Parts and Wholes

We also need to be careful in making judgments from marginalized and aggregated data. Imagine a ``fair'' university which accepts the same rate p_field of female applicants and male applicants for each field, for example the best 10% of female applicants and the best 10% of male applicants for education. Let's also say that the total number of female applicants equals the number of male applicants. Assume the following distribution of applicants (with simplified numbers):

Field	rate	f. applicants	f. accepted	m. applicants	m. accepted
Education	10	1000	100	100	10
Social Sciences	20	500	100	300	60
Engineering	30	200	60	1300	390
total		1700	260	1700	460

If we look at the aggregated total distribution the selection of students seems biased. Many more male students (460) are accepted than female (260). The reason is that female students applied more for fields with lower acceptance rate (higher competition). But this information is not shown anymore in the total (projected) distribution, parts (marginals, projections) do not in general determine wholes (overall distributions). A similar example with patients in a hospital can be found in Glymour et. al. [11, pg. 20].