The majority can be above average

Headlines like this are quite frequent in the media:

70% of Drivers rate themselves as ‘above average’

(example)

Usually this is accompanied by a smug observation of how ridiculous this is considering that it’s impossible for the majority to be above average.

The only thing is that this is not impossible at all. It’s actually quite a common situation with averages.

Here’s a quick example using the driving ability scenario that seems to crop up most frequently. Imagine that we have some numerical measure of driving ability that ranges from 1 to 10, and these are the scores of five drivers:

$\{1, 7, 7, 7, 8\}$

The mean average score is $\frac{1+7+7+7+8}{5} = 6$. As we can see, four out of the five drivers have a score above this. It is accurate to say that 80% of these drivers have above average driving skills.

Perhaps in a lot of cases, what’s being described is actually the modal value and not the mean. The modal value or mode average is the value that occurs the most times in the set. However, it’s easy to think of scenarios where the majority is above the modal value as well, e.g.:

$\{3,3,4,6,8\}$

In the above set, the mode is 3 but 60% of the members are above that.

These example sets might seem contrived, but these kinds of distributions are perfectly possible in the real world. In a lot of countries, if you receive the average income, then your income is higher than what the majority of people receive. Similarly, there’s the joke about Bill Gates walking into a bar and making the average person a millionaire.

With a median average you tend to avoid some of these problems, as the median is by definition in the middle of all the values. However, median averages are still guilty of misleading people into thinking they have found the most descriptive or suitable number from a set.

More frivolous examples are that most people have an above average number of legs or eyes (most have 2, some have 1 or 0 and very few have more than 2). The average human being has slightly less than one testicle and slightly less than one ovary (most have 2 of one kind and 0 of the other, a few have one of one kind, a few have none of either, and very few have both). Clearly these averages are not good descriptions of the human body.

We tend to assume most things are normal distributions (a.k.a. bell curve distributions), which can be fully described with the mean and variance. Plenty of things do not have a normal distribution, though. Wealth is usually a power law distribution. Even when something does have a normal distribution, it does not mean the average is a good choice of number for all decisions about it.

For example, a personal peeve of mine is that seats on aeroplanes seem to be designed for the average human height (which is normally distributed). This is fine if you are below average height or the rare person who is exactly the average height, but terrible if you are over it. The average has given a misleading impression of maximum suitability. Whether it’s mean, mode or median doesn’t make any difference.

(Of course, there is a trade-off between cost and comfort, and it seems that passengers are in general unwilling to pay to tip the scales towards comfort. My point is that it would be wrong to assume the average provides the optimum trade-off.)

A more rigorous and perhaps more comical example comes from the US Army in the 1950s. They were trying to design a cockpit for planes that would better suit their pilots:

In 1950, researchers at Wright Air Force Base in Ohio measured more than 4,000 pilots on 140 dimensions of size, including thumb length, crotch height, and the distance from a pilot’s eye to his ear, and then calculated the average for each of these dimensions. Everyone believed this improved calculation of the average pilot would lead to a better-fitting cockpit and reduce the number of crashes.

After a great deal of effort was put into gathering and computing this data, one scientist decided to check how many pilots actually were average. The answer was zero. Not just some small percentage, but actually none: not a single pilot fit the average dimensions. The average-designed cockpit was bad for everyone.

TL;DR Averages are on average misleading and misused.

Thanks to Marc Jones for improving this post.