Editing Normal distribution (section)

{{Short description|Probability distribution}}
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{{Use mdy dates|date=August 2012}}
{{Infobox probability distribution
  | name       = Normal distribution
  | type       = density
  | pdf_image  = [[Image:Normal Distribution PDF.svg|400px|class=skin-invert-image]]
  | pdf_caption = The red curve is the [[#Standard normal distribution|''standard normal distribution'']].
  | cdf_image  = [[Image:Normal Distribution CDF.svg|400px|class=skin-invert-image]]
  | cdf_caption =
  | notation   = <math>\mathcal{N}(\mu,\sigma^2)</math>
  | parameters = <math>\mu\in\R</math> = [[mean]] ([[location parameter|location]])<br /><math>\sigma^2\in\R_{>0}</math> = [[variance]] (squared [[scale parameter|scale]])
  | support    = <math>x\in\R</math>
  | pdf        = <math>\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}</math>
  | cdf        = <math>\Phi\left(\frac{x-\mu}{\sigma}\right) = \frac{1}{2}\left[1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right]</math>
  | quantile   = <math>\mu+\sigma\sqrt{2} \operatorname{erf}^{-1}(2p-1)</math>
  | mean       = <math>\mu</math>
  | median     = <math>\mu</math>
  | mode       = <math>\mu</math>
  | variance   = <math>\sigma^2</math>
  | mad        = <math>\sigma\sqrt{2}\,\operatorname{erf}^{-1}(1/2)</math>
  | aad        = <math display="inline">\sigma\sqrt{2/\pi}</math>
  | skewness   = <math>0</math>
  | kurtosis   = <math>0</math> <!-- DO NOT REPLACE THIS WITH THE OLD-STYLE KURTOSIS WHICH IS 3. -->
  | entropy    = <math display="inline">\tfrac{1}{2} \log(2\pi e \sigma^2)</math>
  | mgf        = <math>\exp(\mu t + \sigma^2 t^2 / 2)</math>
  | char       = <math>\exp(i \mu t - \sigma^2 t^2 / 2)</math>
  | fisher     = <math>\mathcal{I}(\mu,\sigma) =\begin {pmatrix} 1/\sigma^2 & 0 \\ 0 & 2/\sigma^2\end{pmatrix}</math>
<math>\mathcal{I}(\mu,\sigma^2) =\begin {pmatrix} 1/\sigma^2 & 0 \\ 0 & 1/(2\sigma^4)\end{pmatrix}</math>
  | KLDiv      = <math>{ 1 \over 2 } \left\{ \left( \frac{\sigma_0}{\sigma_1} \right)^2 + \frac{(\mu_1 - \mu_0)^2}{\sigma_1^2} - 1 + \ln {\sigma_1^2 \over \sigma_0^2} \right\}</math>
  | ES         = <math>\mu + \sigma \frac{\frac{1}{\sqrt{2\pi}} e^{\frac{-\left(q_p\left(\frac{X-\mu}{\sigma}\right)\right)^2}{2}}}{1-p}</math><ref name="Norton-2019">{{cite journal |last1=Norton |first1=Matthew |last2=Khokhlov |first2=Valentyn |last3=Uryasev |first3=Stan |year=2019 |title=Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation |journal=Annals of Operations Research |volume=299 |issue=1–2 |pages=1281–1315 |publisher=Springer |doi=10.1007/s10479-019-03373-1 |arxiv=1811.11301 |s2cid=254231768 |url=http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |access-date=2023-02-27 |archive-date=March 31, 2023 |archive-url=https://web.archive.org/web/20230331230821/http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |url-status=dead }}</ref>
}}
{{Probability fundamentals}}

In [[probability theory]] and [[statistics]], a '''normal distribution''' or '''Gaussian distribution''' is a type of [[continuous probability distribution]] for a [[real number|real-valued]] [[random variable]]. The general form of its [[probability density function]] is<ref name="The Joy of Finite Mathematics">{{Cite book |last=Tsokos |first=Chris |url=https://linkinghub.elsevier.com/retrieve/pii/B9780128029671000073 |title=The Joy of Finite Mathematics |last2=Wooten |first2=Rebecca |date=2016-01-01 |publisher=Academic Press |isbn=978-0-12-802967-1 |editor-last=Tsokos |editor-first=Chris |location=Boston |pages=231–263 |doi=10.1016/b978-0-12-802967-1.00007-3 |editor-last2=Wooten |editor-first2=Rebecca}}</ref><ref name="Mathematics for Physical Science and Engineering">{{Cite book |last=Harris |first=Frank E. |url=https://linkinghub.elsevier.com/retrieve/pii/B9780128010006000183 |title=Mathematics for Physical Science and Engineering |date=2014-01-01 |publisher=Academic Press |isbn=978-0-12-801000-6 |editor-last=Harris |editor-first=Frank E. |location=Boston |pages=663–709 |doi=10.1016/b978-0-12-801000-6.00018-3}}</ref><ref>{{harvtxt|Hoel|1947|loc=[https://archive.org/details/in.ernet.dli.2015.263186/page/n39/mode/2up?q=%22normal+distribution%22 p. 31]}} and {{harvtxt|Mood|1950|loc=[https://archive.org/details/introductiontoth0000alex/page/108/mode/2up?q=%22normal+distribution%22 p. 109]}} give this definition with slightly different notation.</ref>
<math display=block>
f(x) = \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{(x-\mu)^2}{2\sigma^2}}\,.
</math>

The parameter {{tmath|\mu}} is the [[Mean#Mean of a probability distribution|mean]] or [[expected value|expectation]] of the distribution (and also its [[median]] and [[mode (statistics)|mode]]), while the parameter <math display=inline>\sigma^2</math> is the [[variance]]. The [[standard deviation]] of the distribution is {{tmath|\sigma}} (sigma). A random variable with a Gaussian distribution is said to be '''normally distributed''', and is called a '''normal deviate'''.

Normal distributions are important in [[statistics]] and are often used in the [[natural science|natural]] and [[social science]]s to represent real-valued [[random variable]]s whose distributions are not known.<ref>[http://www.encyclopedia.com/topic/Normal_Distribution.aspx#3 ''Normal Distribution''], Gale Encyclopedia of Psychology</ref><ref>{{harvtxt |Casella |Berger |2001 |p=102 }}</ref> Their importance is partly due to the [[central limit theorem]]. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution [[convergence in distribution|converges]] to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as [[measurement error]]s, often have distributions that are nearly normal.<ref>Lyon, A. (2014). [https://aidanlyon.com/normal_distributions.pdf Why are Normal Distributions Normal?], The British Journal for the Philosophy of Science.</ref>

Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any [[linear combination]] of a fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as [[propagation of uncertainty]] and [[least squares]]<ref>{{Cite book |last1=Jorge |first1=Nocedal |title=Numerical Optimization |last2=Stephan |first2=J. Wright |publisher=Springer |year=2006 |isbn=978-0387-30303-1 |edition=2nd |pages=249}}</ref> parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed.

A normal distribution is sometimes informally called a '''bell curve'''.<ref name="www.mathsisfun.com">{{Cite web|title=Normal Distribution|url=https://www.mathsisfun.com/data/standard-normal-distribution.html|access-date=2020-08-15|website=www.mathsisfun.com}}</ref><ref>{{cite web |title=bell curve |website=Merriam-Webster.com Dictionary |url=https://www.merriam-webster.com/dictionary/bell%20curve |access-date=25 May 2025}}</ref> However, many other distributions are [[Bell-shaped function|bell-shaped]] (such as the [[Cauchy distribution|Cauchy]], [[Student's t-distribution|Student's ''t'']], and [[logistic distribution|logistic]] distributions). (For other names, see ''[[#Naming|Naming]]''.)

The [[univariate distribution|univariate probability distribution]] is generalized for [[Vector (mathematics and physics)|vectors]] in the [[multivariate normal distribution]] and for matrices in the [[matrix normal distribution]].