I am currently working on a performance indicator system for the number of reviews due on a bunch of document against the number that were due in that month as a percentage.
There are several areas where these values come from and they are totaled together to get the overall.
+-----------------+------------+-------------+-------+
| Area | Due | Delivered | % |
+-----------------+------------+-------------+-------+
| Area Total | 43 | 18 | 41.8 |
+-----------------+------------+-------------+-------+
| Sub-Area 1 | 10 | 0 | 0 |
| Sub-Area 2 | 10 | 0 | 0 |
| Sub-Area 3 | 0 | 0 | 100 | (100% of the delivery)
| Sub-Area 4 | 8 | 8 | 100 |
+-----------------+------------+-------------+-------+
| SubArea 5 Total | 15 | 10 | 66.7 |
+-----------------+------------+-------------+-------+
| Sub-Area 5.1 | 2 | 0 | 0 |
| Sub-Area 5.2 | 13 | 10 | 76.9 |
+-----------------+------------+-------------+-------+
This is how the totals of the actual values work out, however I am trying to work out the values of the total areas using the percentage of the children and it is not working.
I tried this.
$$ x\% = (sa1 + sa2 + sa3 + sa4 + sa5.1 + sa5.2) / 6 $$
And this
$$ x\% = (sa1 + sa2 + sa3 + sa4 + ((sa5.1 + sa5.2)/2)) / 6 $$
Both of these return different values from each other and from when you are taking the percentage from the actual values.
I understand that average is not the right thing to do for percentages, I am going to use the actual values (which is easy enough) but I am still curious why this doesn't work.
This is a milder version of something called Simpson's paradox. Averages of averages need not behave as you might expect them to.
The most famous recent case was in graduate school admissions at Berkeley. Each individual graduate department accepted women at a higher rate than men. However, when you look at the overall graduate admissions rate, women were accepted at a lower rate than men. The reason is that women were applying disproportionately often to programs like English with a low admissions rate.