The math sex gap revisited: a theory of everyoneShe or more likely he settles some questions raised by two recent papers about the gap that were written in La La Land (that's a meta-country where egalitarianism is a mandatory axiom of all their research).

The first paper that La Griffe du Lion clarifies was written by veteran feminist author

Janet Hyde (click)and we have already discusses it: she claimed that the math gap between the sexes has evaporated. Lion's conclusions are pretty much identical to ours when he or she writes that

- she uses tests with small children for which the gap has not yet fully blossomed
- the tests are too simple and only measure the ability to understand basic things
- she only talks about the mean differences, not the ratios of the variances; the latter are more important in highly selective contexts.

**Emancipation vs math sex gap**

However, La Griffe du Lion also talks about another paper written by four economists led by Paola Sapienza,

Culture, gender, math (click, PDF).On their visit to La La Land, the authors argued that the math sex gap decreases in countries where women are more emancipated. La Griffe du Lion confirms that the correlation exists but he also proves that the origin of the correlation is completely different than the four authors suggested. But he shows much more than that.

He provides us with the evidence that the mean difference and the ratio of variances between the male and female distributions of math talent is universal for all nations in the world. The first graph that I will reproduce shows the percentage of boys and girls in different countries who were able to pass PISA tests at level 5 in 2003 and 2006.

This personal achievement pretty much means that the people are above the 85th percentile of the general population, i.e. that they're more skillful in math than at least 85% of the population. Here is the graph:

Click or shift-click to zoom in. Note that in 2006, the Czech Republic became the country that deviates from the interpolating blue line in the upward directions most intensely among all countries. Three years ago, we had the smartest girls, if you wish ;-), but their percentage at level 5 was still below the boys' percentage.

Click or shift-click to zoom in. Note that in 2006, the Czech Republic became the country that deviates from the interpolating blue line in the upward directions most intensely among all countries. Three years ago, we had the smartest girls, if you wish ;-), but their percentage at level 5 was still below the boys' percentage.

In 2006, the Czech girls were relatively the smartest ones in the world despite (or partly because?) the fact that Czechia's politically correct "GGI" index of emancipation, 0.6712, was the second lowest one after Japan ("second most discriminatory") among all the 12 best-scoring countries (where Czechia and Japan belong, together with Germany, Austria, Australia, New Zealand, Switzerland, Canada, Netherlands, Finland, Belgium, and Korea).

You should notice that all the points are remarkably close to an idealized blue line and this blue line surely doesn't say that the percentages of boys and girls are equal. There exists a two-parameter family of such lines: the shape of the line only depends on the mean difference between the male and female gaussians, expressed in standard deviations; and the ratio of the standard deviations (or variances).

All the countries, regardless of their race, culture, or religion, seem to confirm that the ratio of standard deviations is about 1.15 (boys' distribution is wider) while the mean difference between boys and girls is about 2-3 points (if you normalize boys' abilities like IQ with the average of 100 and the standard deviation of 15). These figures haven't changed for 50 years.

You should notice that all the points are remarkably close to an idealized blue line and this blue line surely doesn't say that the percentages of boys and girls are equal. There exists a two-parameter family of such lines: the shape of the line only depends on the mean difference between the male and female gaussians, expressed in standard deviations; and the ratio of the standard deviations (or variances).

All the countries, regardless of their race, culture, or religion, seem to confirm that the ratio of standard deviations is about 1.15 (boys' distribution is wider) while the mean difference between boys and girls is about 2-3 points (if you normalize boys' abilities like IQ with the average of 100 and the standard deviation of 15). These figures haven't changed for 50 years.

I would like to see the African nations on the graph to check that they are close enough to the curve as well because I have certain doubts that the same math sex gap exists in those nations. But these nations will probably appear near the left lower corner of the graph where it is harder to measure the parameters of the Gaussian...

Now, the appearance of all the nations near the blue line indicates that the differences between the male and female distributions (both mean differences and variance ratios) are universal across the world. If it is so, the deviations from the line should be "noise". Alternatively, you may believe that the ratios of girls and boys depend on the "culture" and "emancipation of women".

These two hypotheses predict a very different character of changes in time. If the boys-girls gap is caused by the local culture or religious traditions, you would predict that between 2003 and 2006, the gap should remain pretty much unchanged. On the other hand, the hypothesis that the deviations from the blue line are just noise predicts that if you draw the change of the male/female percentages between 2003 and 2006 into the x-y plane, they will be chaotically distributed. Here is the graph:

Well, it's clearly noise. Nature wins, nurture loses.

Does La Griffe du Lion show that the correlation found by the four economists (between "emancipation" and the "shrinking male-female gap") doesn't exist? Not at all. He or she confirms that it does exist. But correlation is not causation. More precisely, both of these quantities are correlated not because one of them is the cause but because both of them are effects of something else, namely the general IQ of the nation.

Smarter nations have a higher percentage of men and women whose IQ exceeds 95 (or 100 or 120) while the dumber nations have to probe the "extremely smart tail of the Gaussian" if they want to get to these high IQ values (that are necessary for many purposes in the society). Consequently, the dumber nations will lead to a lower emancipation rate because men's skills will be needed and men dominate in the upper portions of the distribution.

So a higher general IQ of a nation makes the society more emancipated; it makes the male-female gap at any required fixed IQ threshold shrink (because the required IQ is not that selective in a smart nation, and the tails of the Gaussian where the differences matter don't have to be probed). The previous sentence can be tested by drawing the right graph:

Well, it's clearly noise. Nature wins, nurture loses.

Does La Griffe du Lion show that the correlation found by the four economists (between "emancipation" and the "shrinking male-female gap") doesn't exist? Not at all. He or she confirms that it does exist. But correlation is not causation. More precisely, both of these quantities are correlated not because one of them is the cause but because both of them are effects of something else, namely the general IQ of the nation.

Smarter nations have a higher percentage of men and women whose IQ exceeds 95 (or 100 or 120) while the dumber nations have to probe the "extremely smart tail of the Gaussian" if they want to get to these high IQ values (that are necessary for many purposes in the society). Consequently, the dumber nations will lead to a lower emancipation rate because men's skills will be needed and men dominate in the upper portions of the distribution.

So a higher general IQ of a nation makes the society more emancipated; it makes the male-female gap at any required fixed IQ threshold shrink (because the required IQ is not that selective in a smart nation, and the tails of the Gaussian where the differences matter don't have to be probed). The previous sentence can be tested by drawing the right graph:

The correlation is clearly there.

That's also the explanation of the correlation found by the four economists. Finally, La Griffe du Lion is able to isolate the contributions of nature and nurture to the particular math sex gap as measured by PISA at level 5 in 2003 and 2006. What he gets is the following:

To find this graph, La Griffe du Lion assumed that the female ratio is equal to the "natural" component plus the "cultural" component that is Taylor-expanded as a function of "discrimination against women", namely (1-GGI) where GGI is an international cultural coefficient between 0 (macho society) and 1 (full emancipation). A least squares fit was performed to find the coefficients.

I think that La Griffe du Lion made a mistake: only the portion of the graph above the "female=male" line should be drawn in blue because for "female=male", there would be no differences, neither natural nor cultural. But even if you fix this bug, the contribution of culture and nurture will be negligible.

And that's the memo.

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