The meaning of means

I've just been reading an engaging book about statistics—The Tiger That Isn't,by Michael Blastland and Andrew Dilnot—and it's got me thinking a little harder, as statistics always ought to, about familiar things I thought I already knew everything about.

For instance, how much do you think the mean national income was in the UK, after tax and benefits, for two childless people living as a couple in 2005/6? I'm willing to bet that many will be surprised by the answer, which is £23,000—equivalent to £11,500 each (most people I've tested thought it would be higher). Still, statistics are often surprising, and the wise reader learns to expect this. What's much more interesting is just how limited this mean is as an index of national incomes.

Blastland and Dilnot use the analogy (made famous by the Dutch economist Jan Pen) of a procession of the world's population, in which people are exactly as tall as they are wealthy. Within this procession, a person of average wealth will also be of average height; remembering that height is a measurement for which the mean is a useful index, as it roughly follows the "normal" distribution of a bell-curve. Global wealth is such, however, that it is not until 80% of humanity have walked diminutively past that you first see a person of average height. Then, in the last few minutes, giants stride into view, each one many miles high, dragging the mean inexorably towards them.

National earnings, similarly, are dragged up by huge wealth at the top, so that only about a third of the population earn the mean income or more. Far more telling is the median—the number that exactly 50% of the population lie above, and 50% lie below. In the UK, this figure is just £18,800 for childless couples, (i.e. £9,400 per person) almost 20 per cent below the mean. In other words, half of the UK's childless couples have less than £1,567 a month coming into their bank accounts in total; less than £784 each.

In terms of income, the mode—the most frequent single result—is less useful even than the mean (it happens to be around £14,000). Sometimes, however, it too comes into its own. Take feet, in the sense of the things on the end of our legs. Because some people have one or no feet, the global mean is something like 1.998: not a very useful number. But, because there are only three possible answers to the question "how many feet do you have?" (even in the case of deformities, I think it's safe to say you either have one foot, two, or none) the median result is also not very useful, because it's one: the middle result of three. Here, finally, the mode comes into its own, and reminds us that most people have two feet: another triumph for statistics.

EDIT: I feel I ought to acknowledge, as highlighted in the comments below, my hastily incorrect interpretation of median in the last paragraph, which of course refers to the middle result from a given sample and has nothing to do with the "middle" out of three possible results. I should also add that this error is my own, and is no reflection of Blastland's and Dilnot's perfectly lucid book…