Nearly everyone is above average

Most people have a higher than average number of legs.

The vast majority of people have two legs. But some people have no legs or one leg.  So the average number of legs will be just slightly less than 2, meaning most people have an above average number of legs. (I didn’t come up with this illustration, but I can’t remember where I saw it.)

Except in Lake Wobegon, it’s not possible for everyone to be above average. But this example shows it is possible for nearly everyone to be above average. And when you’re counting numbers of social connections, nearly everyone is below average. What’s not possible is for a majority of folks to be above or below the median.

Most people expect half the population to be above and below average in every context. This is because they confuse mean and median. And even among people who understand the difference between mean and median, there is a strong tendency to implicitly assume symmetry. And when distributions are symmetric, the mean and the median are the same.

10 thoughts on “Nearly everyone is above average”

1. What’s not possible is for a majority of folks to be above or below the median.

Well, that’s a little misleading, especially when it’s connected to what you say next:

Most people expect half the population to be above and below average in every context. This is because they confuse mean and median.

Correct, it’s not possible for a majority to be above or below the median. But it’s possible for the vast majority to be (above or equal to) or (below or equal to) the median

In the same example, with the legs, the median is 2. Yet we don’t expect half the population to have fewer than two legs — the vast majority are at the median, few are below, and essentially none are above (there probably are one or two who have three legs, actually).

The main thing more than a confusion of mean and median: it’s that most people don’t really understand either one.

2. John Venier

In a similar context, a statistics professor I once had pointed out a seeming impossibility. There can be two groups such that when some members of one group are relocated to the other group, both groups’ means are raised.

The specific example he cited was moving people from Texas to Oklahoma to increase both states’ mean IQ. I swear I’m not making it up :-) In case you’re wondering, he was Polish himself.

Another interesting related statistical example is comparing CDC cancer rates for deaths due to prostate, breast, and colorectal cancer in the US. Among men, prostate cancer causes more deaths per capita (25.4 / 100,000 in 2004) than colorectal cancer (21.6 / 100,000 in 2004). Among women, breast cancer causes more deaths (24.4 / 100,000 in 2004) than colorectal cancer (15.2 / 100,000 in 2004). But in the whole population, colorectal cancer causes more deaths per capita (17.9 / 100,000 in 2004) than either prostate (about 12.7 / 100,000 in 2004)or breast cancer (about 12.2 / 100,000 in 2004). Be careful when you look this up though, since the rates on the CDC web site for breast and prostate cancer are the gender-specific rates, even in the “Male and Female” section.

3. In case you’re wondering, he was Polish himself.

Is this a total non sequitur, or am I just missing something?

4. I suppose the connection is a Polish person telling an Aggie joke — the way I’ve heard the joke above is Aggies moving to Oklahoma — and Polish jokes are similar to Aggie jokes.

For readers unfamiliar with Aggie jokes, an “Aggie” is a Texas A&M student. The punch line of an Aggie joke nearly always has something to do with an Aggie being stupid. The Aggies I know don’t take offense at Aggie jokes; nobody takes them that seriously. About the only people I hear telling Aggie jokes are Aggies.

5. John Venier

Is this a total non sequitur, or am I just missing something?

I probably phrased it poorly. I did think it was ironic that he was Polish given the existence of similar jokes in the US about Poles — and the irony may have been intentional on the part of the professor. But mostly what I meant to convey was that he was neither a Texan nor an Oklahoman, and thus didn’t have a dog in the fight. Which was kind of odd, but that’s probably how the joke was told to him in the first place. Two of the five students were Texans and I don’t think any of the others was Oklahoman, so maybe that had something to do with it. Also he was teaching in a school in Texas.

I probably should have written:

In case you’re wondering, he himself was Polish.

or even:

In case you’re wondering, he was Polish.

or with less color:

In case you’re wondering, he was neither a Texan nor an Oklahoman.

Or with less color, and revealing a little about the solution:

In case you’re wondering, he was not a Texan.

Or a more literal approach:

In case you’re wondering, he was not a Texan, which you might have assumed given that his statement implies that dumber than average Texans are still smarter than smarter than average Oklahomans. Nor was he an Oklahoman depricating his origins or current residence. But since his statement implies something about the relative IQs of Texans and Oklahomans, you might have been wondering if his origins had some bearing on the way the joke was told, i.e. that the group leaves from Texas and arrives in Oklahoma, especially since the group identities are arbitrary and he could just as well have chosen any other two groups.

Anyway, I thought by specifying his nationality it was a succint way of conveying the above, with a hint of irony due to Polish jokes in the US. I try hard to keep in mind that blogs are often read by folks from all over the world, but the main point of my statement wasn’t about Polish jokes in the US, so I didn’t expand that bit of US culture.

Well — on to “Aggie” jokes. My experience with them matches John Cook’s. When I was a kid in Texas they were common playground currency. Oddly enough, they were also popular in Iowa, far removed from Texas, but I suspect not far from Aggies. Iowa State and A&M both think of themselves as the premiere agricultral schools in the US.

What I think is even more interesting is that these jokes seem to exist in many forms in the world, or at least the European-dominated world, based on my own very limited experience. The clusters seem to be “’cause they’re so dang stupid” which often overlaps with “’cause they’re so dang uncultured/rustic” and “’cause they’re so dang cheap” which often overlaps with “’cause they’re so dang uppity/rich”.

For example, when I was in the Netherlands I heard hundreds of “Belgian jokes” which are identical with Aggie jokes and very similar to Polish jokes.

6. Daniel Lemire

There can be two groups such that when some members of one group are relocated to the other group, both groups’ means are raised.

Right:
A=2,3
B=0,1

Relocate 2 from A to B.

Neat.

7. Yes, John Venier, the French also tell “Belgian jokes.” In New York, they make jokes about people from New Jersey. In Niger, West Africa, where I have lived for years, each ethnic group has a special “joking relationship” with certain other ethnic groups. I imagine this kind of thing goes on all over the world.

8. Blaise F Egan

In the UK when I was a child it was for common for the English to make “Irish jokes”. They are now deemed politically incorrect and seem to have disappeared. Again, when I was at school in the 70s a textbook we used mentioned that the French made jokes about the Normans. I later found out that the Spanish made fun of Andalusians and when I went Cairns about ten years ago they were making jokes about the Queenslanders!

9. Statements about averages can be extremely misleading when the shape of the distribution is omitted. For example, the average American has roughly one ovary and one testicle.

10. You may have seen the example with the average number of legs in Hans Rosling’s The Joy of Stats, at least it occurs in this documentary.