Wikipedia section about the UC Berkeley gender bias
According to Statistics, by David Freedman
Technical note. Table 2 is hard to read because it compares twelve admis- sions rates. A statistician might summarize table 2 by computing one overall ad- missions rate for men and another for women, but adjusting for the sex difference in application rates. The procedure would be to take some kind of average ad- mission rate separately for the men and women. An ordinary average ignores the differences in size among the departments. Instead, a weighted average of the admission rates could be used, the weights being the total number of applicants (male and female) to each department; see table 3.
The book adjusts for sex difference and different sizes of departments by calculating the weighted average, where the weights are total number of applications(both male and female) to a department.
My question is, say department A has 2 seats, 1 male and 100 female apply, and 1 male and 1 female are selected.
So, % of male selected = 100% % of female selected = 1%
for calculating the adjusted percentage for male,
we will have
1.00*101 as the weighted term for department A = 101 selections, which is absurd, as department A has only 2 seat. -- 1
Is it possible to prove that the calculation done by David Freedman in the book is sound and produces correct result. If so, how does one account for anomaly presented in equation 1?
Edit: Based on comments of steven gregory, I now know the anomaly is not anomaly, the rate of selection in different departments of both male and female are assigned same weight for comparison.
Now the question is the new weighing scheme used by the author the only way?
If original weights for rate of selection for males for dept A is x, and weight for rate of selection for females of dept A is y, why don't we use $\frac{x+y}{2 }$as the new weighing scheme when comparing the rate of selections of male and female?
Why is it better/worse than the weighing scheme used by Freedman in the book?