TABLE OF CONTENTS This section deals with combining tables which can lead to counterintuitive results. The paradox was first discussed in a paper by E. H. Simpson which was published in 1951.
Example 3.2. Last semester at Albright College in a certain sophomore level math course the following data were observed:
| Total # | #A s | Chance of an A | |
|---|---|---|---|
| Female Students
| 200 | 62 | .31 |
| Male Students | 120 | 54 | .45 |
Assuming no difference in abilities it appears there is a gender bias in grading. But the affirmative action official at the College observes the following breakdown by professor:
| Professor I: | |||
|---|---|---|---|
| Total # | #A s | Chance of an A | |
| Female Students | 40 | 22 | .55 |
| Male Students | 100 | 50 | .50 |
| Professor II: | |||
|---|---|---|---|
| Total # | #A s | Chance of an A | |
| Female Students | 160 | 40 | .25 |
| Male Students | 20 | 4 | .20 |
(Check the sums in the tables above.)
There is a greater overall chance of a male student making an A, but with each professor, females had a greater chance of getting an A. On an intuitive basis we see that a much higher percentage of female students enrolled in Professor II s class than in Professor I s class and Professor II is clearly a harder grader.
Let s look at the situation on a probabilistic basis. If
then rewriting the above in terms of conditional probabilities
even though the direction of the inequality is reversed...
