The strange power of the idea of “average”
“While nothing is more uncertain than a single life, nothing is more certain than the average duration of a thousand lives.” The statement is often attributed to the 19th-century mathematician Elizur Wright, who not coincidentally was a life insurance geek. But buried in the aphorism is a humdrum word concealing a powerful idea: the “average”.
The idea of taking an average — that is, of adding up (say) a hundred lifespans and dividing the total by a hundred, to produce the arithmetic mean — seems absurdly simple. But Stephen Stigler, a historian of statistics, reckons it is the most radical statistical operation ever devised. I am inclined to agree. The mean has a strange power over the way we think, and not always a benign one.
We do not know who invented the arithmetic mean. The statistician Churchill Eisenhart once tried to trace its history. It was originally used as a way of combining various observations that should be identical, but were not — for example, estimates of the direction of magnetic north.
In 1635 the mathematician Henry Gellibrand used the word “meane” to describe the midpoint of a lowest and highest number — not the same thing — but by 1668, a person known as “DB” was quoted in the Transactions of the Royal Society describing “taking the mean” of five values casually enough to make it clear that the concept was by then established.
Why is this such a powerful idea? As Prof Stigler puts it in The Seven Pillars of Statistical Wisdom, “you can actually gain information by throwing information away”. This is true in the straightforward sense that too many numbers become confusing: more than 50m people died last year, but if I could somehow show you a hundred-mile long printout of all their ages at death, you might struggle to learn much from it.
But the mean also eliminates errors. In the context that Gellibrand and DB were writing, taking an average cancels out mistakes in the original observations. This was by no means obvious. When confronted with contradictory measurements, the instinct of mathematicians had been to figure out which one was best and to dismiss the rest. But taking the average was a far better way to eliminate error.
And yet in this method lies a trap, because not all variation is error. The trap was sprung in the 1830s by the hugely influential statistician Adolphe Quetelet, who was an astronomer as well as a founder of the idea of “social physics” — using statistical techniques to understand humans and their societies.
Quetelet asked us to imagine that a thousand sculptors had made a copy of a famous statue of a gladiator. Each copy would have some errors or imperfections — but on average, they would be a perfect copy. So far, so good. But then, Quetelet continued, if we measured a thousand real soldiers and averaged their body measurements, we could get the ideal, perfect soldier. “L’homme moyen” — the average man — was Quetelet’s benchmark for perfection. (What about “la femme moyenne”? Well, quite.)
But as Todd Rose points out in The End of Average, Quetelet’s logic isn’t just andro-centric. It’s nonsense. A tall copy of a statue may be an error, but a tall soldier is not — and may well benefit from having superior reach.
Quetelet did not lack for critics. His contemporary, the mathematician and proto-economist AA Cournot, correctly argued that the Average Man probably didn’t exist. Victorian statistician Francis Galton was fascinated by averages, yet asserted: “No statistician dreams of combining objects into the same generic group that do not cluster towards a common centre . . . if we do so the result is monstrous and meaningless.”
But this idea of the average as perfection did not die. In an age of mass production it was too convenient. Production-line managers and modernist architects found it easy to design for the average person.
In 1943, the sculptor Abram Belskie and obstetrician Robert Dickinson turned Quetelet’s metaphor into reality, carving statues of “Normman” and “Norma”, based on the average measurements of 15,000 young adults. The US press loved Norma in particular: she was regarded as female perfection, at least from the male perspective. (It would be a stretch to describe the statue as “monstrous”.) Yet a competition to find an actual woman who matched her dimensions did not succeed. Being precisely average is not the same thing as being perfect — but it is just as rare.
We no longer have to fall into this trap. It is still hard to personalise rather than standardise — but it is not impossible. Drugs are not best evaluated by the average effect over thousands of patients. Social care will fail if it is designed only for the average recipient, or education for the average pupil. A forecasting model that is correct on average may be a very dangerous model indeed.
Elizur Wright was quite correct to declare that a single life is uncertain; but we should never leap to the conclusion that an average life is ideal.
Written for and first published in the Financial Times on 5 July 2019.