Gaussian Distribution

The normal distribution or Gaussian distribution is a continuous probability distribution that often gives a good description of data that cluster around the mean. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known as the Gaussian function or bell curve. The Gaussian distribution is one of many things named after Carl Friedrich Gauss, who used it to analyze astronomical data, and determined the formula for its probability density function. However, Gauss was not the first to study this distribution or the formula for its density function that had been done earlier by Abraham de Moivre.

The normal distribution is often used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the heights of adult males in the United States are roughly normally distributed, with a mean of about 70 inches (1.8 m). Most men have a height close to the mean, though a small number of outliers have a height significantly above or below the mean. A histogram of male heights will appear similar to a bell curve, with the correspondence becoming closer if more data are used. By the central limit theorem, under certain conditions the sum of a number of random variables with finite means and variances approaches a normal distribution as the number of variables increases. For this reason, the normal distribution is commonly encountered in practice, and is used throughout statistics, natural science, and social science as a simple model for complex phenomena. For example, the observational error in an experiment is usually assumed to follow a normal distribution, and the propagation of uncertainty is computed using this assumption.

The simplest case of a normal distribution is known as the standard normal distribution, described by the probability density function

The constant 1/√2π in this expression ensures that the total area under the curve φ (x) is equal to one, and 1/2 in the exponent makes the width of the curve (measured as half of the distance between the inflection points of the curve) also equal to one. It is traditional in statistics to denote this function with the Greek letter φ(phi) whereas density functions for all other distributions are usually denoted with letters f or p. The alternative glyph φ is also used quite often, however within this article we reserve φ to denote characteristic functions.