Arithmetic mean

The arithmetic mean seems so fundamental that its emergence as a formal concept startles: humanity used numbers for millennia before anyone thought to average them systematically. The transformation began in seventeenth-century England, where the convergence of mortality data, commercial insurance, and nascent statistical thinking created the adjacent possible for conceiving of populations in aggregate terms.

Before averages, each person and each measurement existed individually. A haberdasher named John Graunt changed this in 1662 with his *Natural and Political Observations on the London Bills of Mortality*, the founding document of demography. Graunt applied what he called "the Mathematiques of my Shop-Arithmetique"—division for ratios, addition and subtraction for tracking changes, and crucially, averaged annual valuations to smooth out random variations. From parish death records, he constructed the first life table, transforming individual deaths into statistical patterns.

The adjacent possible for this conceptual leap required several elements. Long division, refined over centuries from Arabic mathematics through European computation, provided the mechanical capability to calculate means. Double-entry bookkeeping, widespread in commercial England, habituated merchants to aggregate thinking. The Bills of Mortality themselves—weekly parish records of deaths by cause—created the raw data that demanded summarization. And the Royal Society, founded in 1660, provided institutional validation for empirical analysis of social phenomena.

The geographical specificity of the arithmetic mean's emergence reflects England's particular circumstances. London's population, swollen to perhaps 400,000 by the 1660s, generated mortality data at scales that demanded statistical treatment. The insurance industry, centered on coffee houses like Lloyd's, needed methods to price annuities and life insurance. And English empiricism, championed by Francis Bacon and institutionalized in the Royal Society, encouraged the application of mathematical methods to practical problems.

What makes this story remarkable is who accomplished it. Statistical methods emerged from relatively low-status people working on practical problems. Graunt was a haberdasher by trade, not a university mathematician. Edmund Halley, though properly trained as an astronomer, developed his life tables in 1693 while working on the unfashionable topic of annuity pricing. The concept of averaging developed not from pure mathematical insight but from commercial necessity.

The revolution that averages enabled cannot be overstated. Before Graunt, life insurance was sold at fixed prices regardless of age—the actuarial absurdity that a twenty-year-old and a sixty-year-old would pay identical premiums for identical coverage. You cannot price life insurance rationally without life expectancy, and you cannot calculate life expectancy without the concept of an average. Halley's 1693 actuarial tables put annuity sales on sound mathematical footing for the first time, enabling the insurance industry that would eventually underwrite global commerce.

The mean enabled statistical thinking to spread across domains. William Petty's "political arithmetic" applied averaging to national economic analysis. Subsequent generations extended the technique to astronomy (averaging multiple observations to reduce measurement error), physics, medicine, and eventually social science. The normal distribution, central to modern statistics, describes how individual measurements cluster around their mean.

Yet the average also enabled characteristic errors. Adolphe Quetelet's nineteenth-century "average man" treated statistical constructs as ideal types, confusing description with prescription. The contemporary critique that averages obscure important variation—that no one is actually average—echoes through debates about educational standards, medical treatments, and public policy.

By 2026, averaging pervades every quantitative domain, from sports statistics to climate modeling to machine learning loss functions. The concept appears so natural that its historical emergence seems almost incomprehensible—how could humanity have calculated for thousands of years without systematically averaging? The answer lies in the adjacent possible: until mortality data demanded summarization and commercial insurance required pricing, there was no pressing need to conceive of populations in aggregate terms. John Graunt's shop arithmetic, applied to parish death records, opened a door that mathematics had not thought to approach.