The invention of the average

Economics, medicine, social and natural sciences would all be impossible without the concept of the ‘average’. Why then did it take historians so long to trace its origin to the seventeenth century?

An orrery designed by the Danish astronomer Tycho Brahe, c1630.
An orrery designed by the Danish astronomer Tycho Brahe, c1630. Credit: Heritage Image Partnership Ltd / Alamy Stock Photo

‘The past is a foreign country: they do things differently there.’ Historians love this quotation from L.P. Hartley, but it is easy to forget just how different the past was from the present. The latest reminder comes in an article by John E. Crowley, published in the June issue of the American Historical Review, ‘How Averages Became Normal’. It’s a remarkable article, exemplary in its clarity, in its learning lightly carried, and above all because it is profound and provocative. So, though it is conventional to review books, not articles, here’s a review of an article.

As it happens I know Jack Crowley – more than twenty-five years ago we were colleagues together in Nova Scotia. He is a rare case of an historian who is numerate, literate, and interested in visual evidence. He has worked on lighting and mirrors in the eighteenth century, on representations of sugar production (and so of slavery), on inheritance in eighteenth-century South Carolina, on the work ethic in eighteenth-century America, on paintings of British imperial landscapes. There’s not much of a pattern here: he works on eighteenth-century topics that catch his attention. But this idiosyncratic range of interests somehow prepared him to grasp something that no one else (or almost no one else) had grasped before: there were no averages before the seventeenth century.

According to Crowley, the best discussion of the subject before his own was Churchill Eisenhart’s 1971 American Statistical Association Presidential address, ‘The Development of the Concept of the Best Mean of a Set of Measurements from Antiquity to the Present Day’, a text which exists only as a typescript, but which has recently made its way onto the internet. Whether we attribute the discovery of the invention of the average to Eisenhart or to Crowley doesn’t really matter: as R.K. Merton demonstrated, all great discoveries can usually be claimed by more than one person. But in Merton’s sociological account of discovery, competing claims tend to be nearly contemporary. How can we explain the gap of more than fifty years between Eisenhart’s ignored discovery and Crowley’s remarkable rediscovery?

Why do averages matter? Averages, in Crowley’s formulation, synthesise singularities into generalities; and prediction (outside the narrow world of gambling on cards or dice, where the odds are predetermined) depends on the production of such generalities. Without prediction there can be no applied or social sciences. If you don’t have the concept of an average then you really aren’t going to be able to do economics, demography, evidence-based medicine, or any other social or natural science which deals with quantitative information which varies.

Thus, take life insurance. Until late in the seventeenth century life insurance was sold at a fixed price, like bread or beer. No matter how old you were you paid the same for the insurance, and you received the same pay out as anyone else purchasing that amount of insurance. You can’t do life insurance, as we understand it, until you have the concept of a life expectancy; and you can’t have the concept of a life expectancy until you have the concept of an average. Edmund Halley, he of Halley’s comet, was the first to put the sale of annuities (in effect a form of life insurance) on a sound basis (1693).

Or take economics. In 1970 Moses Finley asked why the ancient Greeks never invented the discipline of economics. His preferred explanation was that slavery got in the way, but he also remarked: ‘I would be prepared to argue that without the concept of relevant “laws” (or ‘statistical uniformities’ if one prefers) it is not possible to have a concept of “the economy”.’ And to that we can now add: Without the concept of an average it is not possible to have a concept of a statistical uniformity. The ancient Greeks couldn’t invent economics because they hadn’t invented averages.

The word ‘average’ occurs 133 times in Adam Smith’s Wealth of Nations (1776). Smith simply takes the concept for granted, and all the historians of political economy have done the same. Compare Smith with Cantillon, whose Essai sur la nature du commerce en générale was written between 1730 and 1734. Cantillon never uses a word corresponding to ‘average’, but he certainly has the concept – he often uses the term en générale to convey it; and he says prices in the marketplace vary from day to day with a continuous flux and reflux – this tidal metaphor implies an average price or ‘intrinsic value’ concealed beneath the daily variation. This ‘intrinsic value’ Smith called ‘the central price, to which the prices of all commodities are continually gravitating’. But he also calls it ‘the average or ordinary price’. For economic theory to take off, averages have to become ordinary, and, in principle, calculable.

Averages are so commonplace in our world, so straightforward and obvious, that it is difficult to get one’s head around the notion that the concept is an invention. Alasdair MacIntyre wrote: ‘Facts, like telescopes and wigs for gentlemen, were a seventeenth-century invention.’ Facts (which we also take for granted) and averages were invented around the same time. Hand in hand they transformed the intellectual universe, and mark the key moment of transition from ancient and medieval mental tools to modern ones.

To produce averages you need to add up a set of figures and then divide by the number of figures in the set. In simple terms, averages require long division. Long division isn’t an elementary skill. It’s impossible if you use roman numerals, but you can do it with hindu-arabic numbers or with an abacus. It was taught as part of relatively advanced commercial arithmetic in Renaissance Italy, but in the seventeenth century most graduates of Oxford and Cambridge would have been incapable of it. They knew their Euclid, but their arithmetic was solely concerned with addition, subtraction, and multiplication. William Petty, who is often counted the founder of political arithmetic, wasn’t interested in calculating averages: he preferred to produce guesstimates, which he thought ought to be good enough. Real averages, outside a few usages by mathematicians trying to average out measurements that differed from instrument to instrument and day to day, seem to begin with the life expectancy tables of John Graunt (1662).

Let’s pause for a moment over the mathematicians. Tycho Brahe averaged out his astronomical measurements, but didn’t explain to his readers what he was doing. Edward Wright (1610) averaged measurements of magnetic declination in the hope of using them to calculate latitude. But the mathematical manipulation of figures in this way was hardly respectable. Galileo said that the universe ‘is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures’. In other words, for Galileo, science ought to have the deductive certainty of geometry. It never occurred to Galileo to say that the universe was written in the language of arithmetic, of number, and his own measurements (including those which ‘proved’ the law of fall) were haphazard and inaccurate. The rise of the average thus marks the triumph of inductive over deductive knowledge.

Crowley provides an elegant (and beautifully illustrated) history of long division within mathematical education; a brief history of the rise of inductive knowledge; and a learned discourse on the dissemination of the word ‘average’, which derives from a word for the share of unexpected costs (such as wastage) to be born by the partners in an enterprise. The word ‘average’, as opposed to alternatives such as ‘mean’ and ‘medium’, took off in the period between 1690 and 1720. With the South Sea Company crisis of 1720, the first great commercial bubble, the word entered everyday speech.

Crowley thus explains why averages didn’t exist until the late seventeenth century. But why is it only now that they have found their historian? Ian Hacking’s great book The Emergence of Probability was published in 1975. ‘Probability’, Hacking argued, was a fundamentally new concept in the seventeenth century. By 1988 it was obvious to MacIntyre that facts, too, were a seventeenth-century invention. Historians have gone on to write about the histories of other key terms – hypothesis, evidence, experiment, and so on. But nobody had anything useful to say in print about averages. I don’t mean in any way to downplay the brilliance of Crowley’s article; but it’s easy to think of quite a long list of people who ought to have written it before him: Hacking, of course; Lorraine Daston, who followed in Hacking’s footsteps; historians of statistics, such as Stephen Stigler; the various historians of ‘the fact’ (Poovey, Shapin, Shapiro, myself). None saw the significance of the average.

I fear the explanation for this delay is obvious. Averages were not invented by Pascal, Leibniz, or Newton. They were invented by relatively low-status people (Graunt was a haberdasher), or by proper mathematicians, such as Halley, but when working on low-status topics, such as magnetic declination or life expectancy. The history of the average has been ignored because it was invisible; and it was invisible because to explore it you need to read elementary mathematical textbooks, texts on navigation, and endless pamphlets on the South Sea bubble. The historians of science and the historians of philosophy have passed it by because they were looking for intellectual breakthroughs in the publications of major intellectuals, and they assumed they already knew who those people were. In other words they were trapped by what literary scholars call ‘the canon’.

Indeed, one might say that Crowley’s discovery isn’t only delayed, but outmoded. From the early seventies until the early noughties there was a linguistic, textual, or cultural turn in historical writing; but now we have what is called ‘the new materialism’. An exemplary case: consider Lisa Jardine’s evolution from Erasmus Man of Letters (1993) to Worldly Goods (1998). Objects have replaced words: Crowley’s own work on creature comforts is an example of this. Averages may be objective, but they aren’t material.

Jack Crowley was trained as a social historian, not a historian of science or philosophy. Social historians don’t have a canon. He likes statistics. The language of commerce is something he has studied. He is at home with visual evidence: to understand the difficulties of long division you have to look at how it was taught in early modern textbooks, and how calculations were actually performed on paper. You have to see it on the page. And so it is precisely because his background was not the history of science, or philosophy, or mathematics (though he is evidently familiar with all those subjects), that he could see what nobody else could see: averages, like telescopes and wigs for gentlemen, are a seventeenth-century invention. And because we couldn’t see their novelty, we couldn’t even begin to think about their importance.


David Wootton