female distribution means obtained by the least squares fit in diversity space are in standard deviation units of the male distribution. In order to compare our results with others, they were converted to "effect size" as defined above in Note 4. Here's how we did it.

Bars denote laboratory units throughout. The effect size, d, as defined above is:

where Δ is the (male-female) mean difference, σ_F and σ_M are the standard deviations of the female and male distributions, respectively; f_F and f_M are the female and male fractions of the sample population, respectively.

Let δ be the difference (male - female) between the distribution mean in units of the male-distribution standard deviation. (This is the quantity obtained from the least squares fit in diversity space.) Then we may write:

From (1) and (2) we obtain the desired relation between the effect size and d, i.e.,

The ratio of standard deviations, σ _F/σ_M, is dimensionless. Values of the ratio and δ are obtained from the least squares fit.

In the special case where men and women are equally represented in the sample, (3) simplifies to

We used (5) to make the conversion to effect size. The difference between d and δ was less than 5% in all cases.

[8] Icelandic girls performed anomalously well in both PISA 2003 and 2006. A more detailed look at Icelandic data, however, reveals that only in rural Iceland were girls' mean scores higher than boys'. In the Reykjavik metropolitan area the math performance of girls and boys was much like that found in other countries. No satisfactory explanation of the Icelandic anomaly has yet been put forth.

[9] IQ and Global Inequality, R. Lynn and T. Vanhanen, Washington Summit Publishers, 2006.

[10] The Smart Fraction Theory of IQ and the Wealth of Nations, La Griffe du Lion, Vol. 4 No. 1 2004.

[11] Smart Fraction Theory II:  La Griffe du Lion, Vol. 6 No. 2, 2004