You are investigating whether women in a given company are being discriminated against. You look at the salaries of women and find this:
Company 1
Female |
|
Under 50K |
400 |
Over 50K |
200 |
600 |
At another company, you find this:
Company 2
Female |
|
Under 50K |
300 |
Over 50K |
300 |
600 |
What would you conclude about the first company? It looks like the first company discriminates while the second doesn't, right?
Let's look at it from another point of view. Let's just consider all the people who are making good money, starting with the first company:
Company 1
Male |
Female |
||
Over 50K |
100 |
200 |
300 |
Notice that twice as many women as men are making good money. If we look at the second company, we see more or less the same story:
Company 2
Male |
Female |
||
high pay |
200 |
300 |
500 |
So, in both companies, it looks like more women than men are making good money. But from the first two tables, it appears that in Company 1 most women are making low pay, while in Company 2, its about half and half.
Now let's look at the men in both companies:
Company 1
Male |
Female |
||
Under 50K |
200 |
400 |
600 |
Over 50K |
100 |
200 |
300 |
300 |
600 |
900 |
Is there discrimination at Company 1? The biggest group of people is females who make less than 50K!
Company 2
Male |
Female |
||
low pay |
100 |
300 |
400 |
high pay |
200 |
300 |
500 |
300 |
600 |
900 |
How about at Company 2?
If we convert to column percentages we get:
Company 1
Male |
Female |
|
Under 50K |
66.67% |
66.67% |
Over 50K |
33.33% |
33.33% |
100.00% |
100.00% |
Company 2
Male |
Female |
|
low pay |
33.33% |
50.00% |
high pay |
66.67% |
50.00% |
100.00% |
100.00% |
Obviously, it is Company 2 that has lower salaries for women. So it seems like we should always look at the percentages, right?
Have a closer look at the raw frequencies for company 2. What if we had sampled three times as many low paying people as high paying people? Then the results would have looked this:
Company 2
Male |
Female |
||
low pay |
300 |
900 |
1200 |
high pay |
200 |
150 |
350 |
500 |
1050 |
1550 |
And the percentages are these:
Company 2
Male |
Female |
|
low pay |
60.00% |
85.71% |
high pay |
40.00% |
14.29% |
100.00% |
100.00% |
These percentages are different! Somehow, we need to get a handle on the fact that there are different numbers of males and females, AND different numbers of high and low paying jobs. We can't see the pattern in the data because of these different sizes of the groups. A way to deal with this is given in the next handout.