# Fundamental Errors in Kane and Mertz’s Alleged Debunking
of Greater Male Variability in Mathematics Performance

Rosalind Arden and Theodore P. Hill

###### Abstract

Kane and Mertz’s 2012 AMS Notices article “Debunking Myths about Gender and
Mathematics Performance” claims to have debunked the greater male variability
hypothesis with respect to mathematics abilities. The logical and statistical
arguments supporting their claim, however, which are being widely cited in the
scientific literature, contain fundamental errors. The methodology is
critically flawed, the main logical premise is false, and the article omits
reference to numerous published scientific research articles that contradict
its findings. Most critically, Kane and Mertz’s final conclusion that their
data are inconsistent with the greater male variability hypothesis is wrong.
The goal of the present note is to correct the scientific record with respect
to those claims.

##  1 Introduction

The American Mathematical Society’s sole article of record addressing the
controversial “hypothesis” of greater male variability (GMV) is the widely
cited paper “Debunking Myths About Gender and Mathematics Performance” by
husband and wife team Jonathan Kane and Janet Mertz [25].

That feature-length article which appeared in the Notices of the AMS, “the
world’s most widely read magazine aimed at professional mathematicians” seems
to promise, through its sensational title, a dramatic trouncing of false
collective beliefs regarding gender gaps in mathematics performance. Yet
surprisingly the body of the article makes no mention of myths, but rather
presents an investigation of at least five hypotheses (italics theirs) based
on data from children’s performance on standard tests. Kane and Mertz conclude
that most of these hypotheses are “inconsistent” with their findings, thus
apparently “debunking” these pernicious myths. The initial spotlight is
focused on greater male variability, suggesting it is the first and foremost
“myth” to be exposed.

Unfortunately, the Kane-Mertz article is seriously flawed in virtually every
aspect of the GMV discussion. The goal of the present note is not to argue the
validity or invalidity of GMV in general or with respect to cognitive or
mathematical abilities in humans. Rather it is to correct the scientific
record with respect to the basic errors in their article. Perhaps this might
also inspire others to take a closer look at the rest of their
hypothesis/myths, especially given their concern about how resources are spent
in these various endeavors.

##  2 Misleading History

The lead paragraph of [25] introduces the GMV hypothesis as follows:

> “the greater male variability hypothesis, originally proposed by Ellis in
> 1894 and reiterated in 2005 by Lawrence Summers when he was president of
> Harvard University, states that variability in intellectual abilities is
> intrinsically greater among males” [25, p. 10, italics in original].

This statement is misleading in both its history and its formulation. First,
the GMV hypothesis dates back to Charles Darwin, a fact explicitly
acknowledged by variability expert Stephanie Shields in [34], the only
historical article cited in [25]. And second, the classical GMV hypothesis
pertains to traits in many species throughout the animal kingdom, not only to
mathematics performance or even general cognitive abilities in humans. As also
noted in [34], the working definition of the hypothesis has changed back and
forth over the years, but the authors of this note know of no other source
where GMV is interpreted exclusively in terms of variability of mathematics
ability or performance.

By only mentioning “intellectual abilities”, Kane and Mertz explicitly
misrepresent Havelock Ellis, who gathered the scientific literature on sex
differences in variability “in order to study the issue directly and in-depth”
[34, p. 775]. In Ellis’s classic text he devoted an entire chapter to
“Variational Aspects of Men”, in which he clearly states “Both the physical
and the mental characters of men show wider limits of variation than do the
physical and mental characters of women” [12, p. 358].

Similarly, by referring only to intellectual variability, Kane and Mertz also
misrepresent Harvard President Larry Summers, who in fact said

> “It does appear that on many, many different human attributes – height,
> weight, propensity for criminality, overall IQ, mathematical ability,
> scientific ability – there is relatively clear evidence that whatever the
> difference in means – which can be debated – there is a difference in the
> standard deviation, and variability of a male and a female population” [38,
> p. 3].

The standard form of the GMV hypothesis, as recorded by Darwin, is simply that
throughout the animal kingdom, males are generally more variable than females
for many traits. It does not say that there is greater male variability in
every trait of every species. In the special case where the species is human,
the GMV hypothesis simply says that for many traits, both physical and
cognitive, male variability is greater than female variability.

##  3 Flawed Methodology

Kane and Mertz claim they “tested the greater male variance hypothesis” with
respect to mathematics performance in humans, specifically referring to Fields
medalists and “empowerment as reflected by percentage of women in technical,
management, and government positions” [25, p. 12]. That is, their experiments
were designed to draw conclusions about adult humans, not infants or children.

In the description of their method, however, Kane and Mertz clearly state that
“most measures of mathematics performance here are based on the TIMSS, a
quadrenial study that includes a mathematics assessment” of fourth and eighth
graders from numerous countries [25, p. 11]. Kane and Mertz’s conclusions are
thus mostly based on tests of pre- and early-adolescent children, not adults,
and it has been established that at those ages boys and girls follow different
developmental trajectories in many traits such as school performance [7] and
height [33].

For example, Arden and Plomin [1] addressed questions about over- and under-
representation of each gender at the low and high extremes of measures of
cognitive abilities by studying sex differences in variance of test scores
across childhood. Among other conclusions, they found that “From age 2 to age
4, girls in our study were highly significantly over-represented in the top
tail” [1, p. 44]. Thus, employing the same logic and methodology as [25] to
extrapolate data from tests done on children from age 2 to 4 to conclusions
about adults, this finding of Arden and Plomin would imply that among adults,
women are highly significantly over-represented in the top tail of
intelligence. Similarly, extrapolating predicted heights of adults from data
about the fourth and eighth graders that [25] studied would conclude that
adult women are taller on average than men. To draw reasonable inferences
about gender differences in variability (or any other traits) among human
adults from tests on children requires serious formal justification, and this
is missing in [25].

Similarly, the working definition of the GMV hypothesis explicitly stated in
the Kane and Mertz article is that “variability in intellectual abilities is
intrinsically greater among males” [25, p. 12, emphasis added], yet as noted
above their conclusions are based solely on standard tests of mathematical
ability. To infer that mathematical ability alone is a reasonable measure of
overall intrinsic intellectual ability is highly debatable and requires
justification. This, too, is entirely missing in [25].

##  4 Crucial Logical Error

To quantify the notion of gender differences in variability of a collection of
data, the standard parameter used is the so-called variance ratio VR, which,
by convention, is defined as the male variance of the data divided by the
female variance [25, p. 11]. There is greater male variability in a particular
trait of a given sexually-dimorphic species if the VR for that trait is
greater than one. In Kane and Mertz’s main logical argument, they assert

> “we tested the greater male variance hypothesis [with respect to mathematics
> performance in humans]. If true, the variance ratios (VRs) for all countries
> should be greater than unity and similar in value” [25, p. 13].

This logical argument is repeated almost verbatim in a “Doing the Math”
section of the American Academy of Arts and Sciences flagship publication
Science article highlighting their work: “If the greater male variability
hypothesis …is true, then that variability would persist, consistently, across
all 86 countries” [26, p. 2]. In short, the main logical premise of [25] is
this:

  1. (P)

If there is GMV for humans worldwide, i.e., if VR $>1$ for a particular trait,
then the corresponding VRs for each country should all be greater than 1 and
similar in value.

This premise (P) is false from a purely statistical standpoint – it violates a
standard statistical fact, namely, the formula for the variance of a finite
weighted mixture of distributions, as will be seen in the next two examples.

The first example is hypothetical and illustrates how a union of countries (in
this case only two) could exhibit greater male variability as a whole, even
though not all of the individual countries do. That is, $\mathit{VR}>1$ for
the union, but not for each country.

######  Example 4.1.

A population consists of two countries $C_{1}$ and $C_{2}$, with equal numbers
of people in each, divided equally in each among men and women. Scores on a
certain test administered to everyone in the overall population result in
means $m_{1}$ and $m_{2}$ and standard deviations $\sigma_{1}$ and
$\sigma_{2}$ for the men in countries $C_{1}$ and $C_{2}$, respectively, and
means $f_{1}$ and $f_{2}$ and standard deviations $\hat{\sigma}_{1}$ and
$\hat{\sigma}_{2}$ for women. Applying the standard formula (e.g., equation
(1.21) in [18]) for the moments of finite weighted mixtures of distributions,
the variance of the men’s scores in the overall population is given by
$({2(\sigma_{1}^{2}+\sigma_{2}^{2})+(m_{1}-m_{2})^{2})}/4$ and that of women
is $({2(\hat{\sigma}_{1}^{2}+\hat{\sigma}_{2}^{2})+(f_{1}-f_{2})^{2})}/4$.
Letting $\mathit{VR}$ denote the variance ratio in the overall population and
$\mathit{VR}_{i}$ the variance ratios in $C_{i}$, $i=1,2$, it follows
immediately that

(4.1)
$\mathit{VR_{1}}=\frac{\sigma_{1}^{2}}{\hat{\sigma}_{1}^{2}},\mathit{VR_{2}}=\frac{\sigma_{2}^{2}}{\hat{\sigma}_{2}^{2}},\mathit{VR}=\frac{2(\sigma_{1}^{2}+\sigma_{2}^{2})+(m_{1}-m_{2})^{2}}{2(\hat{\sigma}_{1}^{2}+\hat{\sigma}_{2}^{2})+(f_{1}-f_{2})^{2}}.$

For example, if
$m_{1}=m_{2}=102,\sigma_{1}^{2}=5,\sigma_{2}^{2}=1,f_{1}=f_{2}=103,\hat{\sigma}_{1}^{2}=1,\hat{\sigma}_{2}^{2}=2$,
then equation (4.1) implies that $\mathit{VR_{1}}=5,\mathit{VR_{2}}=0.5$, and
$\mathit{VR}=2$, so there is greater male variability in the overall
population, but greater female variability in $C_{2}$. This contradicts (P).

The second example is also based on humans, but in contrast to the previous
example, uses real data and concerns the physical trait of height. (Human
height is one of science’s most studied and documented measurements and has
been recorded and analyzed in great detail, over time and geographic location,
in part because height is easy to measure and is an indicator of important
factors such as nutrition and genetics.)

######  Example 4.2.

Roser, Appel and Ritchie [33] list the mean height of men worldwide at 178.4
cm with a standard deviation of 7.59 cm and women’s mean height at 164.7 cm
with a standard deviation of 7.07 cm. The variance ratio for adult human
height worldwide is therefore $\mathit{VR}>1.07$, implying greater male
variability. The variance ratios by country and birth year, on the other hand,
are “all over the place” and range from less than 0.5 to greater than 2.5 [19,
20]. Thus there is greater variability worldwide in heights of men than
heights of women, even though the VR’s for height vary significantly among
countries. This also contradicts (P).

No justification for (P) is presented in [25], and as just seen in the two
previous examples, (P) is false prima facie.

##  5 Erroneous Main Conclusion

A logical argument can be wrong, yet its conclusion correct, but this is not
the case in [25]. Even though the authors themselves explicitly concede that
they found 8% greater male variability in their analysis of mathematics
performance as indicated by certain test scores worldwide - “These findings
agree well with the VR of 1.08 reported from a large meta-analysis involving
data from 242 studies involving over 1 million Americans” [25, p. 13] \- their
article maintains “These findings are inconsistent with the greater male
variability hypothesis” [25, pp. 14]. This conclusion was repeated in an even
stronger form in Science, where Mertz asserted

  1. (C)

“We have pretty clear data debunking the greater male variability hypothesis”
[26, p. 1].

One of the main objectives of [25] is to claim that their data proves that
greater male variability (with respect to mathematics ability in particular)
is “bunk” - a myth or a fairy tale which if true, could perhaps help explain
the preponderance of male Fields medalists. Yet greater male variability also
leads to more males being found in the left or lower tail of a distribution,
thus perhaps contributing to the fact that seventy per cent of the children
eligible for special education are boys [35, p. 97], and that boys and men are
overrepresented at the lowest levels of IQ [11, p. 42], [24, p. 529]. Greater
male variability has implications for both tails of the distribution, and this
important fact is essentially ignored in [25].

Kane and Mertz also conclude that the non-uniformity in VR’s they found is
largely an artifact of “a complex variety of sociocultural factors rather than
intrinsic differences” [25, p. 11], i.e., cultural factors as opposed to
“innate, biologically determined differences between the sexes” [25, p. 10].

This same secondary conclusion is repeated in the AAAS Science article which
reports that “cross-cultural analysis seems to rule out several causal
candidates, including coeducational schools, low standards of living, and
innate variability among boys” [26]. (Note, however, that it is only innate
variability that is emphatically “debunked”).

Kane and Mertz again repeat this claim in Scientific American:

> “The finding that males’ variance exceeds females’ in some countries but is
> less than females’ in others and that both range ‘all over the place
> suggests it can’t be biologically innate, unless you want to say that human
> genetics is different in different countries,’ Mertz argues. ‘The vast
> majority of the differences between male and female performance must reflect
> social and cultural factors.’ [3, p. 4, emphasis added]”.

But evidence shows that human genetics does differ within and among countries,
even within the continent of Europe [32]. In Kane and Mertz’s study, Taiwan
and Tunisia are seen to have the extreme variance ratios in 2007 TIMSS scores
for eighth graders, namely, 1.31 for Taiwan and 0.91 for Tunisia [25, Table
2]. How do Kane and Mertz conclude that the significantly different VR’s they
found for these two countries are primarily artifacts of sociocultural factors
rather than, say, a more balanced combination of sociocultural and innate
biological factors? This is an empirical question that has been studied in
some depth (e.g., [36]); it is not a matter of “what you want to say”. Yet
based on [25], Scientific American concludes “Now that the greater male
variability hypothesis has fallen short, nature is not looking as important as
scientists once thought” [3, p. 5].

##  6 Failure to Cite Published Counterevidence

The Kane and Mertz article [25] fails to report that their own GMV historical
reference clearly states that in 19th century studies “The biological evidence
overwhelmingly favored males as the more variable sex” [34, p. 773, emphasis
added]. Similarly, their article [25] includes 53 references, but omits scores
of research studies that already had reported greater male variability in many
different contexts, both cognitive and otherwise.

Even for the very special case of mathematics performance, Kane and Mertz fail
to report that their same historical reference states that “Research from the
1960s and 1970s …[indicates that] scores on tests of mathematical ability show
a consistent trend in the direction of greater male variability” [34, p. 793].
As for more recent decades, Kane and Mertz also failed to cite more than a
dozen previously-published studies supporting GMV in mathematical or
quantitative cognitive abilities (emphasis added in the following):

> “Finally, it should be noted that the boys’ SAT-M scores had a larger
> variance than the girls’” [5, p. 1031].

> “The important exception to the rule of vanishing gender differences is that
> the well-documented gender gap at the upper levels of performance on high
> school mathematics has remained constant over the past 27 years” [13, p.
> 95].

> “Not only do males reliably score higher on the SAT-M (mean difference
> approximately .5 standard deviation), they also display a greater
> variability on such measures” [30, p. 328].

> “Males were consistently more variable than females in quantitative
> reasoning, spatial visualization, spelling, and general knowledge” [14, p.
> 61].

> “The current finding that males were more variable than females in math and
> spatial abilities in some countries is consistent with the findings of
> greater male variability in these abilities in the United States” [15, p.
> 90].

> “Examination of the ratios of male score variance to female score variance
> (VR values) in Table 2 [including Mathematics] reveals that the variance of
> male scores is larger than that of female scores” [22, pp. 43–44].

> “As in mathematics, the variance of the total scores among boys [in science]
> was generally larger than that among girls across all participating
> countries” [4, p. 371].

> “A nationally representative UK sample of over 320,000 school pupils aged
> 11–12 years was assessed on …separate nationally standardized tests for
> verbal, quantitative, and non-verbal reasoning …for all three tests there
> were substantial sex differences in the standard deviation of scores, with
> greater variance among boys” [37, pp. 463,475].

> “Males are more variable on most measures of quantitative and visuospatial
> ability, which necessarily results in more males at both high- and low-
> ability extremes” [21, p. 1].

> “Despite the modest differences at the center of the distribution, the
> greater variability of male scores resulted in large asymmetries at the
> tails, with males out-numbering females by a ratio of 7 to 1 in the top 1%
> on tests of mathematics and spatial reasoning …greater male variance is
> observed even prior to the onset of preschool” [8, pp. 220–221].

> “This indicates that the differences that we observed – for example, in the
> overrepresentation of males at the extremes of the distributions for
> quantitative reasoning …With one exception …all variance ratios were greater
> than 1.0” [29, p. 395].

> “The hypothesis of greater male variability was supported in most domains
> [including Arithmetic]” [9, p. 475].

> “[T]he methods are applied to estimate the residual variance in test scores
> for male and female students on the mathematics portion of the 2007 Arizona
> Instrument to Measure Standards …We find that male students exhibit greater
> residual as well as raw variability for this data set” [31, pp. 2937, 2947].

Kane and Mertz also omit an important report that had been published by the
College Board. This official historical summary of 20 years of results of
SAT-M, the mathematics portion of the Scholastic Aptitude Test, shows greater
male variability in every single year, as seen in Figure 1.

Note that the male variances are not much larger than those of females, but
the mean values of males are also noticeably higher. This combination can have
significant effects on the right tails since, as Feingold pointed out, “what
might appear to be trivial group differences in both variability and central
tendency can cumulate to yield very appreciable differences between the groups
in numbers of extreme scorers” [16, p. 11]; see also Example 3.6 in [17].

Figure 1: Table B3 from College Board Report 2006-5 [27, p. 25].

##  7 Summary

Recall that it is not the goal of the present note to argue the validity or
invalidity of greater male variability in general or with respect to cognitive
or mathematical abilities in humans, but simply to correct the scientific
record concerning the faulty arguments and conclusions in [25]. However, the
interested reader should note that since the publication of their paper [25]
in 2012, overwhelming additional new evidence of GMV in various traits and
species has been published; in particular, here are quotes from several very
recent mega-studies. The first even uses data from the same PISA tests used by
Kane and Mertz:

> “Twelve databases from IEA [International Association for the Evaluation of
> Educational Achievement] and PISA [Program for International Student
> Assessment] were used to analyze gender differences within an international
> perspective from 1995 to 2015 …The ‘greater male variability hypothesis’ is
> confirmed” [2, p. 1].

The second may perhaps help explain the possibility that “variability in
intellectual abilities is intrinsically greater among males” [25, pp. 10] is
true, and may have biological underpinnings:

> “the largest-ever mega-analysis of sex differences in variability of brain
> structure, based on international data spanning nine decades of life …The
> present study included a large lifespan sample and robustly confirmed
> previous findings of greater male variance in brain structure in humans. We
> found greater male variance in all brain measures, including subcortical
> volumes and regional cortical surface area and thickness, at both the upper
> and the lower end of the distributions” [39, pp. 6,23].

As emphasized above, GMV for humans refers to many traits, and strong evidence
supporting this phenomenon has also appeared recently:

> “The principal finding is that human intrasex variability is significantly
> higher in males, and consequently constitutes a fundamental sex difference
> …The data presented here show that human greater male intrasex variability
> is not limited to intelligence test scores, and suggest that generally
> greater intrasex variability among males is a fundamental aspect of the
> differences between sexes” [28, pp. 220–221, emphasis added].

(Evidence of GMV in cognitive abilities has also recently been reported for
the first time in non-human [10] and even non-mammalian species [6].)

The arguments above pointed out fatal methodological and logical errors in the
widely cited Kane and Mertz article [25], and that their main conclusion (C)
is false. Their data does not debunk the greater male variability hypothesis,
even in the special case of mathematics performance. On the contrary, the
preponderance of new evidence suggests exactly the opposite and the reader is
encouraged to look at Appendix A of [23], which surveys GMV studies published
since 2000. The Notices of the AMS is looked up to by scientists in many
fields, and should be held to especially high standards, including corrections
of published and widely propagated errors. The goal of this note is to help
correct the scientific record and illuminate errors involving the greater male
variability hypothesis.

## References

  * [1] Rosalind Arden and Robert Plomin.  Sex differences in variance of intelligence across childhood.  Personality and Individual Differences, 41:39–48, 2006.  doi:10.1016/j.paid.2005.11.027.
  * [2] Ariane Baye and Christian Monseur.  Gender differences in variability and extreme scores in an international context.  Large Scale Assessments in Education, 4(1):1–16, 2016.  doi:10.1186/s40536-015-0015-x.
  * [3] Sharon Begley.  Anything boys can do….  Scientific American, January 2012.  URL: https://www.scientificamerican.com/article/anything-boys-can-do/.
  * [4] Michal Beller and Naomi Gafni.  The 1991 International Assessment of Educational Progress in Mathematics and Sciences: The gender differences perspective.  Journal of Educational Psychology, 88(2):365–377, 1996.  doi:10.1037/0022-0663.88.2.365.
  * [5] Camilla P. Benbow and Julian C. Stanley.  Sex differences in mathematical ability: More facts.  Science, 222(4475):1029–1031, 1983.  URL: https://my.vanderbilt.edu/smpy/files/2013/02/ScienceMoreFacts.pdf.
  * [6] Carrie L. Branch, Benjamin R. Sonnenberg, Angela M. Pitera, Lauren M. Benedict, Dovid Y. Kozlovsky, Eli S. Bridge, and Vladimir V. Pravosudov.  Testing the greater male variability phenomenon: Male mountain chickadees exhibit larger variation in reversal learning performance compared with females.  Proceedings of the Royal Society B, 287:20200895, 2020.  URL: https://royalsocietypublishing.org/doi/full/10.1098/rspb.2020.0895, doi:10.1098/rspb.2020.0895.
  * [7] Dorota Buczylowska, Pola Ronniger, Jessica Melzer, and Franz Petermann.  Sex similarities and differences in intelligence in children aged two to eight: Analysis of SON-R 2 8 scores.  Journal of Intelligence, 7(2), 2019.  URL: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6630280/, doi:10.3390/jintelligence7020011.
  * [8] Stephen J. Ceci, Wendy M. Williams, and Susan M. Barnett.  Women’s underrepresentation in science: Sociocultural and biological considerations.  Psychological Bulletin, 135(2):218–261, 2009.  doi:10.1037/a0014412.
  * [9] Hsinyi Chen, Mei-Fang Chen, Te-Sheng Chang, Yuan-Shun Lee, and Hsin-Ping Chen.  Gender reality on multi-domains of school-age children in Taiwan: A developmental approach.  Personality and Individual Differences, 48(4):475–480, 2010.  URL: http://www.sciencedirect.com/science/article/pii/S0191886909004905, doi:10.1016/j.paid.2009.11.027.
  * [10] Alex R. DeCasien, Chet C. Sherwood, Steven J. Schapiro, and James P. Higham.  Greater variability in chimpanzee (pan troglodytes) brain structure among males.  Proceedings of the Royal Society B, 287:20192858, 2020.  URL: https://royalsocietypublishing.org/doi/pdf/10.1098/rspb.2019.2858, doi:10.1098/rspb.2019.2858.
  * [11] Dominika Dykiert, Catharine R. Gale, and Ian J. Deary.  Are apparent sex differences in mean IQ scores created in part by sample restriction and increased male variance?  Intelligence, 37(1):42–47, 2009.  doi:10.1016/j.intell.2008.06.002.
  * [12] Havelock Ellis.  Man and woman: A study of human secondary sexual characters.  Scribner’s, New York, 1894.
  * [13] Alan Feingold.  Cognitive gender differences are disappearing.  American Psychologist, 43(2):95–103, 1988.  URL: https://psycnet.apa.org/record/1988-18577-001, doi:10.1037/0003-066X.43.2.95.
  * [14] Alan Feingold.  Sex differences in variability in intellectual abilities: A new look at an old controversy.  Review of Educational Research, 62(1):61–84, 1992.  URL: http://www.jstor.org/stable/1170716, doi:10.2307/1170716.
  * [15] Alan Feingold.  Gender differences in variability in intellectual abilities: A cross-cultural perspective.  Sex Roles, 30(1/2):81–92, 1994.  doi:10.1007/BF01420741.
  * [16] Alan Feingold.  The additive effects of differences in central tendency and variability are important in comparisons between groups.  American Psychologist, 50(1):5–13, 1995.  doi:10.1037/0003-066X.50.1.5.
  * [17] Ronald F. Fox and Theodore P. Hill.  A note on over- and under-representation among populations with normally-distributed traits, 2021.  URL: https://arxiv.org/abs/2103.09324, arXiv:2103.09324.
  * [18] Sylvia Frühwirth-Schnatter.  Finite Mixture and Markov Switching Models.  Springer, 2006.  Chaper 1.2.4, pp. 10–11.
  * [19] Jaume Garcia and Climent Quintana-Domeque.  The evolution of adult height in Europe: A brief note.  Technical Report Working Paper 1002, Universitat Pompeu Fabra, 2006.  URL: https://econ-papers.upf.edu/papers/1002.pdf.
  * [20] Jaume Garcia and Climent Quintana-Domeque.  The evolution of adult height in Europe: A brief note.  Economics and Human Biology, 5:340–349, 2007.  URL: https://www.sciencedirect.com/journal/economics-and-human-biology/vol/5/issue/2, doi:10.1016/j.ehb.2007.02.002.
  * [21] Diane F. Halpern, Camilla P. Benbow, David C. Geary, Ruben C. Gur, Janet S. Hyde, and Morton Ann Gernsbacher.  The science of sex differences in science and mathematics.  Psychological Science in the Public Interest, 8(1):1–51, 2007.  doi:10.1111/j.1529-1006.2007.00032.x.
  * [22] Larry V. Hedges and Amy Nowell.  Differences in mental test scores, variability, and numbers of high-scoring individuals.  Science, New Series, 269(5220):41–45, 1995.  doi:10.1126/science.7604277.
  * [23] Theodore P. Hill.  An evolutionary theory for the variability hypothesis, 2021.  Preprint, arXiv:1703.04184v4 [q-bio.PE].  URL: https://arxiv.org/abs/1703.04184.
  * [24] Wendy Johnson, Andrew Carothers, and Ian Deary.  Sex differences in variability in general intelligence: A new look at the old question.  Perspectives on Psychological Science, 3(6):518–531, 2008.  doi:10.1111/j.1745-6924.2008.00096.x.
  * [25] Jonathan M. Kane and Janet E. Mertz.  Debunking myths about gender and mathematics performance.  Notices of the AMS, 59(1):10–21, 2012.  URL: https://advance.uncc.edu/sites/advance.uncc.edu/files/media/debunking-myths-about-gender-and-math.pdf, doi:10.1090/noti790.
  * [26] Rachel Kaufman.  It doesn’t add up.  Science, December 16 2011.  URL: https://www.sciencemag.org/careers/2011/12/it-doesnt-add, doi:doi:10.1126/science.caredit.a1100139.
  * [27] Jennifer L. Kobrin, Viji Sathy, and Emily J. Shaw.  A historical view of subgroup performance differences on the SAT Reasoning Test.  Technical Report Research Report No. 2006-5, College Board, 2006.  URL: https://files.eric.ed.gov/fulltext/ED562569.pdf.
  * [28] Anne-Catherine Lehre, Knut Lehre, Petter Laake, and Niels Danbolt.  Greater intrasex phenotype variability in males than in females is a fundamental aspect of the gender differences in humans.  Developmental Psychobiology, 51:198–206, 2009.  doi:10.1002/dev.20358.
  * [29] David F. Lohman and Joni M. Lakin.  Consistencies in sex differences on the Cognitive Abilities Test across countries, grades, test forms, and cohorts.  British Journal of Educational Psychology, 79:389–407, 2009.  doi:10.1348/000709908X354609.
  * [30] David Lubinski and Lloyd G. Humphreys.  A broadly based analysis of mathematical giftedness.  Intelligence, 14:327–355, 1990.  URL: https://www.sciencedirect.com/science/article/abs/pii/016028969090022L, doi:10.1016/0160-2896(90)90022-L.
  * [31] Guillermo Mendez and Sharon Lohr.  Estimating residual variance in random forest regression.  Computational Statistics & Data Analysis, 55(11):2937–2950, 2011\.  URL: http://www.sciencedirect.com/science/article/pii/S0167947311001514, doi:10.1016/j.csda.2011.04.022.
  * [32] Valentina Moskvina, Michael Smith, Dobril Ivanov, Douglas Blackwood, David StClair, Christina Hultman, Draga Toncheva, Michael Gill, Aiden Corvin, Colm O’Dushlaine, Derek W. Morris, Naomi R. Wray, Patrick Sullivan, Carlos Pato, Michele T. Pato, Pamela Sklar, Shaun Purcell, Peter Holmans, Michael C. O Donovan, Michael J. Owen, George Kirov, and International Schizophrenia Consortium.  Genetic differences between five European populations.  Human Heredity, 70:141–149, 2010.  doi:10.1159/000313854.
  * [33] Max Roser, Cameron Appel, and Hannah Ritchie.  Human height.  https://ourworldindata.org/human-height, 2019.  Last accessed January 22, 2021.
  * [34] Stephanie A. Shields.  The variability hypothesis: The history of a biological model of sex differences in intelligence.  Signs, 7(4):769–797, 1982.  URL: http://www.jstor.org/stable/3173639.
  * [35] Karl J. Skårbrevik.  Gender differences among students found eligible for special education.  European Journal of Special Needs Education, 17(2):97–107, 2002\.  doi:10.1080/08856250210129038a.
  * [36] Steve Stewart-Williams and Lewis Halsey.  Men, women, and science: Why the differences and what should be done?  European Journal of Psychology, 35(1):3–39, 2021.  URL: https://journals.sagepub.com/doi/10.1177/0890207020962326?icid=int.sj-abstract.similar-articles.1, doi:10.1177/0890207020962326.
  * [37] Steve Strand, Ian J. Deary, and Pauline Smith.  Sex differences in Cognitive Abilities Test scores: A UK national picture.  British Journal of Educational Psychology, 76:463–480, 2006.  doi:10.1348/000709905X50906.
  * [38] Lawrence Summers.  Remarks at NBER Conference on Diversifying the Science & Engineering Workforce, Cambridge, MA, January 14, 2005.  http://web.archive.org/web/20080130023006/http://www.president.harvard.edu/speeches/2005/nber.html, 2005\.  Accessed Nov 13, 2017.
  * [39] Lara M. Wierenga, Gaelle E. Doucet, Danai Dima, Ingrid Agartz, Moji Aghajani, Thteophilus N. Akudjedu, Anton Albajes-Eizagirre, Dag Alnas, Kathryn Alpert, Ole A. Andreassen, Alan Anticevic, Philip Asherson, Tobias Banaschewski, Nuria Bargallo, Sarah Baumeister, Ramona Baur-Streubel, Alessandro Bertolino, Aurora Bonvino, Dorret Boomsma, Stefan Borgwardt, Josiane Bourque, Anouk den Braber, Daniel Brandeis, Alan Breier, Henry Brodaty, Rachel M. Brouwer, Jan K. Buitelaar, Geraldo F. Busatto, Vince D. Calhoun, Erick J. Canales-Rodr guez, Dara Cannon, Xavier Caseras, Francisco X. Castellanos, Tiffany M. Chaim-Avancini, Christopher R. K. Ching, Vincent P. Clark, Patricia J. Conrod, Annette Conzelmann, Fabrice Crivello, Christopher G. Davey, Erin W. Dickie, Stefan Ehrlich, Dennis van t Ent, Simon E. Fisher, Jean-Paul Fouche, Barbara Franke, Paola Fuentes-Claramonte, Eco J. C. de Geus, Annabella Di Giorgio, David C. Glahn, Ian H. Gotlib, Hans J. Grabe, Oliver Gruber, Patricia Gruner, Raquel E. Gur, Ruben C. Gur, Tiril P. Gurholt, Lieuwe de Haan, Beathe Haatveit, Ben J. Harrison, Catharina A. Hartman, Sean N. Hatton, Dirk J. Heslenfeld, Odile A. van den Heuvel, Ian B. Hickie, Pieter J. Hoekstra, Sarah Hohmann, Avram J. Holmes, Martine Hoogman, Norbert Hosten, Fleur M. Howells, Hilleke E. Hulshoff Pol, Chaim Huyser, Neda Jahanshad, Anthony C. James, Jiyang Jiang, Erik G. Jönsson, John A .Joska, Andrew J. Kalnin, Karolinska Schizophrenia Project (KaSP) Consortium, Marieke Klein, Laura Koenders, Knut K. Kolskår, Bernd Krämer, Jonna Kuntsi, Jim Lagopoulos, Luisa Lazaro, Irina S. Lebedeva, Phil H. Lee, Christine Lochner, Marise W. J. Machielsen, Sophie Maingault, Nicholas G. Martin, Ignacio Martínez-Zalacaín, David Mataix-Cols, Bernard Mazoyer, Brenna C. McDonald, Colm McDonald, Andrew M. McIntosh, Katie L. McMahon, Genevieve McPhilemy, Dennis van der Meer, Jos M. Menchön, Jilly Naaijen, Lars Nyberg, Jaap Oosterlaan, Yannis Paloyelis, Paul Pauli, Giulio Pergola, Edith Pomarol-Clotet, Maria J. Portella, Joaquim Radua, Andreas Reif, Geneviève Richard, Joshua L. Roffman, Pedro G. P. Rosa, Matthew D. Sacchet, Perminder S. Sachdev, Raymond Salvador, Salvador Sarró, Theodore D. Satterthwaite, Andrew J. Saykin, Mauricio H. Serpa, Kang Sim, Andrew Simmons, Jordan W. Smoller, Iris E. Sommer, Carles Soriano-Mas, Dan J. Stein, Lachlan T. Strike, Philip R. Szeszko, Henk S. Temmingh, Sophia Thomopoulos, Alexander S. Tomyshev, Julian N. Trollor, Anne Uhlmann, Ilya M. Veer, Dick J. Veltman, Aristotle Voineskos, Henry Völzke, Henrik Walter, Lei Wang, Yang Wang, Bernd Weber, Wei Wen, John D. West, Lars T. Westlye, Heather C. Whalley, Steven C. R. Williams, Katharina Wittfeld, Daniel H. Wolf, Margaret J. Wright, Yuliya N. Yoncheva, Marcus V. Zanetti, Georg C. Ziegler, Greig I. de Zubicaray, Paul M. Thompson, Eveline A. Crone, Sophia Frangou, and Christian K. Tamnes.  Greater male than female variability in regional brain structure across the lifespan.  Human Brain Mapping, pages 1–30, 2020.  doi:10.1002/hbm.25204.