The normal distribution

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The normal distribution is a continuous probability distribution that has a bell-shaped probability density function:



It is usually considered to be the most prominent probability distribution in statistics partly because it arises in a very large number of contexts as a result of the central limit theorem and partly because it is relatively tractable analytically.


The normal distribution is also called the Gaussian distribution. The unit normal (or standard normal) distribution is .


Characteristics of the normal distribution are set out below:




Distribution name

Normal distribution

Common notation


 = scale parameter ()

 = location parameter


Probability density function

Cumulative distribution function




(Excess) kurtosis

Characteristic function

Other comments

The inverse unit normal distribution function (i.e. its quantile function) is commonly written  (also in some texts  and the unit normal density function is commonly written .  is also called the probit function.


The error function distribution is , where  is now an inverse scale parameter .


The median and mode of a normal distribution are .


The truncated first moments of  are:



where  and  are the pdf and cdf of the unit normal distribution respectively.


The mean excess function of a standard normal distribution is thus


The central moments of the normal distribution are:


Nematrian web functions


Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “normal”. For details of other supported probability distributions see here.


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