Normally distributed numbers

The curve arrived before the man it is named after. Abraham de Moivre, a Huguenot who left France after the revocation of the Edict of Nantes and spent the rest of his life tutoring in London coffee houses, wanted a way to approximate the enormous binomial coefficients that appear when you ask how likely a fair coin is to land heads within a given range in a thousand tosses. In a Latin pamphlet dated 12 November 1733, reprinted in the 1738 second edition of The Doctrine of Chances, he produced the approximation — an exponential of a negative square — along with the observation that the spread grows with the square root of the number of trials. Pierre-Simon Laplace generalised the argument into what is now the central limit theorem, publishing the decisive memoir in 1810.

Carl Friedrich Gauss reached the same function from the other direction. In Theoria Motus Corporum Coelestium (1809) his concern was astronomical measurement — he had made his name in 1801 by computing where the newly lost asteroid Ceres would reappear — and he asked which law of errors would make the familiar arithmetic mean the best estimate of a quantity. The answer was the same bell-shaped curve, which he tied to the method of least squares. Adrien-Marie Legendre had published least squares first, in 1805, and the priority dispute that followed was bitter; Gauss insisted he had been using the method since 1795. Stephen Stigler later made the episode a specimen of what he called the law of eponymy.

The word "normal" came later still, used independently in the 1870s by Charles Sanders Peirce, Francis Galton and Wilhelm Lexis, then cemented by Karl Pearson — who publicly regretted it, since it implies every other distribution is abnormal. Galton supplied the physical demonstration: the quincunx, a board of staggered pins down which lead shot fell into columns and piled up into a bell.

Generating such values on a computer is newer. In 1958 George Box and Mervin Muller published a two-page note in the Annals of Mathematical Statistics showing that a pair of uniform random numbers becomes a pair of independent normal ones with a logarithm, a square root and a cosine. That transform is what this page runs.

Key properties

• The density is f(x) = (1/(σ√(2π)))·e^(−(x−μ)²/(2σ²)); it is symmetric about μ, so the mean, median and mode all coincide there.
• About 68.27% of values fall within one standard deviation of the mean, 95.45% within two and 99.73% within three.
• The quartiles sit at μ ± 0.6745σ, which makes the interquartile range roughly 1.349σ.
• Skewness and excess kurtosis are both exactly zero, and among all distributions with a given mean and variance the normal has the largest differential entropy.
• Adding independent normal variables gives another normal one: N(μ₁, σ₁²) + N(μ₂, σ₂²) = N(μ₁+μ₂, σ₁²+σ₂²).
• The cumulative distribution function has no elementary closed form; it is written with the error function as Φ(x) = ½·[1 + erf(x/√2)].
• Box–Muller turns independent uniforms U₁, U₂ on (0,1) into two independent standard normals, √(−2·ln U₁)·cos(2πU₂) and √(−2·ln U₁)·sin(2πU₂); this page uses the cosine branch.
• By the central limit theorem, the standardised sum of many independent variables with finite variance tends to this distribution whatever the variables themselves look like.

Where they turn up

• Measurement error in physical experiments is routinely modelled as normal, which is why least squares — the fitting method Gauss justified with this curve — underlies most curve fitting and regression.
• Thermal (Johnson–Nyquist) noise in a resistor is Gaussian, and "additive white Gaussian noise" is the standard channel model in communications engineering.
• The displacement of a Brownian particle after a fixed time is normally distributed, the result Einstein derived in 1905 and the defining property of the Wiener process.
• Statistical process control and the Six Sigma programmes built on it measure tolerance in standard deviations from the mean — a management convention that assumes a roughly normal process.
• IQ scores are normal by construction rather than by discovery: tests such as the Wechsler scales are normed so that scores have mean 100 and standard deviation 15.
• The Black–Scholes option-pricing model assumes normally distributed log-returns; observed market returns have markedly fatter tails, so the assumption is known to understate extreme moves.