User:Gmaxwell/mf compsci

It has been asserted that the small gender bias in skills contributing to success cannot account for the non-trivial gender bias in computer science. While I think that raw math skills are not too essential for computer science as it is currently taught (as an applied-java-webtoys trade school all too frequently), I don't think it is at all controversial that computer science requires interests, aptitudes, and opportunities beyond the norm in several areas. If this were not true, we would expect computer scientists to be more common in the general population. It is not unreasonable to assume that the underlying interests, aptitudes, and opportunities have small gender biases outside of the control of the practitioners in a field and that, even by chance, in some fields these biases tend to favor one gender or another.

Here I will show that even very small gender biases in the preconditions for membership in a population can result in significant gender imbalances in that population.

Difference in mean
Let us assume 1)that a particular aptitude above average is required in each of some set of skills, 2) that the skills are uncorrelated, and 3) that in each of these areas of skill, men have a mean performance greater than that of women by some small multiple of the standard deviation.

Assuming a normal distribution of skill, the probability of having a skill over some cutoff can be found by integrating $$\tfrac{1}{\sqrt{2\pi}}\, e^{-\,x^2\!/2}$$ over x from the cutoff to infinity. A difference in performance as a function of standard deviation is equivalent to offsetting the cutoff.

The number of women with the skill would be $${{{\it p_{women}}^}\over{{\it p_{women}}^+ {\it p_{men}}^}}$$

Combining these we find a function on s, the number of contributing skills, c, the cutoff as a multiple of the standard deviation in male performance, and d, the difference in average performance between men and women as a multiple of the standard deviation:

$$ {\it P_{women}}={{\left({{\sqrt{\pi}}\over{\sqrt{2}}}-{{\sqrt{2}\, \sqrt{\pi}\,\mathrm{erf}\left({{\sqrt{2}\,d+\sqrt{2}\,c}\over{2}} \right)}\over{2}}\right)^{s}}\over{\pi^\,\left({{ \left({{\sqrt{\pi}}\over{\sqrt{2}}}-{{\sqrt{2}\,\sqrt{\pi}\, \mathrm{erf}\left({{\sqrt{2}\,d+\sqrt{2}\,c}\over{2}}\right)}\over{2 }}\right)^{s}}\over{\pi^}}+{{\left({{\sqrt{\pi} }\over{\sqrt{2}}}-{{\sqrt{2}\,\sqrt{\pi}\,\mathrm{erf}\left({{\sqrt{2 }\,c}\over{2}}\right)}\over{2}}\right)^{s}}\over{\pi^ }}\right)}} $$

Solving this for d=0.15 (the number found by Hyde, J; Fennema, E; Lamon, S. “Gender differences in mathematics performance: A meta-analysis.” Psychological Bulletin. Vol 107(2), Mar 1990, 139-155 for the overall average female disadvantage in mathematics performance), s∈[1,2,3,4,5,6], c∈[0.5,1,1.5,2,2.5] (top 31%, 16%, 7%, 2%, and 0.6% respectively) we find:

Expected percentage of women in a field given s required skills with an aptitude of at least +c standard deviations in each assuming that women under-perform men by 0.15sd on average in these skills So we find that significant gender imbalances in a population may naturally arise from very small differences in average skill levels, when that population is being selected from individuals with atypically high skill levels, and especially when several skills with a small gender bias are all required.

This result is intuitive when you can consider that the rate of diminishment of the normal curve increases as you move further away from the mean and that the probability of having all of a set of skills that are often unrelated is quite unlikely (i.e. 50%^4=6%).

In order to completely explain the uncommonness of women in computer science, it would require three unrelated slightly male-biased skills which are found in only 7% of the male population (c=1.5; s=3). I won't attempt to speculate what these skills might be, but considering the fairly low rate of computer scientists in the worldwide male population, this possible explanation doesn't sound completely unreasonable to me.

If we were to accept the premise that math performance in the top percentile were require for participation in computer science then the bias of 0.15sd explains about half of the observed gender gap.

Difference in deviation
While not mentioned in the cited presentation, it has been argued that the gender imbalance in some areas is due not to a difference in the average performance between men and women but instead due to a difference in the variance: If men have a greater variance in some skill, then we would expect significant gender imbalances in groups which consist of only top performers (i.e. graduate programs) as well as in groups which consist of only bottom performers. This is possible even when, on average, men and women perform the same due to the cut-off and the slope of the derivative of the gaussian curve.

The effect of a change in the standard deviation of the distribution can be calculated like the above, but the difference has a multiplicative effect on the the cutoff rather than an additive one, so the effect is much stronger. I do not have a readily available citation for the typical differences in variance between men and women in the general population on intellectual tasks, so I'll chart with several possible differences instead of different numbers of contributing skills.

Although this only considers the case of one mandatory skill, if multiple skills are required they would have the same impressive effect; likewise, this effect is additive with any difference in the average performance. My browser doesn't seem to be able to render tables in 4-dimensions.

Expected percentage of women in a field assuming the members are required to have performance in some skill at c standard deviations over the norm or better and that women's performance at the skill is on average the same as men but with a lower standard deviation by the specified factor

In Deary, I.J.; Irwing, P.; Der, G; Bates, T.C. (2005). "Brother–sister differences in the g factor in intelligence: Analysis of full, opposite-sex siblings from the NLSY1979". Intelligence 35:451-456 the ratio of the standard deviation of male scores to that of female scores on the armed forces qualification test was 1.11, and was as high as 1.41 and 1.38 on the Mechanical Comprehension and Electronics Information sections of the ASVAB (Armed Services Vocational Aptitude Battery).

Although I believe the skills gap is likely just one factor out of many, these calculations demonstrate that the measured gender-dependence in relevant skills may, in fact, be a significant contributor to the gender gap in computer science—or, at the very least, the use of the overlapped distribution graphs in the presentation was a mathematically unsound argument, as it actually supports the hypothesis that a skills gap explains a non-trivial part of the gender gap.

--Gmaxwell (talk) 03:26, 18 October 2009 (UTC)