Math Quiz Five

In a certain (obviously fictional) small company, there are three grades of employees. The table below gives the average salary for the men and women in each grade (where the numbers in parentheses give the numbers of employees).

                Average monthly salary

             |  Men       |  Women
 --------------------------------------
 Grade 1     |  1000 (2)  |  1150 (11)
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 Grade 2     |  1444 (8)  |  1656 (5)
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 Grade 3     |  2902 (10) |  2960 (3)

Since the women in each grade are better paid than the men, you might expect that this would still be the case if any combination of grades is considered.

However, it is easily verified with a calculator that the table below correctly gives the average salaries (rounded to the nearest integer in a few cases) for all such combinations.

                Average monthly salary

             |  Men       |  Women
 --------------------------------------
 Grade 1,2   |  1355 (10) |  1308 (16)
 --------------------------------------
 Grade 1,3   |  2585 (12) |  1538 (14)
 --------------------------------------
 Grade 2,3   |  2254 (18) |  2145 (8)
 --------------------------------------
 Grade 1,2,3 |  2129 (20) |  1569 (19)

How come the men in each row of this table are on better average pay than the women?

There's no unique answer, so feel free to comment and discuss below.

There is an increasing amount of males from first through third and a decreasing amount of females. Note that the differences in teachers is consistent (+6, -6; +2, -2). The total second grade male salary is 11552, while the females' total is a mere 8280. The total of male third grade teachers is 29020; females: 8880. The difference between third and second is 17,468 and 600 respectively. The reason the men have a better average is due to inconsistent population distribution.

This seems to be a very nice example of how statistics can be deceiving. The men are on average in a much higher pay grade than the woman (There are 20 men and 19 women working there, and the average pay grade is 2.4 for men and 1.6 for women.) In addition to that, somebody in grade 3 makes on average almost twice as much as in grade 2. So on a second look, the numbers are hardly surprising. As for the reasons for this, that's anybody's guess. If found in some kind of employment statistics, it could either mean that the women are generally less qualified than the men or that they don't have access to the higher paid jobs or...

I like the idea that the employees are teachers. It is not unusual to find more females teaching younger pupils. Your observations are very much along the right lines, though the points you make aren't sufficient to guarantee the situation presented. BTW, the numbers were invented, but the idea was inspired by a similar situation found embedded in a set of data provided for a statistics question set as part of a university course.

CategoryMath