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The distribution of a random variable X for which the probability density function f is given by



The parameters μ and σ2 are, respectively, the mean and variance of the distribution. The distribution is denoted by N(μ, σ2). If the random variable X has such a distribution, then this is denoted by X ∼ N(μ, σ2) and the random variable may be referred to as a normal variable.

The graph of f(x) approaches the x-axis extremely quickly, and is effectively zero if |x−μ| < 3σ (hence the three-sigma rule). In fact, P(|X−μ| < 2σ)≈95.5% and P(|X−μ| < 3σ)≈99.7%. The first derivation of the form of f is believed to be that of de Moivre in 1733. The description 'normal distribution' was used by Galton in 1889, whereas 'Gaussian distribution' was used by Karl Pearson in 1905.

The normal distribution is the basis of a large proportion of statistical analysis. Its importance and ubiquity are largely a consequence of the Central Limit Theorem, which implies that averaging almost always leads to a bell-shaped distribution (hence the name 'normal'). See bell-curve.



Normal distribution. The diagram illustrates the probability density function of a normal random variable X having expectation μ and variance σ2. The distribution has mean, median, and mode at x=μ, where the density function has value 1/(σ√π). Note that almost all the distribution (99.7%) lies within 3σ of the central value.


The standard normal distribution has mean 0 and variance 1. A random variable with this distribution is often denoted by Z and we write Z ∼ N(0, 1). Its probability density function is usually denoted by ϕ and is given by



If X has a general normal distribution N(μ, σ2) then Z, defined by the standardizing transformation



has a standard normal distribution. It follows that the graph of the probability density function of X is obtained from the corresponding graph for Z by a stretch parallel to the z-axis, with centre at the origin and scale-factor σ, followed by a translation along the z-axis by μ.



Standard normal distribution. The distribution is centred on 0, with 99.7% falling between −3 and 3 and 95% falling between −1.96 and 1.96.


The cumulative probability function of Z is usually denoted by Φ and tables of values of Φ(z) are commonly available (see The Standard Normal Distribution Function; see also Upper-Tail Percentage Points for the Standard Normal Distribution). These tables usually give Φ(z) only for z>0, since values for negative values of z can be found using Φ(z)=1−Φ(−z).The tables can be used to find cumulative probabilities for X ∼ N(μ, σ2) via the standardizing transformation given above, since, for example,



As an example, if X ∼ N(7, 25) then the probability of X taking a value between 5 and 10 is given by



The normal distribution plays a central part in the theory of errors that was developed by Gauss. In the theory of errors, the error function (erf) is defined by erf(x)=2Φ(x√2)−1.An important property of the normal distribution is that any linear combination of independent normal variables is normal: if



are independent, and a and b are constants, thenaX1+bX2~N(aμ1+bμ2,a2σ21+b2σ22),with the obvious generalization to n independent normal variables. Many distributions can be approximated by a normal distribution for suitably large values of the relevant parameters. See also binomial distribution; chi-squared distribution; Poisson distribution; t-distribution.