I run into this issue frequently at work. I need a good answer that will make sense to non-math people on why they cannot average percents in the manner below. I think in pictures and am not very good at explaining in plain English!
Week 1: Total Sales = $3000, Shift One Sales = $950, Percent of Total Sales are 31.67%
Week 2: Total Sales = $2500, Shift One Sales = $800, Percent of Total Sales are 32.00%
Week 3: Total Sales = $3200, Shift One Sales = $960, Percent of Total Sales are 30.00%
To calculate the percent of Shift One Sales to Total Sales for the three week period, they should be taking the sum of Shift One Sales divided by the sum of Total Sales. Instead, they keep adding up the three percents and dividing it by 3. I can show them how this gives them different numbers (31.15% vs. 31.22%), but cannot explain in plain English why they should use method one if they want an accurate number. Does any one have a good explanation?
Method two doesn't work because some weeks count more than others. I suggest you use a more revealing example, like this one:
Week 1: Total Sales = $2000, Shift One Sales = $1500, Percent of Total Sales are 75.00%
Week 2: Total Sales = $8000, Shift One Sales = $2000, Percent of Total Sales are 25.00%
Average of percents: 50%
Average by taking Sums: 35%
Some years ago, I had to persuade a Dean at my college
that averaging percents is blatantly wrong.
I submitted an example which finally convinced him.
At a particular unversity, the Deans are evaluated annually.
To be retained, he/she must have the approval of at least
80% of the Faculty.
There are 100 professors, 4 of whom form the Math Department.
Two of the Math professors voted against a particular Dean;
the rest of the Faculty gave its approval.
Although 98 out of 100 approved of this Dean, he was fired.
In the Math Department, only two out of four approved of him.
. . He had a 50% rating.
The other 96 professors approved him.
. . He had a 100% rating.
Therefore, his average rating is only: .
See what happens when we average percents?
That is very helpful - and descriptive of the issue! If a person is outside the normal "range" they select, they can potentially be flagged as someone who needs watching. The problem I had was they were starting with the wrong answer! I appreciate the help.
Also, when we talk about percents, we are always taking the percent OF something. If we just add up the percents and ignore the fact that what we're taking the percents OF are different things, we're removing the percents from their contexts. Percents do not live in a universe by themselves - they're always tied to a quantity we are taking the percent OF. This part is just as important as the percent itself.