The paradox in which considering the rates of occurrence in a two-way table of the characteristics in two groups separately can lead to the opposite conclusion from when the two groups are combined. For this to occur, it is necessary that there be a substantial difference in the proportions of one category within two groups. For example, if 80% of students applying for science are accepted and 40% of students applying for arts are accepted, irrespective of gender in both cases, then there is no discrimination. However, if 75% of boys apply for science courses and only 30% of girls apply for science, the overall success rate of applications for boys and girls would look discriminatory regarding gender. With the above proportions, and 1000 boys and 1000 girls altogether, the two-way table would be:
That is, overall 70% of boys would be accepted compared with only 52% of girls, but the rates of acceptance from each group by type of course were identical.
Conversely, it may be the case that a drug trial seems to suggest, for a group as a whole, that the drug reduces mortality rates. However, when the data are stratified by gender, say, it may prove to be the case that the drug increases mortality rates for both genders. An example of such data is presented below.
| Accept | Fail |
|---|
Boys | 700 | 300 |
Girls | 520 | 420 |
| WHOLE GROUP | MEN | WOMEN |
|---|
| Recover | Die | Mortality | Recover | Die | Mortality | Recover | Die | Mortality |
|---|
Treated | 20 | 20 | 50% | 2 | 8 | 80% | 18 | 12 | 40% |
Control | 16 | 24 | 60% | 9 | 21 | 70% | 7 | 3 | 30% |
Columns 2–4 suggest the drug trial was a success, reducing the mortality rate, while columns 5–7 and 8–10 show it was separately detrimental to both men and women. This paradox is possible here, as men and women have very different mortality rates, and the treated and control groups are not evenly distributed by gender. So, the seeming improvement only reflects that women treated with the drug have a better mortality rate than men who are untreated.