You probably think that all that stuff about normal distributions is pretty boring. "I suppose it's alright in a statistics course, but nobody would ever have any other interest in it." Well, perhaps that's a bit hasty. Maybe it can explain a few things in everyday life that we never thought about.
If you spend any time reading the paper or listening to television, you know that firms are often accused of discriminating against women or minority groups in terms of hiring, promotion, salary, and so on. And there is a great deal of evidence that we do discriminate. Some of it is no doubt conscious, some of it is just plain thoughtless, and some of it may come from the best of intentions. In fact, some job discrimination may result directly from an attempt to be absolutely fair and to set a sex blind criterion.
Suppose that we look at admissions to graduate school, just to take the pressure off business. My actual belief would be that women outscore men on most dimensions along which we choose graduate students. But suppose that I'm wrong and that men actually outscore women by a teenie amount. Suppose further that a department has a very fair way of combining GPA, GRE scores, letters of recommendation, etc. into one nice simple score. They convert to T scores (Mean 50 and St. Dev = 10) and discover that of all their applicants, women have a mean of 50, but men have a mean of 51. No one in their right mind would think that a 1 point difference (which comes out to a tenth of a standard deviation) would make the slightest difference to anyone. If fact, it is so trivial that the department decides to ignore it completely. Besides, they're only looking for the best students, anyway, and the best students aren't getting scores down near 50. Suppose further that we are looking at a field where admission is very competitive, such as clinical psychology. My own department gets between 400 and 500 applications for about 8 slots, which means we only take about 2 percent of the applicants. (I'm not a clinical psychologist, by the way, or I probably wouldn't have made it to graduate school.) So suppose we decide that we will make offers to everyone who scores above a 70, which is two standard deviations above the mean. How many will we admit out of 500 applicants?
You should be able to solve that with a calculator and a table, and in fact you could probably come very close in your head. (I'd guess about 2.3%) But if you want to be part of the 21st century, you don't want to do things like that in your head, especially when you're signed on to the world wide web. Instead, just click on http://psych.colorado.edu/~mcclella/java/zcalc.html and go to Gary McClelland's Java program which will calculate it for you. (At least till I get my own page written to replace his.). In the lower right of his page you can enter z and get back the probability--which turns out to be .023.
But I digress. We just saw that setting a cutoff at 70 would get us about 2.3% of the applicants, and if some of those turn us down we should be OK. And since there's such a teenie weenie difference between males and females, we should be fine. Well, shouldn't we. Well, no. Following our rules we will admit about 1.26 men for every woman, which certainly doesn't look like gender-unbiased admissions to anyone on the outside.
How could this happen when we're trying to be so fair??? Well, the problem is partly that we are selecting people at the extreme end of the distribution. If we took everyone who scored over 50, we would take about 51 men to every 50 women, for a ratio of 1.02:1, and no one would object. But what's true about the center of the scale is not true about the extremes. We don't take very many people way up there at the ends, but the areas above the cutoff are more different than we would have supposed.
I thought of this example because of a very nice demonstration program written by David Lane at Rice. That program allows you to set the distributions for males and females wherever you wish, and then to see what happens as you change the cutoff. I recommend that you try it out. It can be found at http://www.ruf.rice.edu/~lane/stat_sim/group_diff.html
This is a nice illustration of the fact that many of the things that bombard us every day are really of a statistical nature, but we rarely think of them that way. Statisticians have relevance.<
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