Sex disparities at the margin?

There was an interesting question on the xkcd forums asking why there's such a preponderance of men over women in the comedy industry. Standup comedy seemed to be the focus as the most male dominated part of the field, and it got me thinking:

Are preference differences exaggerated at the margin? If 80% of office workers are men, that's one thing since office work is a very common career. But if 80% of a very small career field like stand up comedians are men, does that simple statistic really tell you much about whether or not sexism or acculturation is the cause of the disparity?

I think small absolute differences in preferences might lead to a large disparity in outcomes at the margin, where there are both few jobs to go around and a small proportion of the population searching for them.



For example, say you have a population of 100,000 in which 50% are male. 2% of the male population and 1% of the female population are interested in pursuing a career as stand up comedians but there are only 10 positions available. What is the likelihood a fair lottery would assign a large majority, say 8 or more, of the jobs to men?

(I admit, I had to look up how to do the math. This page was quite handy, and so was Excel >.>  )

Population:  100,000
Male:  50,000
Female:  50,000
Male interest rate:  0.02
Female interest rate:  0.01
Males interested:  1000
Females interested:  500
Total interested:  1500
Jobs available:  10

P(10m,0f)  0.017,  Cumulative  0.017
P(9m,1f)   0.086,  0.103
P(8m,2f)   0.195,  0.298
P(7m,3f)   0.261,  0.559
P(6m,4f)   0.228,  0.788
P(5m,5f)   0.137,  0.924
P(4m,6f)   0.057,  0.981
P(3m,7f)   0.016,  0.997
P(2m,8f)   0.003,  0.9997
P(1m,2f)   0.0003,  0.99998
P(0m,10f),  0.00016,  1

So, there's a 30% chance of a fair, random lottery assigning 8 or more men jobs in  comedy even though men only compose 2/3 of the population that is seeking those jobs. And, since only 1.5% of the population is interested in the jobs, a 1% higher rate of interest in men leads to 2/3 of the candidates being male.

If a larger proportion of the population is interested with the same absolute difference in male and female preferences, say 20% of men and 19% of women, you only end up with 6.4% likelihood of ending up with 8 or more positions going to men. (If 20% of men and 10% of women are looking for comedy jobs you get the same distribution as 2% and 1%.)

Also, though I'm not sure how to do the math without brute forcing it, I think there's a low likelihood of having men get 80% or more of the jobs if there are 1000 jobs available for a group of 1000 men and 500 women.

So, you can have a large gender disparity in outcome with a balanced general population if there are a small number of jobs available, only a small fraction of the population is interested in the jobs, and there is a small absolute difference in the number of men interested compared to women. Exogenous factors like sexism and socialization aren't necessarily implicated.

Now, the question is, which is more important on the margin: The difference in preferences relative to population as a whole, the difference relative their respective gender, or the difference relative those with the same preferences. In a population of 100,000 people, if 1000 men and 500 women want to become comedians, are men 1% more likely to want to become comedians or are they 100% more likely?

I think there is a case for treating this as men being 1% more likely, they have a small difference in preference compared to women. This is an important distinction because it makes the case of socialization, "male" verses "female" occupations, much weaker. Yes, there are twice as many men seeking this occupation, but in society as a whole there are very, very few people of either gender seeking this job. So, if there aren't many people seeking this career, isn't it likely that there isn't a strong or at least dominant attitude that this position is reserved for one gender? Perhaps the general consensus is that this job isn't fit for anybody, socialization in the case of marginal  occupations might be focused more along class or race lines instead of gender.

Though this isn't a terribly compelling argument, either. There certainly are social norms regarding at least some marginal positions, the presidency is an example where the deck is seriously stacked in favor of a male WASP and there is only one at a time. But is that the case for every marginal position?

I think it's much more reasonable, and a stronger argument, to say that when a certain preference is rare, small disparities between genders can lead to large differences in outcomes. If men are only 1% more likely to want to be comedians than women, but only 1.5% of the population wants to become a comedian, that can lead to a large disparity. However, if men are 1% more likely to want to be comedians but 19.5% of the population wants to become a comedian, the results are much more likely to be balanced. An absolute difference of 1% in a preference seems plausible without external influences like sexism, socialization or acculturation. Men and women are similar in many ways, but they're also different in some ways.

If you observe large gender (or race, etc) disparities in occupations that have large numbers of openings and large interest from the general population it's quite reasonable to look seriously at sexism or socialization. But when the number of positions are small and general interest is low, discrimination might not be the primary culprit and other factors should probably warrant serious consideration.