Normal distribution

(Or Gaussian distribution.) The fundamental frequency distribution of statistical analysis.

A continuous variate x is said to have a normal distribution or to be normally distributed if it possesses a density function f(x) that satisfies the equation

where μ is the arithmetic mean (or first moment) and σ is the standard deviation. About two- thirds of the total area under the curve is included between x = μ − σ and x = μ + σ. The corresponding frequency distribution of vectors is the normal circular distribution in which the frequencies of vector deviations are represented by a series of circles centered on a vector mean. When applied to error distribution, this function is the normal law of errors, and the distribution is called the normal curve of error. Although discovered by DeMoivre, the normal distribution is usually called the Gaussian distribution. In early anthropometric studies and also investigations of random errors in physical measurements, the variates exhibited the normal distribution so faithfully that this distribution was mistakenly assumed to be the governing principle of nearly all random phenomena and was therefore given the name "normal." While less universal than formerly believed, the normal distribution does have remarkable breadth of application, inasmuch as the distribution of averages computed from repeated random samples of almost any population tends more and more nearly toward the "normal" form as the sample size increases. Formulated in precise terms, this proposition is known as the central limit theorem.