Simpson's paradox: UC Berkeley gender bias

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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.
enter image description here



enter image description here



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 $fracx+y2 $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?




statistics




share|cite|improve this question





















  • $dfrac1(101)101=1$ and $dfrac1(101)101=1$
    – steven gregory
    Aug 1 at 7:56










  • @stevengregory the final result seems reasonable, but it is the result of intermediate step that makes me uneasy.
    – q126y
    Aug 1 at 8:02






  • 1




    Don't think of it as $dfrac.62 times 933 + .63 times 585 cdots4526$ but as $.62 dfrac9334526 + .63 dfrac5854526 + cdots$
    – steven gregory
    Aug 1 at 8:29










  • @stevengregory that is very helpful. Previously, different weights were given to male&female selection rates. Now both male and female acceptance rates for a given department are given same weight. Now the question is the new weights that we are assigning is it valid/justifiable? Why do we not assign weight for dept A as $fracno., of, seats, in ,dept ,Atotal ,no. ,of ,seats ,in ,all ,depts$ ? How do we decide which weighing scheme is valid/better?
    – q126y
    Aug 1 at 9:47











  • @stevengregory say old weight for male for dept A is x and for female for dept A is y, why not take the new weight as x+y and divide the whole fraction by 2(since sum of old male and females weights were 1 each). One reason I can think is this weighing scheme doesn't remove the bias, if dept A had more weight for females, the new weight will be more for dept A. How do we know the new weighing scheme used by Freedman is free of such bias/problems
    – q126y
    Aug 1 at 10:12















up vote
0
down vote

favorite












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.
enter image description here



enter image description here



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 $fracx+y2 $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?




statistics




share|cite|improve this question





















  • $dfrac1(101)101=1$ and $dfrac1(101)101=1$
    – steven gregory
    Aug 1 at 7:56










  • @stevengregory the final result seems reasonable, but it is the result of intermediate step that makes me uneasy.
    – q126y
    Aug 1 at 8:02






  • 1




    Don't think of it as $dfrac.62 times 933 + .63 times 585 cdots4526$ but as $.62 dfrac9334526 + .63 dfrac5854526 + cdots$
    – steven gregory
    Aug 1 at 8:29










  • @stevengregory that is very helpful. Previously, different weights were given to male&female selection rates. Now both male and female acceptance rates for a given department are given same weight. Now the question is the new weights that we are assigning is it valid/justifiable? Why do we not assign weight for dept A as $fracno., of, seats, in ,dept ,Atotal ,no. ,of ,seats ,in ,all ,depts$ ? How do we decide which weighing scheme is valid/better?
    – q126y
    Aug 1 at 9:47











  • @stevengregory say old weight for male for dept A is x and for female for dept A is y, why not take the new weight as x+y and divide the whole fraction by 2(since sum of old male and females weights were 1 each). One reason I can think is this weighing scheme doesn't remove the bias, if dept A had more weight for females, the new weight will be more for dept A. How do we know the new weighing scheme used by Freedman is free of such bias/problems
    – q126y
    Aug 1 at 10:12













up vote
0
down vote

favorite









up vote
0
down vote

favorite











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.
enter image description here



enter image description here



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 $fracx+y2 $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?




statistics




share|cite|improve this question













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.
enter image description here



enter image description here



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 $fracx+y2 $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?





statistics





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 1 at 11:28
























asked Aug 1 at 7:43









q126y

1666




1666











  • $dfrac1(101)101=1$ and $dfrac1(101)101=1$
    – steven gregory
    Aug 1 at 7:56










  • @stevengregory the final result seems reasonable, but it is the result of intermediate step that makes me uneasy.
    – q126y
    Aug 1 at 8:02






  • 1




    Don't think of it as $dfrac.62 times 933 + .63 times 585 cdots4526$ but as $.62 dfrac9334526 + .63 dfrac5854526 + cdots$
    – steven gregory
    Aug 1 at 8:29










  • @stevengregory that is very helpful. Previously, different weights were given to male&female selection rates. Now both male and female acceptance rates for a given department are given same weight. Now the question is the new weights that we are assigning is it valid/justifiable? Why do we not assign weight for dept A as $fracno., of, seats, in ,dept ,Atotal ,no. ,of ,seats ,in ,all ,depts$ ? How do we decide which weighing scheme is valid/better?
    – q126y
    Aug 1 at 9:47











  • @stevengregory say old weight for male for dept A is x and for female for dept A is y, why not take the new weight as x+y and divide the whole fraction by 2(since sum of old male and females weights were 1 each). One reason I can think is this weighing scheme doesn't remove the bias, if dept A had more weight for females, the new weight will be more for dept A. How do we know the new weighing scheme used by Freedman is free of such bias/problems
    – q126y
    Aug 1 at 10:12

















  • $dfrac1(101)101=1$ and $dfrac1(101)101=1$
    – steven gregory
    Aug 1 at 7:56










  • @stevengregory the final result seems reasonable, but it is the result of intermediate step that makes me uneasy.
    – q126y
    Aug 1 at 8:02






  • 1




    Don't think of it as $dfrac.62 times 933 + .63 times 585 cdots4526$ but as $.62 dfrac9334526 + .63 dfrac5854526 + cdots$
    – steven gregory
    Aug 1 at 8:29










  • @stevengregory that is very helpful. Previously, different weights were given to male&female selection rates. Now both male and female acceptance rates for a given department are given same weight. Now the question is the new weights that we are assigning is it valid/justifiable? Why do we not assign weight for dept A as $fracno., of, seats, in ,dept ,Atotal ,no. ,of ,seats ,in ,all ,depts$ ? How do we decide which weighing scheme is valid/better?
    – q126y
    Aug 1 at 9:47











  • @stevengregory say old weight for male for dept A is x and for female for dept A is y, why not take the new weight as x+y and divide the whole fraction by 2(since sum of old male and females weights were 1 each). One reason I can think is this weighing scheme doesn't remove the bias, if dept A had more weight for females, the new weight will be more for dept A. How do we know the new weighing scheme used by Freedman is free of such bias/problems
    – q126y
    Aug 1 at 10:12
















$dfrac1(101)101=1$ and $dfrac1(101)101=1$
– steven gregory
Aug 1 at 7:56




$dfrac1(101)101=1$ and $dfrac1(101)101=1$
– steven gregory
Aug 1 at 7:56












@stevengregory the final result seems reasonable, but it is the result of intermediate step that makes me uneasy.
– q126y
Aug 1 at 8:02




@stevengregory the final result seems reasonable, but it is the result of intermediate step that makes me uneasy.
– q126y
Aug 1 at 8:02




1




1




Don't think of it as $dfrac.62 times 933 + .63 times 585 cdots4526$ but as $.62 dfrac9334526 + .63 dfrac5854526 + cdots$
– steven gregory
Aug 1 at 8:29




Don't think of it as $dfrac.62 times 933 + .63 times 585 cdots4526$ but as $.62 dfrac9334526 + .63 dfrac5854526 + cdots$
– steven gregory
Aug 1 at 8:29












@stevengregory that is very helpful. Previously, different weights were given to male&female selection rates. Now both male and female acceptance rates for a given department are given same weight. Now the question is the new weights that we are assigning is it valid/justifiable? Why do we not assign weight for dept A as $fracno., of, seats, in ,dept ,Atotal ,no. ,of ,seats ,in ,all ,depts$ ? How do we decide which weighing scheme is valid/better?
– q126y
Aug 1 at 9:47





@stevengregory that is very helpful. Previously, different weights were given to male&female selection rates. Now both male and female acceptance rates for a given department are given same weight. Now the question is the new weights that we are assigning is it valid/justifiable? Why do we not assign weight for dept A as $fracno., of, seats, in ,dept ,Atotal ,no. ,of ,seats ,in ,all ,depts$ ? How do we decide which weighing scheme is valid/better?
– q126y
Aug 1 at 9:47













@stevengregory say old weight for male for dept A is x and for female for dept A is y, why not take the new weight as x+y and divide the whole fraction by 2(since sum of old male and females weights were 1 each). One reason I can think is this weighing scheme doesn't remove the bias, if dept A had more weight for females, the new weight will be more for dept A. How do we know the new weighing scheme used by Freedman is free of such bias/problems
– q126y
Aug 1 at 10:12





@stevengregory say old weight for male for dept A is x and for female for dept A is y, why not take the new weight as x+y and divide the whole fraction by 2(since sum of old male and females weights were 1 each). One reason I can think is this weighing scheme doesn't remove the bias, if dept A had more weight for females, the new weight will be more for dept A. How do we know the new weighing scheme used by Freedman is free of such bias/problems
– q126y
Aug 1 at 10:12
















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