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Portrait of The normal distribution

Science · Mathematics

The normal distribution

1733–1809

Mathematics — probability and statistics

The symmetrical bell-shaped probability distribution defined by just two parameters, its mean and its standard deviation. Abraham de Moivre derived it in 1733 as an approximation to the binomial distribution; Laplace and Carl Friedrich Gauss developed it for the theory of errors in astronomical measurement, and it is usually called the Gaussian distribution outside Britain. Its central importance comes from the central limit theorem: the sum of many independent random influences tends towards this shape whatever their individual distributions, which is why heights, measurement errors and examination marks so often follow it.

How to spot them

  • ▸The symmetrical bell curve, defined by a mean and a standard deviation
  • ▸Mean, median and mode all coincide at the peak
  • ▸The 68–95–99.7 rule: that percentage of data within one, two and three standard deviations
  • ▸The central limit theorem is why it turns up everywhere
  • ▸Was printed on the German ten-mark note beside the man usually credited with it

Key facts

De Moivre's 1733 approximation to the binomial distribution for large numbers of trials (1733)

Abraham de Moivre, in a privately circulated pamphlet, later in The Doctrine of Chances

The theorem that sums of many independent random effects tend to one particular shape

The central limit theorem, named by George Pólya in 1920

The rule of thumb 68–95–99.7, counting standard deviations from the mean

The proportion of values within one, two and three standard deviations

A German ten-mark banknote showing a bell curve beside a portrait of its mathematician (1991)

The Deutsche Mark note bearing Carl Friedrich Gauss, in circulation from 1991 to 2001

Quetelet's 'average man', built by applying astronomers' error theory to human measurements (1835)

Adolphe Quetelet's l'homme moyen, the beginning of social statistics

Galton's quincunx, a board of pins down which beads fall into a bell-shaped heap

Francis Galton's bean machine, a physical demonstration of the central limit theorem

The version with mean 0 and standard deviation 1, tabulated at the back of every textbook

The standard form, and the z-score that converts any value into it

The curve a manufacturer aims to keep six standard deviations inside the tolerance limits

Six Sigma, the quality-control methodology developed at Motorola in the 1980s

Key ideas

  • Two numbers fix the whole curve: the mean locates the peak and the standard deviation sets the width. Subtracting the mean and dividing by the standard deviation gives the standard form and a z-score.
  • The central limit theorem is the reason for its ubiquity: add up enough independent random contributions, of almost any kind, and the total is approximately bell-shaped.
  • It is a poor model for anything with fat tails — incomes, city sizes, market crashes — where power laws or the log-normal distribution are better. Misapplying it was a notable ingredient of the 2008 financial crisis.

Quick-fire facts

  • •Abraham de Moivre (1667–1754), a Huguenot refugee in London, derived the curve in 1733 as a limit of the binomial distribution
  • •Carl Friedrich Gauss used it in 1809 in Theoria motus corporum coelestium, for errors in astronomical observation; Laplace supplied the central limit theorem
  • •Outside Britain it is usually called the Gaussian distribution; Karl Pearson popularised the English name in the 1890s
  • •Its density involves both the circle constant and the base of natural logarithms, in the factor 1/(σ√(2π))
  • •Mean, median and mode coincide; skewness is zero and kurtosis is three
  • •About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three
  • •Adolphe Quetelet applied it to human measurements from 1835, inventing the 'average man' and what became the body mass index
  • •Francis Galton built the quincunx or bean machine to demonstrate it, and coined 'regression to the mean' while studying heights
  • •A German ten-mark note issued from 1991 carried Gauss's portrait alongside the curve and its formula
  • •Five standard deviations, about one chance in 3.5 million, is the conventional threshold for claiming a discovery in particle physics
  • •The log-normal distribution, whose logarithm is normally distributed, models quantities that cannot be negative, such as incomes

Drill it — 9 cards

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Sources