## Eye Color and Math Ability

When you talk over the phone with an adult stranger, you can generally determine if this person is male or female. From this, I conclude that the voice characteristics are often noticeably different for males and females. There are many other characteristics that have a different distribution by gender — for example, height.

My question is: “Can we find a trait such that the distributions are the same for both genders?”

Trying to find the answer, I remembered what we learned in high school about the genetics of eye color. I checked the Internet on the subject and discovered that the story is somewhat more complicated than what I studied 30 years ago, but still we can say that eye color is defined by several genes, which are located on non-sex chromosomes. That means, your eye color depends on the genes your parents have and doesn’t depend on your sex. A boy and a girl from the same parents have the same chances for any particular eye color.

Since eye coloring has nothing to do with gender, women and men are equal in the eyes of eye colors.

Does that mean that if we check the distribution of eye color for the world population, the distribution histogram will be the same for men and women? That sounds like a logical conclusion, right? I would argue that this is not necessarily the case.

Let me remind you that the distribution of eye color depends on the country. China has an unprecedented gender imbalance, with 6% more men than women in its population. As the eye color of Chinese people is mostly dark brown, this creates an extra pool for a randomly chosen man in the world to have a darker eye color than for a randomly chosen woman. If we exclude China from consideration, we can still have different distributions. For example, in Russia the life expectancy for women is 15 years longer than the life expectancy for men. Consequently, Russia has 14% less men than women, while globally the male/female sex ratio is 1.01. Therefore, eye colors common in Russia will contribute to female eye colors more than those of male.

What if we consider only one country? Let us look at the US. Immigrants to the US are mostly males. If the distribution of eye color for immigrants is different than the distribution for non-immigrants, then male immigrants contribute more to the eye color distribution than female immigrants.

There are so many factors impacting eye color distribution, that it isn’t clear whether it’s possible to find a group of people other than siblings in which the distribution of eye color would be the same for women and men.

We see that eye color distribution, which theoretically doesn’t depend on gender, when measured in a large population can produce different distributions for men and women.

Recently I wrote a theoretical essay titled “Math Career Predictor”, where I assumed that the distribution of math ability is different for men and women. In reality, there is no good way to measure math ability, hence we do not have enough data to draw a complete picture. For the purposes of this discussion let us assume that we can measure the math ability and that Nature is fair and gave girls and boys the same math ability. My example with eye color shows that if we start measuring we might still see different distributions in math ability in boys and girls.

My conclusion is that if we measure some ability and the distribution is different for boys and girls, or for any other groups for that matter, we can’t just conclude that boys and girls are different in that ability. For some distributions, like voice, we probably can prove that the difference is significant, but for other characteristics, different distribution graphs are not enough; we need to understand the bigger picture before drawing conclusions.

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1. #### Misha:

Some related food for thought

2. #### TruePath:

Interesting point but I think one can go much further in deconstructing how silly and totally unrelated to actual facts this whole discussion has become.

First of all I’d just like to point out that there is no reason that anyone, especially mathematicians should care about the distribution of abilities/accomplishment by gender (or race or anything else) except maybe as a vague PR concern. Certainly they shouldn’t think it reflects on their ability in any fashion. I mean I’m a member of the set of all people who really suck at math + me but that doesn’t give me reason to believe I’m bad at math.

What presumably matters is how conditioning on one’s gender changes your expectation of your mathematical accomplishment. Now if one knew nothing except one’s gender finding out that there was some correlation between gender and mathematical accomplishment. However, anyone who has made it to math grad school (or even taken a test in elementary school) has so much information about their own ability I doubt conditioning on some vague weak correlation should ever substantially adjust that expectation. Moreover, there is no reason to think that conditioning on gender has to go the same way for math grad students as it does for the population at large. It’s totally possible that P( x has high math ‘ability’ | x is a woman ) P ( x has high math ‘ability’ | x is in math grad school). In fact so long as you think that women have less encouragement to do mathematics or tend to have fewer role models you should probably think that conditioning on being female increases the expectation of ‘ability’ given that one has made it to grad school regardless of what is true for the population at large.

It’s simply a mathematical fallacy to infer something about female grad students or other distinguished sub-population based on any statistical claim about women at large. Since any given person knows a great deal about their own ability by the time they pass third grade any general result really shouldn’t substantially affect any person’s self-judgment no matter what it says about gender and math.

Second of all there isn’t even really a meaningful notion of ‘innate math ability.’ Certainly there may be some innate concept of intelligence/brightness but that is only very indirectly related to revealed math ability. There are many people who are quite smart, most are very bad at mathematics because they simply aren’t interested or don’t want to bother to do the work to learn mathematics.

Does being less interested in math count as having less math ability? If not then you can conclude almost nothing about ability from any test or survey of accomplishments. Math ability would become a notion even more ethereal than g (factor supposedly explaning correlations between IQ like tests) with no way to prove the concept even is well defined much less measure it. On the other hand if you want to say that being less interested in mathematics counts as having less mathematical ability the notion ceases to resemble our intuitive concept at all.

After all if I’m better at other subjects/pursuits I will be less interested in doing math. Taking the same person and making them a better artists would reduce their mathematical ability even if we changed nothing else about them. Hell, giving someone better social skills would increase the relative attractiveness of spending time with friends rather than hiding and reading your math book and thus would ‘reduce’ your math ability. On this definition not only is it unclear if having more/less mathematical ability is a good thing for a group (depends on the reasons) but person A might have more mathematical ability than person B even by dint of the fact that person A always does better than B at everything they have equal practice with (thus giving A more options that draw them away from math).

Moreover, this hints at the broader incoherence of most of the nature/nurture debate that assumes that there is a definite answer and that answer maps onto our intuitive idea of ability. Many possible explanations don’t even make sense in these terms. For instance, there are studies showing that girls tend to experience more peer pressure from their female classmates in their pre-college years both to do well in school and to please teachers while boys experience some pressure along the opposite lines (not to look like they are too studious or the teachers pet). Given the way math is taught in pre-college courses I wouldn’t be surprised if listening too much to the pre-college teachers isn’t a serious hinderance to entering real mathematics while kids who ignore the teacher and try to figure it out on their own have an advantage. Now suppose (totally hypothetically) that it turns out this greater social pressure to be studious has some biological basis. Does that mean girls are innately worse at math? The question doesn’t even really make sense on this explanation.

I don’t offer the above explanation because I think it’s particularly probable (one possibility amoung a multitude) but only to point out that there are many explanations on which the whole discussion simply doesn’t turn out to really make sense.

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Anyway I thought you might be interested in these points since you seem to be interested in acutally looking at this from a logical/fact based point of view. However, as I’ve discovered making these kind of logically based arguments doesn’t get through to even most mathematicians on the subject because people don’t parse these claims as factive claims that can be right or wrong. Rather most people parse statements about statistical differences the same way they would parse statistical statements comparing their sports team and it’s rival.

I mean if one was to say, “My team averages twice as many points as yours does each game,” people interpret that as a coded way of saying “my team rocks, yours sucks,” even though it’s totally compatible with my team just focusing on defense. That seems to be more or less how these claims and discussions are interpreted with people so talking about the literal meaning of the claims and what consequences those might have doesn’t really penetrate.

3. #### esmeyny:

“Since any given person knows a great deal about their own ability by the time they pass third grade any general result really shouldn’t substantially affect any person’s self-judgment no matter what it says about gender and math.”

Why are you talking about “shouldn’t” when simple studies shows that common prejudices (i.e. blanket judgements based on right or wrong statistics) *do* influence results?

My favourite study is a math test with Asian-American female students. Half of them were subtly reminded before the test that they were Asian (and thus good at math) and half of them were reminded before the test that they were female. The difference in results of the two groups was significant.

4. #### Christ Schlacta:

Consider the eye colours of individuals based on aggregate births rather than existing populace. Many people’s eye colours change by the end of puberty, so whenever applicable, consider the final eye colour instead of initial colour. Weight the total in such a way that country specific gender differences don’t affect the global eye colour averages.