# Normal distribution

Parameters Probability density functionThe green line is the standard normal distribution Cumulative distribution functionColors match the image above ${\displaystyle \mu }$ location (real)${\displaystyle \sigma ^{2}>0}$ squared scale (real) ${\displaystyle x\in \mathbb {R} \!}$ ${\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}\;\exp \left(-{\frac {\left(x-\mu \right)^{2}}{2\sigma ^{2}}}\right)\!}$ ${\displaystyle {\frac {1}{2}}\left(1+\mathrm {erf} \,{\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\!}$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \sigma ^{2}}$ 0 0 ${\displaystyle \ln \left(\sigma {\sqrt {2\,\pi \,e}}\right)\!}$ ${\displaystyle M_{X}(t)=\exp \left(\mu \,t+{\frac {\sigma ^{2}t^{2}}{2}}\right)}$ ${\displaystyle \chi _{X}(t)=\exp \left(\mu \,i\,t-{\frac {\sigma ^{2}t^{2}}{2}}\right)}$

The normal distribution is a probability distribution. It is also called Gaussian distribution because it was first discovered by Carl Friedrich Gauss.[1] The normal distribution is a continuous probability distribution that is very important in many fields of science.

Normal distributions are a family of distributions of the same general form. These distributions differ in their location and scale parameters: the mean ("average") of the distribution defines its location, and the standard deviation ("variability") defines the scale. These two parameters are represented by the symbols ${\displaystyle \mu }$ and ${\displaystyle \sigma }$, respectively.[2]

The standard normal distribution (also known as the Z distribution) is the normal distribution with a mean of zero and a standard deviation of one (the green curves in the plots to the right).[2] It is often called the bell curve, because the graph of its probability density looks like a bell.

Many values follow a normal distribution. This is because of the central limit theorem, which says that if an event is the sum of identical but random events, it will be normally distributed.[3] Some examples include:[4]

• Height
• Test scores
• Measurement errors
• Light intensity (so-called Gaussian beams, as in laser light)
• Intelligence is probably normally distributed. There is a problem with accurately defining or measuring it, though.
• Insurance companies use normal distributions to model certain average cases.

## References

1. Kirkwood, Betty R; Sterne, Jonathan AC (2003). Essential Medical Statistics. Blackwell Science Ltd.CS1 maint: multiple names: authors list (link)
2. "List of Probability and Statistics Symbols". Math Vault. 2020-04-26. Retrieved 2020-08-15.
3. Weisstein, Eric W. "Normal Distribution". mathworld.wolfram.com. Retrieved 2020-08-15.
4. "Normal Distribution". www.mathsisfun.com. Retrieved 2020-08-15.