# What is meant by Gauss, distributed

## Normal distribution

*Gaussian normal distribution;* an especially important probability distribution in inferential statistics named after C.F. Gauss. The density function of a normal distribution with the parameters μ and σ^{2}> 0 has the form

for -∞

1. The parameters of the normal distribution are the expectation μ and the variance σ^{2}. Using the standard transformation, normal distributions with any parameter position can be incorporated into the *Standard normal distribution* (μ = 0; σ^{2} = 1) can be transferred. If the random variable X has a normal distribution with the parameters μ and σ^{2}, then the random variable (X-μ) / σ has a standard normal distribution. For the evaluation of the density function or distribution function of the standard normal distribution exist *Tables,* in which values of the distribution function are recorded. The tables of the standard normal distribution can therefore be used to evaluate any normal distribution.

2. *Properties:* When graphically represented, the density function of a normal distribution results in a bell-shaped curve that is symmetrical to the straight line x = μ. The expected value μ coincides with the mode and the median. The bell curve has turning points at the abscissas μ + σ and μ - σ. For a μ-normally distributed random variable X, the following applies (rounded values): P (μ - σ ≤ X ≤ μ + σ) = 0.6827; P (µ - 2σ X µ + 2σ) = 0.9545; P (μ - 3σ ≤ X ≤ μ + 3σ) = 0.9973.

See figure "normal distribution".

3. *Importance:* Approximately normally distributed characteristics are occasionally to be observed in the economy and in the technical-scientific field. For example, if the sample size is large, the sample average (arithmetic mean) can be regarded as approximately normally distributed under suitable conditions even if nothing is known about the distribution of the population. This can be justified by the central limit value theorem. Under certain conditions, the normal distribution is used to approximate many theoretical distributions, such as the binomial distribution, the hypergeometric distribution, the Poisson distribution or the chi-square distribution.

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