The latest issue of Time has an article titled “In praise of the ordinary child” about parents who push their kids too hard to excel, and the psychological harm it can do to children and parents alike.
It’s a good article. But at one point, there’s a box full of statistics, one of which says:
70%
Share of students who consider themselves above average in academic ability—a mathematical impossibility
That is, it’s impossible (says Time) for 70% of a population to be above average.
I’ve seen this idea other places. “Ninety percent of drivers think they’re better than average,” people laugh, amused at the absurdity.
Listen up, Internet. This is a public service announcement. Are you ready? *ahem*
Any percentage of a population can be above average (except 100%).
A simple example: say you have 100 students. Of those, 99 students score an 80 on a test, and the remaining student scores a 78. The average will therefore be slightly below 80, so 99 students will be above average.
That’s a trivial example to prove a point. But it can happen in more substantial ways. Say that wealth distribution for a community is a bell curve from $0 to $100,000. Then a relatively small group of multi-billionaires moves in. This jacks the average way up, and now most (or all) of the people from our original bell curve are below average.
The confusion seems to be between mean (average) and median. The latter refers to the middle of a sample, and it does indeed split the sample into two halves, one below and one above, roughly speaking. I say “roughly speaking” because even with median there are exceptions, though the exceptions are minor.
It’s an easy mistake to make, even for a magazine as venerable as Time, and I wouldn’t have said anything – except they specifically called it a mathematical impossibility. That means they didn’t just get their math wrong; they got their math wrong while claiming to be right with the power of math, and while using that math power to shoot down the supposed academic abilities of other people.
(Of course, I’m not saying 70% of kids are above average academically, just that it isn’t mathematically impossible.)
This has been a public service announcement. Remember, friends don’t let friends write math-impaired.
Brian D. Buckley 
There is also modal average: the most common result in the set. Which opens up a whole new set of X% above/below average shenanigans.
So, in other words, if Lake Wobegon had one below-average child, who they hid away somewhere, the rest could indeed all be above average. 🙂
This reminds me (I may have mentioned this before) of a story that got reported very widely about fifteen or twenty years ago. It was that research showed that the average American man has twelve sexual partners over the course of his life, and the average American woman about six (I’m making up the numbers, but this is the general idea).
I read quite a bit about the survey that the report was based on, and it was obvious that they were only looking at straight sex.
So, sorry, but no.
If you assume approximately equal numbers of men and women, there’s no way this is possible, at least not unless American men are jetting off to other countries en masse to have sex with non-American women. Every checkmark on the Men side of the ledger would have a matching checkmark on the Women side.
But this didn’t stop the report from being very widely reported as fact at the time.
It’s been a quiet week in Lake Wobegon, my hometown, where Mrs. Branford’s sixth-grade math class used their Monday lesson to disprove inflated reports of promiscuity…