Here is an interesting example of why one needs to be careful when combining statistics that have been collected from various sources of unequal size.
The following apochryphal story describes a problem:
Supposedly some university president had commissioned a study of gender bias. The results from the study showed that the institution as a whole was extending offers of faculty positions to a larger proportion of male interviewees than female interviewees. Worried by the possible legal and the almost certain political consequences if such a state of affairs was true and became known, the president ordered the panel to find out which departments were doing the discriminating. The follow-up study showed that none of the departments was favoring men applicants over women applicants but that, instead, the result of the initial study was an artifact of the facts (a) that different departments had different male-female ratios and (b) that the university as a whole had statistics quite different from those of the individual departments.
Is an apparent paradox such as this possible? It turns out that the answer is yes. Here is an example* illustrating the possibility:
Let the [university's Mechanical Engineering department] get 30 male and 20 female applicants and decide to take half of them across the board, and then decide to take one extra woman:
M: 30 applied, 15 get a place (50%)
F: 20 applied, 11 get a place (55%)Let the Literature [Department] get 20 male and 30 female applicants but, strapped for cash and space, decide to take only 10% across the board, plus one more woman:
M: 20 applied, 2 get a place (10%)
F: 30 applied, 4 get a place (13.33%)Both departments decided to take one more woman than would have been par with the men (perhaps to avoid accusations of sexism?) and this reflects in the percentages. But what about the departments combined?
M: 50 applied, 17 get a place (34%)
F: 50 applied, 15 get a place (30%)You can get this when you compare unequal things. In this cooked up example, the department getting more men happens to admit more people across the board than the department getting more women.
Of course, there still could be institutional sexism at work here. If the minutes of the meetings of the relevant authority that allocates funds and/or space across the university leak out and it turns out they've underfunded the traditional girlie subjects because they're girlie subjects, it's still true that no department is to blame—but somebody would be.
Incidentally, who would be to blame in that scenario? The vice chancellor favoring Mech Eng over Literature? The parents and school teachers who were responsible for steering more girls this way and more boys that way? Or the students themselves? Or are they only reacting sensibly to known employment prospects? Or...
The trouble with statistics is that they don't tell the how and why.
*My thanks go to Marijke van Gans for this nice example. Ms. van Gans is a doctoral student in mathematics at the University of Birmingham (in the United Kingdom) and a frequent contributor to the Science and Mathematics Forum of CompuServe. This example was part of a message she posted to that Forum on 2003 February 1.
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Last revised 2003 Apr 19