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GAUSSIAN DISTRIBUTION

DOI: 10.1615/AtoZ.g.gaussian_distribution

The Gaussian distribution is a continuous distribution having the probability density function f(x) expressed as

where μ is the mean and σ is the standard deviation of the distribution. This distribution is also called the normal distribution and a random variable having this distribution is said to be normally distributed. The curve of f(x) is bell-shaped and is symmetric with respect to m as illustrated in Figure 1.

Curve of f(x).

Figure 1. Curve of f(x).

The corresponding probability distribution function F(x) is represented by the integral

which cannot be solved analytically but can be numerically evaluated for any x. Tabulated values of F(x) abound in standard textbooks on statistics. This is an important distribution because a host of random variables of practical interest such as the distribution of height and intelligence among males and females in an adult population, the lifetime of light bulbs and electric batteries, repeat measurements on a standard measurement, etc. are normal, approximately normal or can simply be converted into normal variables. The Gaussian distribution is also a useful approximation of more complicated distributions and it appears in the mathematical proofs of various statistical tests.

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