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LostInParadise's avatar

Can you figure out this apparent paradox?

Asked by LostInParadise (27918points) August 10th, 2015

There was a mathematically interesting lawsuit against the University of California Berkeley graduate school claiming that men were consistently accepted at a higher rate than women. The university looked at the acceptance rates for each department and found that at the department level women were accepted at a slightly higher rate than men. How is this possible? To simplify things suppose there are just two departments with equal numbers of total applicants. Can you come up with numbers that have women at each department being accepted at a higher rate than men but overall men are accepted at a higher rate?

This is an example of what is referred to as Simpson’s paradox. It is not really a paradox and requires no advanced math to see how it can happen.

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11 Answers

ragingloli's avatar

I would need to see the actual numbers and what exactly was being counted.
Do they compare the absolute numbers of women vs men being accepted?
Or do they compare it to how many men and women applied relative to each other.

So, as an example, let’s say that there were 100 applicants each, and in both cases, 20 women were accepted, vs 15 men, with the rest being rejected. In that case, women are more likely to be accepted.
But let us say that of these 100 applicants, there only were 15 men to begin with.
In that case, 100% of male applicants are accepted, vs only 23% of women.

zenvelo's avatar

Not at all difficult when one considers that term rate is ambiguous unless defined.

If a department has 200 applicants and accepts 100, it may get 40 applications from women, and accept 100%, or 40. But it is accepting 60 men, which is an acceptance rate of 37.5% of male applications, but an acceptance rate of 30% of all applications, as opposed to women who are only accepted at a rate of 20% of all applications.

It is how the issue is defined.

Adirondackwannabe's avatar

You can sue over anything, and figures don’t lie, but liars can figure.

LostInParadise's avatar

For each department, acceptance rate of women = women applicants / women acceptances and men acceptance rate = men applicants / men acceptance. Overall women acceptance rate = total women applicants / total women acceptances and the same for men.

What you have to consider is that both the relative number of male and female applicants at the two departments is different as well as the two acceptance rates.

jaytkay's avatar

Another scenario.

—University has 10 departments
—8 very small departments accept women more often
—2 very large departments accept men more often

Zaku's avatar

@LostInParadise In that case, for example:

Art Department:
120 women apply, 80 accepted, acceptance rate = 67%
40 men apply, 26 accepted, acceptance rate = 65%

Math Department:
60 women apply, 52 accepted, acceptance rate = 87%
200 men apply, 170 accepted, acceptance rate = 85%

Overall acceptance rate for women = 132/180 = 82%
Overall acceptance rate for men = 196/240 = 70%

In other words, when more women than men apply to departments with overall lower acceptance rates, women’s overall acceptance rate will be lower than it otherwise would be, because more men applied to departments with higher acceptance rates.

Zaku's avatar

Oh whoops I wrote the percentages for the overall figures wrong! It’s:

Overall acceptance rate for women = 132/180 = 73%
Overall acceptance rate for men = 196/240 = 82%

LostInParadise's avatar

That is correct with the correct interpretation. The acceptance rate for the women was lower because they tended to apply to the departments with lower overall acceptance rates.

There is a really neat biological theory of altruism based on Simpson’s paradox. Biologists have tried to explain in evolutionary terms why a member of a species will act altruistically, expending energy for the benefit of the group. The selfish gene theory of evolution appears to contradict such behavior.

There are two leading theories of altruism. One is kin selection. It says that organisms are wired to help relatives because relatives have many of the same genes. The other theory is much more intriguing. It says that selection occurs on a group level. The group with more altruists will prevail over a group with fewer altruists.

You have to be careful when talking about group level selection. Groups do not reproduce. Individuals do. Suppose that for both groups, the non-altruists have a higher survival rate, but in the group with more altruists the survival rate for everybody is higher than in the other group due to the benefits of having lots of altruists. This in analogous to the Berkeley problem. Just like the men in the problem, the altruists do worse in both groups but overall they do better.

Zaku's avatar

Well, and if your groups are men and women, they aren’t really in competition with each other for survival at the level of college admissions, so I wouldn’t think that’d be relevant to any kind of survival calculation, even if that were what it were about.

I don’t see how the “selfish gene” idea makes much if any real sense. Society and families and friends are all clearly much more effective, constructive and helpful ways to behave than being selfish, at least in most long-term situations. Overall, being helpful to others tends to be helpful to oneself anyway, because people like you back. Often being selfish in the short term backfires in the long term.

But getting away from over-stated survival-oriented theories of causation and human nature, it seems to me that there is a basic human nature to be friendly and help others when their own needs are met, and somewhat more self-oriented when there’s a distress. But more than that, I think the cause of negative, hostile and self-centered behavior tends to be abuse – practically every jerkish behavior, and every narcissist and sociopath and so on that I know enough about to say, seems to have a root cause of having been neglected or abused or otherwise messed with, and so developed an orientation based on that. People in great distress go into survival mode and end up with stuck emotions that mess them up until they get healing, and getting healing requires getting attention for upsets that were covered up in order to survive the trauma in the first place. To get attention, people create drama that somehow is like the original trauma, without realizing what they’re doing (unless they’ve sufficiently figured themselves out). None of that really has a whole lot to do with a strategy for optimum gene propagation, at least not in a direct way. The genetic/evolutionary theories seldom ring true for me at the level of how people actually are.

LostInParadise's avatar

We like to think of ourselves as supremely rational beings not controlled by our genes, but consider a simple question. If you could save the life of one person between a sibling, a cousin and a stranger, which would you choose? We are wired by our selfish genes to favor those whose genes are most similar to our own. This would be an example of the kin selection view of altruism.

The other thing that must be considered is that altruism exists at all levels of organism down to bacteria. It really does beg for an explanation.

Zaku's avatar

It would depend on the sibling, cousin, or stranger in question, etc. Family relationship would not be the determining factor, for me. Even if I chose a cousin over a stranger, it would be because of various things I feel and know about the cousin versus the stranger. And if I had been abused by the family member and/or I knew them to be awful in some way, I might well prefer the stranger. I might (probably would, in my actual cases) feel an emotional attachment to my family members, but I also feel attached to (or otherwise particularly highly value the lives of) friends and others I know (you didn’t mention friends). I don’t agree that “we are wired by our selfish genes to favor those whose genes are most similar to our own”. I would say that attractions to same come from familiarity, attachment to individuals and one’s own known community and culture, xenophobia, etc., and doesn’t necessarily have anything to do with genes.

If my 5th grade class had been threatened by attack, I would have above all tried to save the girl I had a crush on… who was genetically from the opposite side of the planet I was from.

If I want an explanation for altruism, I am not prone to turn first to “survival of the fittest” thinking! I would turn more to my own experience, and ideas from psychology, Jung, and Buddhism, and observe that the nature of pretty much every Earth creature in the absence of threat or hardship, is love.

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