I have data depicting college admission statistics in a combined two-way table and a three-way table (with colleges C1&C2) with the following probabilities:
Overall acceptance: $72.37\%$
Overall acceptance - men only - : $77.5\%$
Overall acceptance - women only - : $68.01\%$
Acceptance given men and C1: $30\%$
Acceptance given women and C1: $66.66\%$
Acceptance given men and C2: $93.33\%$
Acceptance given women and C2: $97.56\%$
Acceptance given C1: $60\%$
Acceptance given C2: $93.6\%$
Question: Does the data depict discrimination against men, women, or not at all?
Attempted Answer:
I would say the data shows discrimination against men. This is because the Acceptance rate given men and C1 is significantly lower than the Acceptance rate given C1 ($30\%$ vs $60\%$).
Also, the Acceptance rate given men and C2 ($93.33\%$) is lower than that of the Acceptance rate given women and C2 ($97.56\%$) and slightly lower than the Acceptance rate given C2 ($93.6\%$)
Even though the overall acceptance rate - men only - ($77.5\%$) is higher than that of women only ($68.01\%$) I would still say the colleges C1&C2 are discriminating against men.
Is this the correct line of thinking? Thanks in advance!
This is not a mathematical but a sociological question.
The “paradox” comes about because more men than women are applying at the college with the much higher overall acceptance rate. Thus more of them get accepted overall, even though both colleges have higher acceptance rates for women than for men.
The college system could be discriminating against men, against women, or neither, depending on the overall interplay of factors.
For instance, it could be that the reason that the college that more men are applying to has higher acceptance rates is that more funding is being directed towards education typically sought by men, so this college can accept more students. This could be because these professions are more needed, or it could be because of discrimination against women in the funding decisions. (Whether the gender-specific professional preferences are inherent or themselves the product of societal discrimination is a separate question.)
Or it could be that, as you suspect, the colleges themselves are discriminating against men.
Or it could be that women and men tend to apply for different subjects within each college, with men applying for more sought-after subjects with lower acceptance rates.
Or it could be that the schools are discriminating against men, preparing them less well for college and thus lowering their acceptance rates.
The list could easily be continued, and of course any combination of these aspects could also obtain.
So I would strongly advise against drawing any far-reaching conclusions about the presence or absence of gender-based discrimination in either direction merely on the basis of some aggregated data that doesn’t even pretend to take into account all sorts of relevant factors.