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{{Short description|Probability distribution}}
{{Redirect|Bell curve}}
{{More citations needed|date=December 2024}}
{{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''.
  | 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>
<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> 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]].

== Definitions ==
=== Standard normal distribution ===
The simplest case of a normal distribution is known as the '''standard normal distribution''' or '''unit normal distribution'''. This is a special case when <math display=inline>\mu=0</math> and <math display=inline>\sigma^2 =1</math>, and it is described by this [[probability density function]] (or density):
<math display="block">\varphi(z) = \frac{e^\frac{-z^2}{2}}{\sqrt{2\pi}}\,.</math>
The variable {{tmath|z}} has a mean of 0 and a variance and standard deviation of 1. The density <math display=inline>\varphi(z)</math> has its peak <math display="inline">\frac{1}{\sqrt{2\pi}}</math> at <math display=inline>z=0</math> and [[inflection point]]s at <math display=inline>z=+1</math> and {{tmath|1=z=-1}}.

Although the density above is most commonly known as the ''standard normal,'' a few authors have used that term to describe other versions of the normal distribution. [[Carl Friedrich Gauss]], for example, once defined the standard normal as
<math display=block>\varphi(z) = \frac{e^{-z^2}}{\sqrt\pi},</math>
which has a variance of {{tmath|\frac{{mset|1}}{{mset|2}}}}, and [[Stephen Stigler]]<ref>{{harvtxt |Stigler |1982 }}</ref> once defined the standard normal as
<math display=block>\varphi(z) = e^{-\pi z^2},</math>
which has a simple functional form and a variance of <math display=inline>\sigma^2 = \frac {1}{2\pi}.</math>

=== General normal distribution ===
Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor {{tmath|\sigma}} (the standard deviation) and then translated by {{tmath|\mu}} (the mean value):

<math display=block>
f(x \mid \mu, \sigma^2) =\frac 1 \sigma \varphi\left(\frac{x-\mu} \sigma \right)\,.
</math>

The probability density must be scaled by <math display=inline>1/\sigma</math> so that the [[integral]] is still&nbsp;1.

If {{tmath|Z}} is a [[standard normal deviate]], then <math display=inline>X=\sigma Z + \mu</math> will have a normal distribution with expected value {{tmath|\mu}} and standard deviation {{tmath|\sigma}}. This is equivalent to saying that the standard normal distribution {{tmath|Z}} can be scaled/stretched by a factor of {{tmath|\sigma}} and shifted by {{tmath|\mu}} to yield a different normal distribution, called {{tmath|X}}. Conversely, if {{tmath|X}} is a normal deviate with parameters {{tmath|\mu}} and <math display=inline>\sigma^2</math>, then this {{tmath|X}} distribution can be re-scaled and shifted via the formula <math display=inline>Z=(X-\mu)/\sigma</math> to convert it to the standard normal distribution. This variate is also called the standardized form of {{tmath|X}}.

=== Notation ===
The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter {{tmath|\phi}} ([[phi]]).<ref>{{harvtxt |Halperin |Hartley |Hoel |1965 |loc=item 7 }}</ref> The alternative form of the Greek letter phi, {{tmath|\varphi}}, is also used quite often.

The normal distribution is often referred to as <math display=inline>N(\mu,\sigma^2)</math> or {{tmath|\mathcal{N}(\mu,\sigma^2)}}.<ref>{{harvtxt |McPherson |1990 |p=110 }}</ref> Thus when a random variable {{tmath|X}} is normally distributed with mean {{tmath|\mu}} and standard deviation {{tmath|\sigma}}, one may write

<math display=block>X \sim \mathcal{N}(\mu,\sigma^2).</math>

=== Alternative parameterizations ===
Some authors advocate using the [[precision (statistics)|precision]] {{tmath|\tau}} as the parameter defining the width of the distribution, instead of the standard deviation {{tmath|\sigma}} or the variance {{tmath|\sigma^2}}. The precision is normally defined as the reciprocal of the variance, {{tmath|1/\sigma^2}}.<ref>{{harvtxt |Bernardo |Smith |2000 |page=121 }}</ref> The formula for the distribution then becomes

<math display=block>f(x) = \sqrt{\frac\tau{2\pi}} e^{-\tau(x-\mu)^2/2}.</math>

This choice is claimed to have advantages in numerical computations when {{tmath|\sigma}} is very close to zero, and simplifies formulas in some contexts, such as in the [[Bayesian statistics|Bayesian inference]] of variables with [[multivariate normal distribution]].

Alternatively, the reciprocal of the standard deviation <math display=inline>\tau'=1/\sigma</math> might be defined as the ''precision'', in which case the expression of the normal distribution becomes

<math display=block>f(x) = \frac{\tau'}{\sqrt{2\pi}} e^{-(\tau')^2(x-\mu)^2/2}.</math>

According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the [[quantile]]s of the distribution.

Normal distributions form an [[exponential family]] with [[natural parameter]]s <math display=inline>\textstyle\theta_1=\frac{\mu}{\sigma^2}</math> and <math display=inline>\textstyle\theta_2=\frac{-1}{2\sigma^2}</math>, and natural statistics ''x'' and ''x''<sup>2</sup>. The dual expectation parameters for normal distribution are {{math|1=''η''<sub>1</sub> = ''μ''}} and {{math|1=''η''<sub>2</sub> = ''μ''<sup>2</sup> + ''σ''<sup>2</sup>}}.

=== Cumulative distribution function ===
The [[cumulative distribution function]] (CDF) of the standard normal distribution, usually denoted with the capital Greek letter {{tmath|\Phi}}, is the integral

<math display=block>\Phi(x) = \frac 1 {\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2} \, dt\,.</math>

=== Error function ===
The related [[error function]] <math display=inline>\operatorname{erf}(x)</math> gives the probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range {{tmath|[-x, x]}}. That is:<!-- SIC! The interval for erf is [−x,+x], NOT [0,x]. -->

<math display=block>\operatorname{erf}(x) = \frac 1 {\sqrt\pi} \int_{-x}^x e^{-t^2} \, dt = \frac 2 {\sqrt\pi} \int_0^x e^{-t^2} \, dt\,.</math>

These integrals cannot be expressed in terms of elementary functions, and are often said to be [[special function]]s. However, many numerical approximations are known; see [[#Numerical approximations for the normal cumulative distribution function and normal quantile function|below]] for more.

The two functions are closely related, namely

<math display=block>\Phi(x) = \frac{1}{2} \left[1 + \operatorname{erf}\left( \frac x {\sqrt 2} \right) \right]\,.</math>

For a generic normal distribution with density {{tmath|f}}, mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, the cumulative distribution function is

<math display="block">
F(x) = \Phi{\left(\frac{x-\mu} \sigma \right)} = \frac{1}{2} \left[1 + \operatorname{erf}\left(\frac{x-\mu}{\sigma \sqrt 2 }\right)\right]\,.
</math>

The complement of the standard normal cumulative distribution function, <math display=inline>Q(x) = 1 - \Phi(x)</math>, is often called the [[Q-function]], especially in engineering texts.<ref>{{cite web|url=http://cnx.org/content/m11537/1.2/|last1=Scott|first1=Clayton|first2=Robert|last2=Nowak|title=The Q-function|work=Connexions|date=August 7, 2003}}</ref><ref>{{cite web|url=http://www.eng.tau.ac.il/~jo/academic/Q.pdf|last=Barak|first=Ohad|title=Q Function and Error Function|publisher=Tel Aviv University|date=April 6, 2006|url-status=dead|archive-url=https://web.archive.org/web/20090325160012/http://www.eng.tau.ac.il/~jo/academic/Q.pdf|archive-date=March 25, 2009|df=mdy-all}}</ref> It gives the probability that the value of a standard normal random variable {{tmath|X}} will exceed {{tmath|x}}: {{tmath|P(X>x)}}. Other definitions of the {{tmath|Q}}-function, all of which are simple transformations of {{tmath|\Phi}}, are also used occasionally.<ref>{{MathWorld |urlname=NormalDistributionFunction |title=Normal Distribution Function }}</ref>

The [[graph of a function|graph]] of the standard normal cumulative distribution function {{tmath|\Phi}} has 2-fold [[rotational symmetry]] around the point (0,1/2); that is, {{tmath|1=\Phi(-x) = 1 - \Phi(x)}}. Its [[antiderivative]] (indefinite integral) can be expressed as follows:
<math display=block>\int \Phi(x)\, dx = x\Phi(x) + \varphi(x) + C.</math>

The cumulative distribution function of the standard normal distribution can be expanded by [[integration by parts]] into a series:

<math display=block>\Phi(x)=\frac{1}{2} + \frac{1}{\sqrt{2\pi}}\cdot e^{-x^2/2} \left[x + \frac{x^3}{3} + \frac{x^5}{3\cdot 5} + \cdots + \frac{x^{2n+1}}{(2n+1)!!} + \cdots\right]\,.</math>

where <math display=inline>!!</math> denotes the [[double factorial]].

An [[asymptotic expansion]] of the cumulative distribution function for large ''x'' can also be derived using integration by parts. For more, see {{slink|Error function#Asymptotic expansion}}.<ref>{{AS ref|26, eqn 26.2.12|932}}</ref>

A quick approximation to the standard normal distribution's cumulative distribution function can be found by using a Taylor series approximation:

<math display=block>\Phi(x) \approx \frac{1}{2}+\frac{1}{\sqrt{2\pi}} \sum_{k=0}^n \frac{(-1)^k x^{(2k+1)}}{2^k k! (2k+1)}\,.</math>

==== Recursive computation with Taylor series expansion ====
The recursive nature of the <math display=inline>e^{ax^2}</math>family of derivatives may be used to easily construct a rapidly converging [[Taylor series]] expansion using recursive entries about any point of known value of the distribution,<math display=inline>\Phi(x_0)</math>:

<math display=block>\Phi(x) = \sum_{n=0}^\infty \frac{\Phi^{(n)}(x_0)}{n!}(x-x_0)^n\,,</math>

where:

<math display=block>\begin{align}
\Phi^{(0)}(x_0) &= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x_0}e^{-t^2/2}\,dt \\
\Phi^{(1)}(x_0) &= \frac{1}{\sqrt{2\pi}}e^{-x_0^2/2} \\
\Phi^{(n)}(x_0) &= -\left(x_0\Phi^{(n-1)}(x_0)+(n-2)\Phi^{(n-2)}(x_0)\right), & n \geq 2\,.
\end{align}</math>

==== Using the Taylor series and Newton's method for the inverse function ====
An application for the above Taylor series expansion is to use [[Newton's method]] to reverse the computation.  That is, if we have a value for the [[cumulative distribution function]], <math display=inline>\Phi(x)</math>, but do not know the x needed to obtain the <math display=inline>\Phi(x)</math>, we can use Newton's method to find x, and use the Taylor series expansion above to minimize the number of computations.  Newton's method is ideal to solve this problem because the first derivative of <math display=inline>\Phi(x)</math>, which is an integral of the normal standard distribution, is the normal standard distribution, and is readily available to use in the Newton's method solution.

To solve, select a known approximate solution, <math display=inline>x_0</math>, to the desired {{tmath|\Phi(x)}}.  <math display=inline>x_0</math> may be a value from a distribution table, or an intelligent estimate followed by a computation of <math display=inline>\Phi(x_0)</math> using any desired means to compute. Use this value of <math display=inline>x_0</math> and the Taylor series expansion above to minimize computations.

Repeat the following process until the difference between the computed  <math display=inline>\Phi(x_{n})</math> and the desired {{tmath|\Phi}}, which we will call <math display=inline>\Phi(\text{desired})</math>, is below a chosen acceptably small error, such as 10<sup>−5</sup>, 10<sup>−15</sup>, etc.:

<math display=block>x_{n+1} = x_n - \frac{\Phi(x_n,x_0,\Phi(x_0))-\Phi(\text{desired})}{\Phi'(x_n)}\,,</math>

where

: <math display=inline>\Phi(x,x_0,\Phi(x_0))</math> is the <math display=inline>\Phi(x)</math> from a Taylor series solution using <math display=inline>x_0</math> and <math display=inline>\Phi(x_0)</math>

<math display=block>\Phi'(x_n)=\frac{1}{\sqrt{2\pi}}e^{-x_n^2/2}\,.</math>

When the repeated computations converge to an error below the chosen acceptably small value, ''x'' will be the value needed to obtain a <math display=inline>\Phi(x)</math> of the desired value, {{tmath|\Phi(\text{desired})}}.

==== Standard deviation and coverage ====
{{Further|Interval estimation|Coverage probability}}
[[File:Standard deviation diagram.svg|thumb|350px|For the normal distribution, the values less than one standard deviation from the mean account for 68.27% of the set; while two standard deviations from the mean account for 95.45%; and three standard deviations account for 99.73%.]]
About 68% of values drawn from a normal distribution are within one standard deviation ''σ'' from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations.<ref name="www.mathsisfun.com" /> This fact is known as the [[68–95–99.7 rule|68–95–99.7 (empirical) rule]], or the ''3-sigma rule''.

More precisely, the probability that a normal deviate lies in the range between <math display=inline>\mu-n\sigma</math> and <math display=inline>\mu+n\sigma</math> is given by
<math display=block>
F(\mu+n\sigma) - F(\mu-n\sigma) = \Phi(n)-\Phi(-n) = \operatorname{erf} \left(\frac{n}{\sqrt{2}}\right).
</math>
To 12 significant digits, the values for <math display=inline>n=1,2,\ldots , 6</math> are:

{| class="wikitable" style="text-align:center;margin-left:24pt"
|-
! {{tmath|n}} !! <math display=inline>p= F(\mu+n\sigma) - F(\mu-n\sigma)</math> !! <math display=inline>1-p</math>!! <math display=inline>\text{or }1\text{ in }(1-p)</math> !! [[OEIS]]
|-
|1 || {{val|0.682689492137}} || {{val|0.317310507863}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|3}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.15148718753}}
|}
|| {{OEIS2C|A178647}}
|-
|2 || {{val|0.954499736104}} || {{val|0.045500263896}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|21}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.9778945080}}
|}
|| {{OEIS2C|A110894}}
|-
|3 || {{val|0.997300203937}} || {{val|0.002699796063}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|370}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.398347345}}
|}
|| {{OEIS2C|A270712}}
|-
|4 || {{val|0.999936657516}} || {{val|0.000063342484}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|15787}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.1927673}}
|}
|-
|5 || {{val|0.999999426697}} || {{val|0.000000573303}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|1744277}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.89362}}
|}
|-
|6 || {{val|0.999999998027}} || {{val|0.000000001973}} ||
{| cellpadding="0" cellspacing="0" style="width: 16em;"
| style="text-align: right; width: 7em;" | {{val|506797345}} || style="text-align: left; width: 9em;" | {{#invoke:Gapnum|main|.897}}
|}
|}

For large {{tmath|n}}, one can use the approximation <math display=inline>1 - p \approx \frac{e^{-n^2/2}}{n\sqrt{\pi/2}}</math>.

==== Quantile function ====
{{Further|Quantile function#Normal distribution}}

The [[quantile function]] of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the [[probit function]], and can be expressed in terms of the inverse [[error function]]:
<math display=block>
\Phi^{-1}(p) = \sqrt2\operatorname{erf}^{-1}(2p - 1), \quad p\in(0,1).
</math>
For a normal random variable with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, the quantile function is
<math display=block>
F^{-1}(p) = \mu + \sigma\Phi^{-1}(p)
= \mu + \sigma\sqrt 2 \operatorname{erf}^{-1}(2p - 1), \quad p\in(0,1).
</math>
The [[quantile]] <math display=inline>\Phi^{-1}(p)</math> of the standard normal distribution is commonly denoted as {{tmath|z_p}}. These values are used in [[hypothesis testing]], construction of [[confidence interval]]s and [[Q–Q plot]]s. A normal random variable {{tmath|X}} will exceed <math display=inline>\mu + z_p\sigma</math> with probability <math display=inline>1-p</math>, and will lie outside the interval <math display=inline>\mu \pm z_p\sigma</math> with probability {{tmath|2(1-p)}}. In particular, the quantile <math display=inline>z_{0.975}</math> is [[1.96]]; therefore a normal random variable will lie outside the interval <math display=inline>\mu \pm 1.96\sigma</math> in only 5% of cases.

The following table gives the quantile <math display=inline>z_p</math> such that {{tmath|X}} will lie in the range <math display=inline>\mu \pm z_p\sigma</math> with a specified probability {{tmath|p}}. These values are useful to determine [[tolerance interval]] for [[Sample mean and sample covariance#Sample mean|sample averages]] and other statistical [[estimator]]s with normal (or [[asymptotic]]ally normal) distributions.<ref>{{Cite book |last=Vaart |first=A. W. van der |url=http://dx.doi.org/10.1017/cbo9780511802256 |title=Asymptotic Statistics |date=1998-10-13 |publisher=Cambridge University Press |doi=10.1017/cbo9780511802256 |isbn=978-0-511-80225-6}}</ref> The following table shows <math display=inline>\sqrt 2 \operatorname{erf}^{-1}(p)=\Phi^{-1}\left(\frac{p+1}{2}\right)</math>, not <math display=inline>\Phi^{-1}(p)</math> as defined above.

{| class="wikitable" style="text-align:left;margin-left:24pt;border:none;"
! {{tmath|p}} !! <math display=inline>z_p</math>
| rowspan="8" style="border:none;"|&nbsp;
! {{tmath|p}} !! <math display=inline>z_p</math>
|-
| 0.80  || {{val|1.281551565545}} || 0.999       || {{val|3.290526731492}}
|-
| 0.90  || {{val|1.644853626951}} || 0.9999      || {{val|3.890591886413}}
|-
| 0.95  || {{val|1.959963984540}} || 0.99999     || {{val|4.417173413469}}
|-
| 0.98  || {{val|2.326347874041}} || 0.999999    || {{val|4.891638475699}}
|-
| 0.99  || {{val|2.575829303549}} || 0.9999999   || {{val|5.326723886384}}
|-
| 0.995 || {{val|2.807033768344}} || 0.99999999  || {{val|5.730728868236}}
|-
| 0.998 || {{val|3.090232306168}} || 0.999999999 || {{val|6.109410204869}}
|}

For small {{tmath|p}}, the quantile function has the useful [[asymptotic expansion]]
<math display=inline>\Phi^{-1}(p)=-\sqrt{\ln\frac{1}{p^2}-\ln\ln\frac{1}{p^2}-\ln(2\pi)}+\mathcal{o}(1).</math>{{citation needed|date=February 2023}}

== Properties ==
The normal distribution is the only distribution whose [[cumulant]]s beyond the first two (i.e., other than the mean and [[variance]]) are zero. It is also the continuous distribution with the [[maximum entropy probability distribution|maximum entropy]] for a specified mean and variance.<ref>{{cite book|last=Cover|first=Thomas M.|author2=Thomas, Joy A.|year=2006|title=Elements of Information Theory|url=https://archive.org/details/elementsinformat00cove|url-access=limited|publisher=John Wiley and Sons|page=[https://archive.org/details/elementsinformat00cove/page/n279 254]|isbn=9780471748816}}</ref><ref>{{cite journal|last1=Park|first1=Sung Y.|last2=Bera|first2=Anil K.|year=2009|title=Maximum Entropy Autoregressive Conditional Heteroskedasticity Model|journal=Journal of Econometrics|pages=219–230|url=http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf|access-date=2011-06-02|doi=10.1016/j.jeconom.2008.12.014|volume=150|issue=2|citeseerx=10.1.1.511.9750|archive-date=March 7, 2016|archive-url=https://web.archive.org/web/20160307144515/http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf|url-status=dead}}</ref> Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.<ref name="Geary RC">Geary RC(1936) The distribution of the "Student's ratio for the non-normal samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184</ref><ref>{{Cite Q|Q55897617|author1=Lukacs, Eugene|author-link1=Eugene Lukacs}}</ref>

The normal distribution is a subclass of the [[elliptical distribution]]s. The normal distribution is [[Symmetric distribution|symmetric]] about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the [[weight]] of a person or the price of a [[share (finance)|share]]. Such variables may be better described by other distributions, such as the [[log-normal distribution]] or the [[Pareto distribution]].

The value of the normal density is practically zero when the value {{tmath|x}} lies more than a few [[standard deviation]]s away from the mean (e.g., a spread of three standard deviations covers all but 0.27% of the total distribution). Therefore, it may not be an appropriate model when one expects a significant fraction of [[outlier]]s—values that lie many standard deviations away from the mean—and least squares and other [[statistical inference]] methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more [[heavy-tailed]] distribution should be assumed and the appropriate [[robust statistics|robust statistical inference]] methods applied.

The Gaussian distribution belongs to the family of [[stable distribution]]s which are the attractors of sums of [[independent, identically distributed]] distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance. It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the [[Cauchy distribution]] and the [[Lévy distribution]].

=== Symmetries and derivatives ===
The normal distribution with density <math display=inline>f(x)</math> (mean {{tmath|\mu}} and variance <math display=inline>\sigma^2 > 0</math>) has the following properties:
* It is symmetric around the point <math display=inline>x=\mu,</math> which is at the same time the [[mode (statistics)|mode]], the [[median]] and the [[mean]] of the distribution.<ref name="Patel">{{harvtxt |Patel |Read |1996 |loc=[2.1.4] }}</ref>
* It is [[unimodal]]: its first [[derivative]] is positive for <math display=inline>x<\mu,</math> negative for <math display=inline>x>\mu,</math> and zero only at <math display=inline>x=\mu.</math>
* The area bounded by the curve and the {{tmath|x}}-axis is unity (i.e. equal to one).
* Its first derivative is <math display=inline>f'(x)=-\frac{x-\mu}{\sigma^2} f(x).</math>
* Its second derivative is <math display=inline>f''(x) =  \frac{(x-\mu)^2 - \sigma^2}{\sigma^4} f(x).</math>
* Its density has two [[inflection point]]s (where the second derivative of {{tmath|f}} is zero and changes sign), located one standard deviation away from the mean, namely at <math display=inline>x=\mu-\sigma</math> and <math display=inline>x=\mu+\sigma.</math><ref name="Patel" />
* Its density is [[logarithmically concave function|log-concave]].<ref name="Patel" />
* Its density is infinitely [[differentiable]], indeed [[supersmooth]] of order 2.<ref>{{harvtxt |Fan |1991 |p=1258 }}</ref>

Furthermore, the density {{tmath|\varphi}} of the standard normal distribution (i.e. <math display=inline>\mu=0</math> and <math display=inline>\sigma=1</math>) also has the following properties:
* Its first derivative is <math display=inline>\varphi'(x)=-x\varphi(x).</math>
* Its second derivative is <math display=inline>\varphi''(x)=(x^2-1)\varphi(x)</math>
* More generally, its {{mvar|n}}th derivative is <math display=inline>\varphi^{(n)}(x) = (-1)^n\operatorname{He}_n(x)\varphi(x),</math> where <math display=inline>\operatorname{He}_n(x)</math> is the {{mvar|n}}th (probabilist) [[Hermite polynomial]].<ref>{{harvtxt |Patel |Read |1996 |loc=[2.1.8] }}</ref>
* The probability that a normally distributed variable {{tmath|X}} with known {{tmath|\mu}} and <math display=inline>\sigma^2</math> is in a particular set, can be calculated by using the fact that the fraction <math display=inline>Z = (X-\mu)/\sigma</math> has a standard normal distribution.

=== Moments ===
{{See also|List of integrals of Gaussian functions}}
The plain and absolute [[moment (mathematics)|moments]] of a variable {{tmath|X}} are the expected values of <math display=inline>X^p</math> and <math display=inline>|X|^p</math>, respectively. If the expected value {{tmath|\mu}} of {{tmath|X}} is zero, these parameters are called ''central moments;'' otherwise, these parameters are called ''non-central moments.'' Usually we are interested only in moments with integer order {{tmath|p}}.

If {{tmath|X}} has a normal distribution, the non-central moments exist and are finite for any {{tmath|p}} whose real part is greater than&nbsp;−1. For any non-negative integer {{tmath|p}}, the plain central moments are:<ref>{{cite book|last1=Papoulis|first1=Athanasios|title=Probability, Random Variables and Stochastic Processes|page=148|edition=4th}}</ref>
<math display=block>
    \operatorname{E}\left[(X-\mu)^p\right] =
      \begin{cases}
        0 & \text{if }p\text{ is odd,} \\
        \sigma^p (p-1)!! & \text{if }p\text{ is even.}
      \end{cases}
</math>
Here <math display=inline>n!!</math> denotes the [[double factorial]], that is, the product of all numbers from {{tmath|n}} to&nbsp;1 that have the same parity as <math display=inline>n.</math>

The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer <math display=inline>p,</math>

<math display=block>\begin{align}
   \operatorname{E}\left[|X - \mu|^p\right] &= \sigma^p (p-1)!! \cdot \begin{cases}
     \sqrt{\frac{2}{\pi}} & \text{if }p\text{ is odd} \\
     1 & \text{if }p\text{ is even}
   \end{cases} \\
   &= \sigma^p \cdot \frac{2^{p/2}\Gamma\left(\frac{p+1} 2 \right)}{\sqrt\pi}.
 \end{align}</math>
The last formula is valid also for any non-integer <math display=inline>p>-1.</math> When the mean <math display=inline>\mu \ne 0,</math> the plain and absolute moments can be expressed in terms of [[confluent hypergeometric function]]s <math display=inline>{}_1F_1</math> and <math display=inline>U.</math><ref>{{cite arXiv|last1=Winkelbauer|first1=Andreas|title=Moments and Absolute Moments of the Normal Distribution |date= 2012|class=math.ST |eprint=1209.4340}}</ref>

<math display="block">\begin{align}
   \operatorname{E}\left[X^p\right] &= \sigma^p\cdot {\left(-i\sqrt 2\right)}^p \, U{\left(-\frac{p}{2}, \frac{1}{2}, -\frac{\mu^2}{2\sigma^2}\right)}, \\
   \operatorname{E}\left[|X|^p \right] &= \sigma^p \cdot 2^{p/2} \frac {\Gamma{\left(\frac{1+p} 2\right)}}{\sqrt\pi} \, {}_1F_1{\left( -\frac{p}{2}, \frac{1}{2}, -\frac{\mu^2}{2\sigma^2} \right)}.
 \end{align}</math>

These expressions remain valid even if {{tmath|p}} is not an integer. See also [[Hermite polynomials#"Negative variance"|generalized Hermite polynomials]].

{| class="wikitable" style="margin: auto;"
|-
! Order !! Non-central moment, <math>\operatorname{E}\left[X^p\right]</math> !! Central moment, <math>\operatorname{E}\left[(X-\mu)^p\right]</math>
|-
| 1
| {{tmath|\mu}}
| {{tmath|0}}
|-
| 2
| <math display=inline>\mu^2+\sigma^2</math>
| <math display=inline>\sigma^2</math>
|-
| 3
| <math display=inline>\mu^3+3\mu\sigma^2</math>
| {{tmath|0}}
|-
| 4
| <math display=inline>\mu^4+6\mu^2\sigma^2+3\sigma^4</math>
| <math display=inline>3\sigma^4</math>
|-
| 5
| <math display=inline>\mu^5+10\mu^3\sigma^2+15\mu\sigma^4</math>
| {{tmath|0}}
|-
| 6
| <math display=inline>\mu^6+15\mu^4\sigma^2+45\mu^2\sigma^4+15\sigma^6</math>
| <math display=inline>15\sigma^6</math>
|-
| 7
| <math display=inline>\mu^7+21\mu^5\sigma^2+105\mu^3\sigma^4+105\mu\sigma^6</math>
| {{tmath|0}}
|-
| 8
| <math display=inline>\mu^8+28\mu^6\sigma^2+210\mu^4\sigma^4+420\mu^2\sigma^6+105\sigma^8</math>
| <math display=inline>105\sigma^8</math>
|}
The expectation of {{tmath|X}} conditioned on the event that {{tmath|X}} lies in an interval <math display=inline>[a,b]</math> is given by
<math display=block>\operatorname{E}\left[X \mid a<X<b \right] = \mu - \sigma^2\frac{f(b)-f(a)}{F(b)-F(a)}\,,</math>
where {{tmath|f}} and {{tmath|F}} respectively are the density and the cumulative distribution function of {{tmath|X}}. For <math display=inline>b=\infty</math> this is known as the [[inverse Mills ratio]]. Note that above, density {{tmath|f}} of {{tmath|X}} is used instead of standard normal density as in inverse Mills ratio, so here we have <math display=inline>\sigma^2</math> instead of {{tmath|\sigma}}.

=== Fourier transform and characteristic function ===
The [[Fourier transform]] of a normal density {{tmath|f}} with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math> is<ref>{{harvtxt |Bryc |1995 |p=23 }}</ref>

<math display=block>
\hat f(t) = \int_{-\infty}^\infty f(x)e^{-itx} \, dx = e^{-i\mu t} e^{- \frac12 (\sigma t)^2}\,,
</math>

where {{tmath|i}} is the [[imaginary unit]]. If the mean <math display=inline>\mu=0</math>, the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the [[frequency domain]], with mean 0 and variance {{tmath|1/\sigma^2}}. In particular, the standard normal distribution {{tmath|\varphi}} is an [[Fourier transform#Eigenfunctions|eigenfunction]] of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable {{tmath|X}} is closely connected to the [[characteristic function (probability theory)|characteristic function]] <math display=inline>\varphi_X(t)</math> of that variable, which is defined as the [[expected value]] of <math display=inline>e^{itX}</math>, as a function of the real variable {{tmath|t}} (the [[frequency]] parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable {{tmath|t}}.<ref>{{harvtxt |Bryc |1995 |p=24 }}</ref> The relation between both is:
<math display=block>\varphi_X(t) = \hat f(-t)\,.</math>

=== Moment- and cumulant-generating functions ===
The [[moment generating function]] of a real random variable {{tmath|X}} is the expected value of <math display=inline>e^{tX}</math>, as a function of the real parameter {{tmath|t}}. For a normal distribution with density {{tmath|f}}, mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, the moment generating function exists and is equal to

<math display=block>M(t) = \operatorname{E}\left[e^{tX}\right] = \hat f(it) = e^{\mu t} e^{\sigma^2 t^2/2}\,.</math>
For any {{tmath|k}}, the coefficient of {{tmath|t^k / k!}} in the moment generating function (expressed as an [[Generating function#Exponential generating function (EGF)|exponential power series]] in {{tmath|t}}) is the normal distribution's expected value {{tmath|E[X^k]}}.

The [[cumulant generating function]] is the logarithm of the moment generating function, namely

<math display=block>g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.</math>

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in  {{tmath|t}}, only the first two [[cumulant]]s are nonzero, namely the mean&nbsp;{{tmath|\mu}} and the variance&nbsp;{{tmath|\sigma^2}}.

Some authors prefer to instead work with the [[characteristic function (probability theory)|characteristic function]] {{math|1=E[''e''{{sup|''itX''}}] = ''e''{{sup|''iμt'' − ''σ''{{sup|2}}''t''{{sup|2}}/2}}}} and {{math|1=ln E[''e''{{sup|''itX''}}] = ''iμt'' − {{sfrac|1|2}}''σ''{{sup|2}}''t''{{sup|2}}}}.

=== Stein operator and class ===
Within [[Stein's method]] the Stein operator and class of a random variable  <math display=inline>X \sim \mathcal{N}(\mu, \sigma^2)</math> are <math display=inline>\mathcal{A}f(x) = \sigma^2 f'(x) - (x-\mu)f(x)</math> and <math display=inline>\mathcal{F}</math> the class of all absolutely continuous functions <math display=inline>f : \R \to \R \mbox{ such that }\mathbb{E}[|f'(X)|]< \infty</math>.

=== Zero-variance limit ===
In the [[limit (mathematics)|limit]] when <math display=inline>\sigma^2</math> tends to zero, the probability density <math display=inline>f(x)</math> eventually tends to zero at any <math display=inline>x\ne \mu</math>, but grows without limit if <math display=inline>x = \mu</math>, while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary [[function (mathematics)|function]] when {{tmath|1=\sigma^2 = 0}}.

However, one can define the normal distribution with zero variance as a [[generalized function]]; specifically, as a [[Dirac delta function]] {{tmath|\delta}} translated by the mean {{tmath|\mu}}, that is <math display=inline>f(x)=\delta(x-\mu).</math>
Its cumulative distribution function is then the [[Heaviside step function]] translated by the mean {{tmath|\mu}}, namely
<math display=block>F(x) =
\begin{cases}
  0 & \text{if }x < \mu \\
  1 & \text{if }x \geq \mu\,.
\end{cases}
</math>

=== Maximum entropy ===
Of all probability distributions over the reals with a specified finite mean {{tmath|\mu}} and finite variance&nbsp;<math display=inline>\sigma^2</math>, the normal distribution <math display=inline>N(\mu,\sigma^2)</math> is the one with [[Maximum entropy probability distribution|maximum entropy]].<ref>{{harvtxt |Cover |Thomas |2006 |p=254 }}</ref> To see this, let {{tmath|X}} be a [[continuous random variable]] with [[probability density]] {{tmath|f(x)}}. The entropy of {{tmath|X}} is defined as<ref>{{cite book|last1=Williams|first1=David|title=Weighing the odds : a course in probability and statistics|url=https://archive.org/details/weighingoddscour00will|url-access=limited|date=2001|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0-521-00618-7|pages=[https://archive.org/details/weighingoddscour00will/page/n219 197]–199|edition=Reprinted.}}</ref><ref>{{cite book|last1=Smith|first1=José M. Bernardo; Adrian F. M.|title=Bayesian theory|url=https://archive.org/details/bayesiantheory00bern_963|url-access=limited|date=2000|publisher=Wiley|location=Chichester [u.a.]|isbn=978-0-471-49464-5|pages=[https://archive.org/details/bayesiantheory00bern_963/page/n224 209], 366|edition=Reprint}}</ref><ref>O'Hagan, A. (1994) ''Kendall's Advanced Theory of statistics, Vol 2B, Bayesian Inference'', Edward Arnold. {{isbn|0-340-52922-9}} (Section 5.40)</ref>
<math display=block>
H(X) = - \int_{-\infty}^\infty f(x)\ln f(x)\, dx\,,
</math>

where <math display=inline>f(x)\log f(x)</math> is understood to be zero whenever {{tmath|1=f(x)=0}}. This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using [[variational calculus]]. A function with three [[Lagrange multipliers]] is defined:

<math display=block>
L=-\int_{-\infty}^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_{-\infty}^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_{-\infty}^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_{-\infty}^\infty f(x)(x-\mu)^2\,dx\right)\,.
</math>

At maximum entropy, a small variation <math display=inline>\delta f(x)</math> about <math display=inline>f(x)</math> will produce a variation <math display=inline>\delta L</math> about {{tmath|L}} which is equal to 0:

<math display=block>
0=\delta L=\int_{-\infty}^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,.
</math>

Since this must hold for any small <math display=inline>\delta f(x)</math>, the factor multiplying <math display=inline>\delta f(x)</math> must be zero, and solving for <math display=inline>f(x)</math> yields:

<math display=block>f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.</math>

The Lagrange constraints that <math display=inline>f(x)</math> is properly normalized and has the specified mean and variance are satisfied if and only if <math display=inline>\lambda_0</math>, <math display=inline>\lambda_1</math>, and <math display=inline>\lambda_2</math> are chosen so that
<math display=block>
f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\,.
</math>
The entropy of a normal distribution <math display=inline>X \sim N(\mu,\sigma^2)</math> is equal to
<math display=block>
H(X)=\tfrac{1}{2}(1+\ln 2\sigma^2\pi)\,,
</math>
which is independent of the mean {{tmath|\mu}}.

=== Other properties ===
{{ordered list
| 1 = If the characteristic function <math display=inline>\phi_X</math> of some random variable {{tmath|X}} is of the form <math display=inline>\phi_X(t) = \exp Q(t)</math> in a neighborhood of zero, where <math display=inline>Q(t)</math> is a [[polynomial]], then the '''Marcinkiewicz theorem''' (named after [[Józef Marcinkiewicz]]) asserts that {{tmath|Q}} can be at most a quadratic polynomial, and therefore {{tmath|X}} is a normal random variable.<ref name="Bryc 1995 35" /> The consequence of this result is that the normal distribution is the only distribution with a finite number (two) of non-zero [[cumulant]]s.
| 2 = If {{tmath|X}} and {{tmath|Y}} are [[jointly normal]] and [[uncorrelated]], then they are [[independence (probability theory)|independent]]. The requirement that {{tmath|X}} and {{tmath|Y}} should be ''jointly'' normal is essential; without it the property does not hold.<ref>[http://www.math.uiuc.edu/~r-ash/Stat/StatLec21-25.pdf UIUC, Lecture 21. ''The Multivariate Normal Distribution''], 21.6:"Individually Gaussian Versus Jointly Gaussian".</ref><ref>Edward L. Melnick and Aaron Tenenbein, "Misspecifications of the Normal Distribution", ''[[The American Statistician]]'', volume 36, number 4 November 1982, pages 372–373</ref><sup>[[Normally distributed and uncorrelated does not imply independent|[proof]]]</sup> For non-normal random variables uncorrelatedness does not imply independence.
| 3 = The [[Kullback–Leibler divergence]] of one normal distribution <math display=inline>X_1 \sim N(\mu_1, \sigma^2_1)</math> from another <math display=inline>X_2 \sim N(\mu_2, \sigma^2_2)</math> is given by:<ref>{{cite web |url=http://www.allisons.org/ll/MML/KL/Normal/|title=Kullback Leibler (KL) Distance of Two Normal (Gaussian) Probability Distributions| website=Allisons.org |date=2007-12-05 |access-date=2017-03-03}}</ref> <math display=block>
    D_\mathrm{KL}( X_1 \parallel X_2 ) = \frac{(\mu_1 - \mu_2)^2}{2\sigma_2^2} + \frac{1}{2}\left( \frac{\sigma_1^2}{\sigma_2^2} - 1 - \ln\frac{\sigma_1^2}{\sigma_2^2} \right)
</math>
The [[Hellinger distance]] between the same distributions is equal to <math display=block>
    H^2(X_1,X_2) = 1 - \sqrt{\frac{2\sigma_1\sigma_2}{\sigma_1^2+\sigma_2^2}} \exp\left(-\frac{1}{4}\frac{(\mu_1-\mu_2)^2}{\sigma_1^2+\sigma_2^2}\right)
</math>
| 4 = The [[Fisher information matrix]] for a normal distribution w.r.t. {{tmath|\mu}} and <math display=inline>\sigma^2</math> is diagonal and takes the form <math display=block>
    \mathcal I (\mu, \sigma^2) = \begin{pmatrix} \frac{1}{\sigma^2} & 0 \\ 0 & \frac{1}{2\sigma^4} \end{pmatrix}
</math>
| 5 = The [[conjugate prior]] of the mean of a normal distribution is another normal distribution.<ref>{{cite web|url=http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture5.pdf|title=Stat260: Bayesian Modeling and Inference: The Conjugate Prior for the Normal Distribution|first=Michael I.|last=Jordan|date=February 8, 2010}}</ref> Specifically, if <math display=inline>x_1, \ldots, x_n</math> are iid <math display=inline>\sim N(\mu, \sigma^2)</math> and the prior is <math display=inline>\mu \sim N(\mu_0 , \sigma^2_0)</math>, then the posterior distribution for the estimator of {{tmath|\mu}} will be <math display=block>
    \mu \mid x_1,\ldots,x_n \sim \mathcal{N}\left( \frac{\frac{\sigma^2}{n}\mu_0 + \sigma_0^2\bar{x}}{\frac{\sigma^2}{n}+\sigma_0^2},\left( \frac{n}{\sigma^2} + \frac{1}{\sigma_0^2} \right)^{-1} \right)
</math>
| 6 = The family of normal distributions not only forms an [[exponential family]] (EF), but in fact forms a [[natural exponential family]] (NEF) with quadratic [[variance function]] ([[NEF-QVF]]). Many properties of normal distributions generalize to properties of NEF-QVF distributions, NEF distributions, or EF distributions generally. NEF-QVF distributions comprises 6 families, including Poisson, Gamma, binomial, and negative binomial distributions, while many of the common families studied in probability and statistics are NEF or EF.
| 7 = In [[information geometry]], the family of normal distributions forms a [[statistical manifold]] with [[constant curvature]] {{tmath|-1}}. The same family is [[flat manifold|flat]] with respect to the (±1)-connections <math display=inline>\nabla^{(e)}</math> and <math display=inline>\nabla^{(m)}</math>.<ref>{{harvtxt |Amari |Nagaoka |2000 }}</ref>
| 8 = If <math display=inline>X_1, \dots, X_n</math> are distributed according to <math display=inline>N(0, \sigma^2)</math>, then <math display=inline>E[\max_i X_i ] \leq \sigma\sqrt{2\ln n}</math>. Note that there is no assumption of independence.<ref>{{Cite web |title=Expectation of the maximum of gaussian random variables |url=https://math.stackexchange.com/a/89147 |access-date=2024-04-07 |website=Mathematics Stack Exchange |language=en}}</ref>
}}

== Related distributions ==

=== Central limit theorem ===
[[File:De moivre-laplace.gif|right|thumb|250px|As the number of discrete events increases, the function begins to resemble a normal distribution.]]
[[File:Dice sum central limit theorem.svg|thumb|250px|Comparison of probability density functions, <math display=inline>p(k)</math> for the sum of {{tmath|n}} fair 6-sided dice to show their convergence to a normal distribution with increasing <math display=inline>na</math>, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).]]
{{Main|Central limit theorem}}

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where <math display=inline>X_1,\ldots ,X_n</math> are [[independent and identically distributed]] random variables with the same arbitrary distribution, zero mean, and variance <math display=inline>\sigma^2</math> and {{tmath|Z}} is their
mean scaled by <math display=inline>\sqrt{n}</math>
<math display=block>Z = \sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n X_i\right)</math>
Then, as {{tmath|n}} increases, the probability distribution of {{tmath|Z}} will tend to the normal distribution with zero mean and variance {{tmath|\sigma^2}}.

The theorem can be extended to variables <math display=inline>(X_i)</math> that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many [[test statistic]]s, [[score (statistics)|scores]], and [[estimator]]s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of [[influence function (statistics)|influence functions]]. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The [[binomial distribution]] <math display=inline>B(n,p)</math> is [[De Moivre–Laplace theorem|approximately normal]] with mean <math display=inline>np</math> and variance <math display=inline>np(1-p)</math> for large {{tmath|n}} and for {{tmath|p}} not too close to 0 or 1.
* The [[Poisson distribution]] with parameter {{tmath|\lambda}} is approximately normal with mean {{tmath|\lambda}} and variance {{tmath|\lambda}}, for large values of {{tmath|\lambda}}.<ref>{{cite web|url=http://www.stat.ucla.edu/~dinov/courses_students.dir/Applets.dir/NormalApprox2PoissonApplet.html|title=Normal Approximation to Poisson Distribution|website=Stat.ucla.edu|access-date=2017-03-03}}</ref>
* The [[chi-squared distribution]] <math display=inline>\chi^2(k)</math> is approximately normal with mean {{tmath|k}} and variance <math display=inline>2k</math>, for large {{tmath|k}}.
* The [[Student's t-distribution]] <math display=inline>t(\nu)</math> is approximately normal with mean 0 and variance 1 when {{tmath|\nu}} is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the [[Berry–Esseen theorem]], improvements of the approximation are given by the [[Edgeworth expansion]]s.

This theorem can also be used to justify modeling the sum of many uniform noise sources as [[Gaussian noise]]. See [[AWGN]].

=== Operations and functions of normal variables ===

[[File:Probabilities of functions of normal vectors.png|thumb|right|'''a:''' Probability density of a function <math display=inline>\cos x^2</math> of a normal variable {{tmath|x}} with <math display=inline>\mu=-2</math> and {{tmath|1=\sigma=3}}. '''b:''' Probability density of a function <math display=inline>x^y</math> of two normal variables {{tmath|x}} and {{tmath|y}}, where <math display=inline>\mu_x=1</math>, <math display=inline>\mu_y=2</math>, <math display=inline>\sigma_x = 0.1</math>, <math display=inline>\sigma_y = 0.2</math>, and {{tmath|1=\rho_{xy} = 0.8}}. '''c:''' Heat map of the joint probability density of two functions of two correlated normal variables {{tmath|x}} and {{tmath|y}}, where <math display=inline>\mu_x = -2</math>, <math display=inline>\mu_y=5</math>, <math display=inline>\sigma_x^2 = 10</math>, <math display=inline>\sigma_y^2 = 20</math>, and <math display=inline>\rho_{xy} = 0.495</math>. '''d:''' Probability density of a function <math display=inline>\sum_{i=1}^4 \vert x_i \vert</math> of 4 iid standard normal variables. These are computed by the numerical method of ray-tracing.<ref name="Das-2021" />]]

The [[probability density]], [[cumulative distribution function|cumulative distribution]], and [[inverse cumulative distribution function|inverse cumulative distribution]] of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing<ref name="Das-2021">{{cite journal | last=Das|first=Abhranil| arxiv=2012.14331| title=A method to integrate and classify normal distributions|journal=Journal of Vision |date=2021|volume=21 |issue=10 |page=1 |doi=10.1167/jov.21.10.1 |pmid=34468706 |pmc=8419883 }}</ref> ([https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions Matlab code]). In the following sections we look at some special cases.

==== Operations on a single normal variable ====
If {{tmath|X}} is distributed normally with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, then
* <math display=inline>aX+b</math>, for any real numbers {{tmath|a}} and {{tmath|b}}, is also normally distributed, with mean <math display=inline>a\mu+b</math> and variance <math display=inline>a^2\sigma^2</math>. That is, the family of normal distributions is closed under [[linear transformations]].
* The exponential of {{tmath|X}} is distributed [[Log-normal distribution|log-normally]]: <math display=inline>e^X \sim \ln(N(\mu, \sigma^2))</math>.
* The standard [[logistic function|sigmoid]] of {{tmath|X}} is [[Logit-normal distribution|logit-normally distributed]]: <math display=inline>\sigma(X) \sim P( \mathcal{N}(\mu,\,\sigma^2) )</math>.
* The absolute value of {{tmath|X}} has [[folded normal distribution]]: <math display=inline>{{\left| X \right| \sim N_f(\mu, \sigma^2)}}</math>. If <math display=inline>\mu = 0</math> this is known as the [[half-normal distribution]].
* The absolute value of normalized residuals, <math display=inline>|X - \mu| / \sigma</math>, has [[chi distribution]] with one degree of freedom: <math display=inline>|X - \mu| / \sigma \sim \chi_1</math>.
* The square of <math display=inline>X/\sigma</math> has the [[noncentral chi-squared distribution]] with one degree of freedom: <math display=inline>X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)</math>. If <math display=inline>\mu = 0</math>, the distribution is called simply [[chi-squared distribution|chi-squared]].
* The log-likelihood of a normal variable {{tmath|x}} is simply the log of its [[probability density function]]: <math display=block>\ln p(x)= -\frac{1}{2} \left(\frac{x-\mu}{\sigma} \right)^2 -\ln \left(\sigma \sqrt{2\pi} \right).</math> Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted [[chi-squared distribution|chi-squared]] variable.
* The distribution of the variable {{tmath|X}} restricted to an interval <math display=inline>[a, b]</math> is called the [[truncated normal distribution]].
* <math display=inline>(X - \mu)^{-2}</math> has a [[Lévy distribution]] with location 0 and scale <math display=inline>\sigma^{-2}</math>.

===== Operations on two independent normal variables =====
* If <math display=inline>X_1</math> and <math display=inline>X_2</math> are two [[independence (probability theory)|independent]] normal random variables, with means <math display=inline>\mu_1</math>, <math display=inline>\mu_2</math> and variances <math display=inline>\sigma_1^2</math>, <math display=inline>\sigma_2^2</math>, then their sum <math display=inline>X_1 + X_2</math> will also be normally distributed,<sup>[[sum of normally distributed random variables|[proof]]]</sup> with mean <math display=inline>\mu_1 + \mu_2</math> and variance <math display=inline>\sigma_1^2 + \sigma_2^2</math>.
* In particular, if {{tmath|X}} and {{tmath|Y}} are independent normal deviates with zero mean and variance <math display=inline>\sigma^2</math>, then <math display=inline>X + Y</math> and <math display=inline>X - Y</math> are also independent and normally distributed, with zero mean and variance <math display=inline>2\sigma^2</math>. This is a special case of the [[polarization identity]].<ref>{{harvtxt |Bryc |1995 |p=27 }}</ref>
* If <math display=inline>X_1</math>, <math display=inline>X_2</math> are two independent normal deviates with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, and {{tmath|a}}, {{tmath|b}} are arbitrary real numbers, then the variable <math display=block>
    X_3 = \frac{aX_1 + bX_2 - (a+b)\mu}{\sqrt{a^2+b^2}} + \mu
</math> is also normally distributed with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>. It follows that the normal distribution is [[stable distribution|stable]] (with exponent <math display=inline>\alpha=2</math>).
* If <math display=inline>X_k \sim \mathcal N(m_k, \sigma_k^2)</math>, <math display=inline>k \in \{ 0, 1 \}</math> are normal distributions, then their normalized [[geometric mean]] <math display=inline>\frac{1}{\int_{\R^n} X_0^{\alpha}(x) X_1^{1 - \alpha}(x) \, \text{d}x} X_0^{\alpha} X_1^{1 - \alpha}</math> is a normal distribution <math display=inline>\mathcal N(m_{\alpha}, \sigma_{\alpha}^2)</math> with <math display=inline>m_{\alpha} = \frac{\alpha m_0 \sigma_1^2 + (1 - \alpha) m_1 \sigma_0^2}{\alpha \sigma_1^2 + (1 - \alpha) \sigma_0^2}</math> and <math display=inline>\sigma_{\alpha}^2 = \frac{\sigma_0^2 \sigma_1^2}{\alpha \sigma_1^2 + (1 - \alpha) \sigma_0^2}</math> (see [https://www.geogebra.org/geometry/qfqnjtsu here] for a visualization).

===== Operations on two independent standard normal variables =====
If <math display=inline>X_1</math> and <math display=inline>X_2</math> are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: <math display=inline>X_1 \pm X_2 \sim \mathcal{N}(0, 2)</math>.
* Their product <math display=inline>Z = X_1 X_2</math> follows the [[product distribution#Independent central-normal distributions|product distribution]]<ref>{{cite web|url = http://mathworld.wolfram.com/NormalProductDistribution.html |title = Normal Product Distribution|work = MathWorld |publisher =wolfram.com| first = Eric W. |last = Weisstein}}</ref> with density function <math display=inline>f_Z(z) = \pi^{-1} K_0(|z|)</math> where <math display=inline>K_0</math> is the [[Macdonald function|modified Bessel function of the second kind]]. This distribution is symmetric around zero, unbounded at <math display=inline>z = 0</math>, and has the [[characteristic function (probability theory)|characteristic function]] <math display=inline>\phi_Z(t) = (1 + t^2)^{-1/2}</math>.
* Their ratio follows the standard [[Cauchy distribution]]: <math display=inline>X_1/ X_2 \sim \operatorname{Cauchy}(0, 1)</math>.
* Their Euclidean norm <math display=inline>\sqrt{X_1^2 + X_2^2}</math> has the [[Rayleigh distribution]].

==== Operations on multiple independent normal variables ====
* Any [[linear combination]] of independent normal deviates is a normal deviate.
* If <math display=inline>X_1, X_2, \ldots, X_n</math> are independent standard normal random variables, then the sum of their squares has the [[chi-squared distribution]] with {{tmath|n}} degrees of freedom <math display=block>X_1^2 + \cdots + X_n^2 \sim \chi_n^2.</math>
* If <math display=inline>X_1, X_2, \ldots, X_n</math> are independent normally distributed random variables with means {{tmath|\mu}} and variances <math display=inline>\sigma^2</math>, then their [[sample mean]] is independent from the sample [[standard deviation]],<ref>{{cite journal|title=A Characterization of the Normal Distribution |last=Lukacs |first=Eugene |journal=[[The Annals of Mathematical Statistics]] |issn=0003-4851 |volume=13|issue=1 |year=1942 |pages=91–3 |jstor=2236166 |doi=10.1214/aoms/1177731647 |doi-access=free}}</ref> which can be demonstrated using [[Basu's theorem]] or [[Cochran's theorem]].<ref>{{cite journal |title=On Some Characterizations of the Normal Distribution | last1=Basu|first1=D. |last2=Laha|first2=R. G.|journal=[[Sankhyā (journal)|Sankhyā]]|issn=0036-4452| volume=13|issue=4|year=1954|pages=359–62| jstor=25048183}}</ref> The ratio of these two quantities will have the [[Student's t-distribution]] with <math display=inline>n-1</math> degrees of freedom: <math display=block>t = \frac{\overline X - \mu}{S/\sqrt{n}} = \frac{\frac{1}{n}(X_1+\cdots+X_n) - \mu}{\sqrt{\frac{1}{n(n-1)}\left[(X_1-\overline X)^2 + \cdots+(X_n-\overline X)^2\right]}} \sim t_{n-1}.</math>
* If <math display=inline>X_1, X_2, \ldots, X_n</math>, <math display=inline>Y_1, Y_2, \ldots, Y_m</math> are independent standard normal random variables, then the ratio of their normalized sums of squares will have the [[F-distribution]] with {{math|(''n'', ''m'')}} degrees of freedom:<ref>{{cite book |title=Testing Statistical Hypotheses |edition=2nd | first=E. L. | last=Lehmann | publisher=Springer |year=1997 | isbn=978-0-387-94919-2| page=199}}</ref> <math display=block>F = \frac{\left(X_1^2+X_2^2+\cdots+X_n^2\right)/n}{\left(Y_1^2+Y_2^2+\cdots+Y_m^2\right)/m} \sim F_{n,m}.</math>

==== Operations on multiple correlated normal variables ====
* A [[quadratic form]] of a normal vector, i.e. a quadratic function <math display=inline>q = \sum x_i^2 + \sum x_j + c</math> of multiple independent or correlated normal variables, is a [[generalized chi-square distribution|generalized chi-square]] variable.

=== Operations on the density function ===
The [[split normal distribution]] is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The [[truncated normal distribution]] results from rescaling a section of a single density function.

=== Infinite divisibility and Cramér's theorem ===
For any positive integer <math display=inline>\text{n}</math>, any normal distribution with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math> is the distribution of the sum of <math display=inline>\text{n}</math> independent normal deviates, each with mean <math display=inline>\frac{\mu}{n}</math> and variance <math display=inline>\frac{\sigma^2}{n}</math>. This property is called [[infinite divisibility (probability)|infinite divisibility]].<ref>{{harvtxt |Patel |Read |1996 |loc=[2.3.6] }}</ref>

Conversely, if <math display=inline>X_1</math> and <math display=inline>X_2</math> are independent random variables and their sum <math display=inline>X_1+X_2</math> has a normal distribution, then both <math display=inline>X_1</math> and <math display=inline>X_2</math> must be normal deviates.<ref>{{harvtxt |Galambos |Simonelli |2004 |loc=Theorem&nbsp;3.5 }}</ref>

This result is known as [[Cramér's decomposition theorem]], and is equivalent to saying that the [[convolution]] of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.<ref name="Bryc 1995 35">{{harvtxt |Bryc |1995 |p=35 }}</ref>

=== The Kac–Bernstein theorem ===
The [[Kac–Bernstein theorem]] states that if <math display="inline">X</math> and {{tmath|Y}} are independent and <math display=inline>X + Y</math> and <math display=inline>X - Y</math> are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.<ref name="Lukacs">{{harvtxt |Lukacs |King |1954 }}</ref><ref>{{cite journal| last1=Quine| first1=M.P. |year=1993|title=On three characterisations of the normal distribution |url=http://www.math.uni.wroc.pl/~pms/publicationsArticle.php?nr=14.2&nrA=8&ppB=257&ppE=263 |journal=Probability and Mathematical Statistics|volume=14 |issue=2 |pages=257–263}}</ref>

More generally, if <math display=inline>X_1, \ldots, X_n</math> are independent random variables, then two distinct linear combinations <math display=inline>\sum{a_kX_k}</math> and <math display=inline>\sum{b_kX_k}</math>will be independent if and only if all <math display=inline>X_k</math> are normal and <math display=inline>\sum{a_kb_k\sigma_k^2=0}</math>, where <math display=inline>\sigma_k^2</math> denotes the variance of <math display=inline>X_k</math>.<ref name="Lukacs" />

=== Extensions ===
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The [[multivariate normal distribution]] describes the Gaussian law in the ''k''-dimensional [[Euclidean space]]. A vector {{math|''X'' ∈ '''R'''<sup>''k''</sup>}} is multivariate-normally distributed if any linear combination of its components {{math|Σ{{su|p=''k''|b=''j''=1}}''a<sub>j</sub> X<sub>j</sub>''}} has a (univariate) normal distribution. The variance of ''X'' is a ''k×k'' symmetric positive-definite matrix&nbsp;''V''. The multivariate normal distribution is a special case of the [[elliptical distribution]]s. As such, its iso-density loci in the ''k'' = 2 case are [[ellipse]]s and in the case of arbitrary ''k'' are [[ellipsoid]]s.
* [[Rectified Gaussian distribution]] a rectified version of normal distribution with all the negative elements reset to 0
* [[Complex normal distribution]] deals with the complex normal vectors. A complex vector {{math|''X'' ∈ '''C'''<sup>''k''</sup>}} is said to be normal if both its real and imaginary components jointly possess a 2''k''-dimensional multivariate normal distribution. The variance-covariance structure of ''X'' is described by two matrices: the ''variance'' matrix&nbsp;Γ, and the ''relation'' matrix&nbsp;''C''.
* [[Matrix normal distribution]] describes the case of normally distributed matrices.
* [[Gaussian process]]es are the normally distributed [[stochastic process]]es. These can be viewed as elements of some infinite-dimensional [[Hilbert space]]&nbsp;''H'', and thus are the analogues of multivariate normal vectors for the case {{math|''k'' {{=}} ∞}}. A random element {{math|''h'' ∈ ''H''}} is said to be normal if for any constant {{math|''a'' ∈ ''H''}} the [[scalar product]] {{math|(''a'', ''h'')}} has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance {{math|operator K: H → H}}''. Several Gaussian processes became popular enough to have their own names:
** [[Wiener process|Brownian motion]],
** [[Brownian bridge]],
** [[Ornstein–Uhlenbeck process]].
* [[Gaussian q-distribution]] is an abstract mathematical construction that represents a [[q-analogue]] of the normal distribution.
* the [[q-Gaussian]] is an analogue of the Gaussian distribution, in the sense that it maximises the [[Tsallis entropy]], and is one type of [[Tsallis distribution]]. This distribution is different from the [[Gaussian q-distribution]] above.
* The [[Kaniadakis Gaussian distribution|Kaniadakis ''κ''-Gaussian distribution]] is a generalization of the Gaussian distribution which arises from the [[Kaniadakis statistics]], being one of the [[Kaniadakis distribution]]s.

A random variable ''X'' has a two-piece normal distribution if it has a distribution

<math display=block>f_X( x ) = N( \mu, \sigma_1^2 )  \text{ if } x \le \mu</math>
<math display=block>f_X( x ) = N( \mu, \sigma_2^2 )  \text{ if } x \ge \mu</math>

where ''μ'' is the mean and ''σ''<sub>1</sub><sup>2</sup> and ''σ''<sub>2</sub><sup>2</sup> are the variances of the distribution to the left and right of the mean respectively.

The mean, variance and third central moment of this distribution have been determined<ref name="John-1982">{{cite journal|last1=John|first1=S|year=1982|title=The three parameter two-piece normal family of distributions and its fitting|journal=Communications in Statistics – Theory and Methods|volume=11|issue=8|pages=879–885|doi=10.1080/03610928208828279}}</ref>

<math display=block>\operatorname{E}( X ) = \mu +  \sqrt{\frac 2 \pi } ( \sigma_2 - \sigma_1 )</math>
<math display=block>\operatorname{V}( X ) = \left( 1 - \frac 2 \pi\right)( \sigma_2 - \sigma_1 )^2 + \sigma_1 \sigma_2</math>
<math display=block>\operatorname{T}( X ) = \sqrt{ \frac 2 \pi}( \sigma_2 - \sigma_1 ) \left[ \left( \frac 4 \pi  - 1 \right) ( \sigma_2 - \sigma_1)^2 + \sigma_1 \sigma_2 \right]</math>

where E(''X''), V(''X'') and T(''X'') are the mean, variance, and third central moment respectively.

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* [[Pearson distribution]] — a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The [[generalized normal distribution]], also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

== Statistical inference ==
=== Estimation of parameters ===
{{See also|Maximum likelihood#Continuous distribution, continuous parameter space|Gaussian function#Estimation of parameters}}

It is often the case that we do not know the parameters of the normal distribution, but instead want to [[Estimation theory|estimate]] them. That is, having a sample <math display=inline>(x_1, \ldots, x_n)</math> from a normal <math display=inline>\mathcal{N}(\mu, \sigma^2)</math> population we would like to learn the approximate values of parameters {{tmath|\mu}} and <math display=inline>\sigma^2</math>. The standard approach to this problem is the [[maximum likelihood]] method, which requires maximization of the ''[[log-likelihood function]]'':{{anchor|Log-likelihood}}
<math display=block>
   \ln\mathcal{L}(\mu,\sigma^2)
     = \sum_{i=1}^n \ln f(x_i\mid\mu,\sigma^2)
     = -\frac{n}{2}\ln(2\pi) - \frac{n}{2}\ln\sigma^2 - \frac{1}{2\sigma^2}\sum_{i=1}^n (x_i-\mu)^2.
</math>
Taking derivatives with respect to {{tmath|\mu}} and <math display=inline>\sigma^2</math> and solving the resulting system of first order conditions yields the ''maximum likelihood estimates'':
<math display=block>
    \hat{\mu} = \overline{x} \equiv \frac{1}{n}\sum_{i=1}^n x_i, \qquad
    \hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \overline{x})^2.
</math>

Then <math display=inline>\ln\mathcal{L}(\hat{\mu},\hat{\sigma}^2)</math> is as follows:

<math display=block>\ln\mathcal{L}(\hat{\mu},\hat{\sigma}^2)
     = (-n/2) [\ln(2 \pi \hat{\sigma}^2)+1]</math>

==== Sample mean ====
{{See also|Standard error of the mean}}

Estimator <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is called the ''[[sample mean]]'', since it is the arithmetic mean of all observations. The statistic <math style="vertical-align:0">\textstyle\overline{x}</math> is [[complete statistic|complete]] and [[sufficient statistic|sufficient]] for {{tmath|\mu}}, and therefore by the [[Lehmann–Scheffé theorem]], <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is the [[uniformly minimum variance unbiased]] (UMVU) estimator.<ref name="Krishnamoorthy">{{harvtxt |Krishnamoorthy |2006 |p=127 }}</ref> In finite samples it is distributed normally:
<math display=block>
    \hat\mu \sim \mathcal{N}(\mu,\sigma^2/n).
</math>
The variance of this estimator is equal to the ''μμ''-element of the inverse [[Fisher information matrix]] <math style="vertical-align:0">\textstyle\mathcal{I}^{-1}</math>. This implies that the estimator is [[efficient estimator|finite-sample efficient]]. Of practical importance is the fact that the [[standard error]] of <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is proportional to <math style="vertical-align:-.3em">\textstyle1/\sqrt{n}</math>, that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in [[Monte Carlo simulation]]s.

From the standpoint of the [[asymptotic theory (statistics)|asymptotic theory]], <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is [[consistent estimator|consistent]], that is, it [[converges in probability]] to {{tmath|\mu}} as <math display=inline>n\rightarrow\infty</math>. The estimator is also [[asymptotic normality|asymptotically normal]], which is a simple corollary of the fact that it is normal in finite samples:
<math display=block>
    \sqrt{n}(\hat\mu-\mu) \,\xrightarrow{d}\, \mathcal{N}(0,\sigma^2).
</math>

==== Sample variance ====
{{See also|Standard deviation#Estimation|Variance#Estimation}}

The estimator <math style="vertical-align:0">\textstyle\hat\sigma^2</math> is called the ''[[sample variance]]'', since it is the variance of the sample (<math display=inline>(x_1, \ldots, x_n)</math>). In practice, another estimator is often used instead of the <math style="vertical-align:0">\textstyle\hat\sigma^2</math>. This other estimator is denoted  <math display=inline>s^2</math>, and is also called the ''sample variance'', which represents a certain ambiguity in terminology; its square root {{tmath|s}} is called the ''sample standard deviation''. The estimator <math display=inline>s^2</math> differs from <math style="vertical-align:0">\textstyle\hat\sigma^2</math> by having {{math|(''n'' − 1)}} instead of&nbsp;''n'' in the denominator (the so-called [[Bessel's correction]]):
<math display=block>
    s^2 = \frac{n}{n-1} \hat\sigma^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2.
</math>
The difference between <math display=inline>s^2</math> and <math style="vertical-align:0">\textstyle\hat\sigma^2</math> becomes negligibly small for large ''n''{{'}}s. In finite samples however, the motivation behind the use of <math display=inline>s^2</math> is that it is an [[unbiased estimator]] of the underlying parameter <math display=inline>\sigma^2</math>, whereas <math style="vertical-align:0">\textstyle\hat\sigma^2</math> is biased. Also, by the Lehmann–Scheffé theorem the estimator <math display=inline>s^2</math> is uniformly minimum variance unbiased ([[UMVU]]),<ref name="Krishnamoorthy" /> which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator <math style="vertical-align:0">\textstyle\hat\sigma^2</math> is better than the <math display=inline>s^2</math> in terms of the [[mean squared error]] (MSE) criterion. In finite samples both <math display=inline>s^2</math> and <math style="vertical-align:0">\textstyle\hat\sigma^2</math> have scaled [[chi-squared distribution]] with {{math|(''n'' − 1)}} degrees of freedom:
<math display=block>
    s^2 \sim \frac{\sigma^2}{n-1} \cdot \chi^2_{n-1}, \qquad
    \hat\sigma^2 \sim \frac{\sigma^2}{n} \cdot \chi^2_{n-1}.
</math>
The first of these expressions shows that the variance of <math display=inline>s^2</math> is equal to <math display=inline>2\sigma^4/(n-1)</math>, which is slightly greater than the ''σσ''-element of the inverse Fisher information matrix <math style="vertical-align:0">\textstyle\mathcal{I}^{-1}</math>, which is <math display=inline>2\sigma^4/n</math>. Thus, <math display=inline>s^2</math> is not an efficient estimator for <math display=inline>\sigma^2</math>, and moreover, since <math display=inline>s^2</math> is UMVU, we can conclude that the finite-sample efficient estimator for <math display=inline>\sigma^2</math> does not exist.

Applying the asymptotic theory, both estimators <math display=inline>s^2</math> and <math style="vertical-align:0">\textstyle\hat\sigma^2</math> are consistent, that is they converge in probability to <math display=inline>\sigma^2</math> as the sample size <math display=inline>n\rightarrow\infty</math>. The two estimators are also both asymptotically normal:
<math display=block>
    \sqrt{n}(\hat\sigma^2 - \sigma^2) \simeq
    \sqrt{n}(s^2-\sigma^2) \,\xrightarrow{d}\, \mathcal{N}(0,2\sigma^4).
</math>
In particular, both estimators are asymptotically efficient for  <math display=inline>\sigma^2</math>.

=== Confidence intervals ===
{{See also|Studentization|3-sigma rule}}

By [[Cochran's theorem]], for normal distributions the sample mean <math style="vertical-align:-.3em">\textstyle\hat\mu</math> and the sample variance ''s''<sup>2</sup> are [[independence (probability theory)|independent]], which means there can be no gain in considering their [[joint distribution]]. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between <math style="vertical-align:-.3em">\textstyle\hat\mu</math> and ''s'' can be employed to construct the so-called ''t-statistic'':
<math display=block>
    t = \frac{\hat\mu-\mu}{s/\sqrt{n}} = \frac{\overline{x}-\mu}{\sqrt{\frac{1}{n(n-1)}\sum(x_i-\overline{x})^2}} \sim t_{n-1}
</math>
This quantity ''t'' has the [[Student's t-distribution]] with {{math|(''n'' − 1)}} degrees of freedom, and it is an [[ancillary statistic]] (independent of the value of the parameters). Inverting the distribution of this ''t''-statistics will allow us to construct the [[confidence interval]] for ''μ'';<ref>{{harvtxt |Krishnamoorthy |2006 |p=130 }}</ref> similarly, inverting the ''χ''<sup>2</sup> distribution of the statistic ''s''<sup>2</sup> will give us the confidence interval for ''σ''<sup>2</sup>:<ref>{{harvtxt |Krishnamoorthy |2006 |p=133 }}</ref>
<math display=block>\mu \in \left[ \hat\mu - t_{n-1,1-\alpha/2} \frac{s}{\sqrt{n}},\,
                                   \hat\mu + t_{n-1,1-\alpha/2} \frac{s}{\sqrt{n}} \right]</math>
<math display=block>\sigma^2 \in \left[ \frac{n-1}{\chi^2_{n-1,1-\alpha/2}}s^2,\,
                                        \frac{n-1}{\chi^2_{n-1,\alpha/2}}s^2\right]</math>
where ''t<sub>k,p</sub>'' and {{SubSup|χ|''k,p''|2}} are the ''p''th [[quantile]]s of the ''t''- and ''χ''<sup>2</sup>-distributions respectively. These confidence intervals are of the ''[[confidence level]]'' {{math|1 − ''α''}}, meaning that the true values ''μ'' and ''σ''<sup>2</sup> fall outside of these intervals with probability (or [[significance level]]) ''α''. In practice people usually take {{math|''α'' {{=}} 5%}}, resulting in the 95% confidence intervals. The confidence interval for ''σ'' can be found by taking the square root of the interval bounds for ''σ''<sup>2</sup>.

Approximate formulas can be derived from the asymptotic distributions of <math style="vertical-align:-.3em">\textstyle\hat\mu</math> and ''s''<sup>2</sup>:
<math display=block>\mu \in
              \left[ \hat\mu - \frac{|z_{\alpha/2}|}{\sqrt n}s,\,
                      \hat\mu + \frac{|z_{\alpha/2}|}{\sqrt n}s \right]</math>
<math display=block>\sigma^2 \in
                   \left[ s^2 - \sqrt{2}\frac{|z_{\alpha/2}|}{\sqrt{n}} s^2 ,\,
                          s^2 + \sqrt{2}\frac{|z_{\alpha/2}|}{\sqrt{n}} s^2 \right]</math>
The approximate formulas become valid for large values of ''n'', and are more convenient for the manual calculation since the standard normal quantiles ''z''<sub>''α''/2</sub> do not depend on ''n''. In particular, the most popular value of {{math|''α'' {{=}} 5%}}, results in {{math|{{!}}''z''<sub>0.025</sub>{{!}} {{=}} [[1.96]]}}.

=== Normality tests ===
{{Main|Normality tests}}

Normality tests assess the likelihood that the given data set {''x''<sub>1</sub>, ..., ''x<sub>n</sub>''} comes from a normal distribution. Typically the [[null hypothesis]] ''H''<sub>0</sub> is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''<sup>2</sup>, versus the alternative ''H<sub>a</sub>'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

'''Diagnostic plots''' are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* [[Q–Q plot]], also known as [[normal probability plot]] or [[rankit]] plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ<sup>−1</sup>(''p<sub>k</sub>''), ''x''<sub>(''k'')</sub>), where plotting points ''p<sub>k</sub>'' are equal to ''p<sub>k</sub>''&nbsp;=&nbsp;(''k''&nbsp;−&nbsp;''α'')/(''n''&nbsp;+&nbsp;1&nbsp;−&nbsp;2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and&nbsp;1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* [[P–P plot]] – similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''<sub>(''k'')</sub>), ''p<sub>k</sub>''), where <math display=inline>\textstyle z_{(k)} = (x_{(k)}-\hat\mu)/\hat\sigma</math>. For normally distributed data this plot should lie on a 45° line between (0,&nbsp;0) and&nbsp;(1,&nbsp;1).
'''Goodness-of-fit tests''':

''Moment-based tests'':
* [[D'Agostino's K-squared test]]
* [[Jarque–Bera test]]
* [[Shapiro–Wilk test]]: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* [[Anderson–Darling test]]
* [[Lilliefors test]] (an adaptation of the [[Kolmogorov–Smirnov test]])

=== Bayesian analysis of the normal distribution ===
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
* Either the mean, or the variance, or neither, may be considered a fixed quantity.
* When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the [[precision (statistics)|precision]], the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
* Both univariate and [[multivariate normal distribution|multivariate]] cases need to be considered.
* Either [[conjugate prior|conjugate]] or [[improper prior|improper]] [[prior distribution]]s may be placed on the unknown variables.
* An additional set of cases occurs in [[Bayesian linear regression]], where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the [[regression coefficient]]s. The resulting analysis is similar to the basic cases of [[independent identically distributed]] data.

The formulas for the non-linear-regression cases are summarized in the [[conjugate prior]] article.

==== Sum of two quadratics ====

===== Scalar form =====
The following auxiliary formula is useful for simplifying the [[posterior distribution|posterior]] update equations, which otherwise become fairly tedious.

<math display=block>a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac{ay+bz}{a+b}\right)^2 + \frac{ab}{a+b}(y-z)^2</math>

This equation rewrites the sum of two quadratics in ''x'' by expanding the squares, grouping the terms in ''x'', and [[completing the square]]. Note the following about the complex constant factors attached to some of the terms:
# The factor <math display=inline>\frac{ay+bz}{a+b}</math> has the form of a [[weighted average]] of ''y'' and ''z''.
# <math display=inline>\frac{ab}{a+b} = \frac{1}{\frac{1}{a}+\frac{1}{b}} = (a^{-1} + b^{-1})^{-1}.</math> This shows that this factor can be thought of as resulting from a situation where the [[Multiplicative inverse|reciprocals]] of quantities ''a'' and ''b'' add directly, so to combine ''a'' and ''b'' themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the [[harmonic mean]], so it is not surprising that <math display=inline>\frac{ab}{a+b}</math> is one-half the [[harmonic mean]] of ''a'' and ''b''.

===== Vector form =====
A similar formula can be written for the sum of two vector quadratics: If '''x''', '''y''', '''z''' are vectors of length ''k'', and '''A''' and '''B''' are [[symmetric matrix|symmetric]], [[invertible matrices]] of size <math display=inline>k\times k</math>, then

<math display=block>
\begin{align}
& (\mathbf{y}-\mathbf{x})'\mathbf{A}(\mathbf{y}-\mathbf{x}) + (\mathbf{x}-\mathbf{z})' \mathbf{B}(\mathbf{x}-\mathbf{z}) \\
= {} & (\mathbf{x} - \mathbf{c})'(\mathbf{A}+\mathbf{B})(\mathbf{x} - \mathbf{c}) + (\mathbf{y} - \mathbf{z})'(\mathbf{A}^{-1} + \mathbf{B}^{-1})^{-1}(\mathbf{y} - \mathbf{z})
\end{align}
</math>

where

<math display=block>\mathbf{c} = (\mathbf{A} + \mathbf{B})^{-1}(\mathbf{A}\mathbf{y} + \mathbf{B} \mathbf{z})</math>

The form '''x'''′ '''A''' '''x''' is called a [[quadratic form]] and is a [[scalar (mathematics)|scalar]]:
<math display=block>\mathbf{x}'\mathbf{A}\mathbf{x} = \sum_{i,j}a_{ij} x_i x_j</math>
In other words, it sums up all possible combinations of products of pairs of elements from '''x''', with a separate coefficient for each. In addition, since <math display=inline>x_i x_j = x_j x_i</math>, only the sum <math display=inline>a_{ij} + a_{ji}</math> matters for any off-diagonal elements of '''A''', and there is no loss of generality in assuming that '''A''' is [[symmetric matrix|symmetric]]. Furthermore, if '''A''' is symmetric, then the form <math display=inline>\mathbf{x}'\mathbf{A}\mathbf{y} = \mathbf{y}'\mathbf{A}\mathbf{x}.</math>

==== Sum of differences from the mean ====
Another useful formula is as follows:
<math display=block>\sum_{i=1}^n (x_i-\mu)^2 = \sum_{i=1}^n (x_i-\bar{x})^2 + n(\bar{x} -\mu)^2</math>
where <math display=inline>\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i.</math>

=== With known variance ===
For a set of [[i.i.d.]] normally distributed data points '''X''' of size ''n'' where each individual point ''x'' follows <math display=inline>x \sim \mathcal{N}(\mu, \sigma^2)</math> with known [[variance]] σ<sup>2</sup>, the [[conjugate prior]] distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the [[precision (statistics)|precision]], i.e. using τ = 1/σ<sup>2</sup>. Then if <math display=inline>x \sim \mathcal{N}(\mu, 1/\tau)</math> and <math display=inline>\mu \sim \mathcal{N}(\mu_0, 1/\tau_0),</math> we proceed as follows.

First, the [[likelihood function]] is (using the formula above for the sum of differences from the mean):

<math display=block>\begin{align}
p(\mathbf{X}\mid\mu,\tau) &= \prod_{i=1}^n \sqrt{\frac{\tau}{2\pi}} \exp\left(-\frac{1}{2}\tau(x_i-\mu)^2\right) \\
&= \left(\frac{\tau}{2\pi}\right)^{n/2} \exp\left(-\frac{1}{2}\tau \sum_{i=1}^n (x_i-\mu)^2\right) \\
&= \left(\frac{\tau}{2\pi}\right)^{n/2} \exp\left[-\frac{1}{2}\tau \left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right)\right].
\end{align}</math>

Then, we proceed as follows:

<math display=block>\begin{align}
p(\mu\mid\mathbf{X}) &\propto p(\mathbf{X}\mid\mu) p(\mu) \\
& = \left(\frac{\tau}{2\pi}\right)^{n/2} \exp\left[-\frac{1}{2}\tau \left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right)\right] \sqrt{\frac{\tau_0}{2\pi}} \exp\left(-\frac{1}{2}\tau_0(\mu-\mu_0)^2\right) \\
&\propto \exp\left(-\frac{1}{2}\left(\tau\left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\
&\propto \exp\left(-\frac{1}{2} \left(n\tau(\bar{x}-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\
&= \exp\left(-\frac{1}{2}(n\tau + \tau_0)\left(\mu - \dfrac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}\right)^2 + \frac{n\tau\tau_0}{n\tau+\tau_0}(\bar{x} - \mu_0)^2\right) \\
&\propto \exp\left(-\frac{1}{2}(n\tau + \tau_0)\left(\mu - \dfrac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}\right)^2\right)
\end{align}</math>

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving&nbsp;''μ''. The result is the [[kernel (statistics)|kernel]] of a normal distribution, with mean <math display=inline>\frac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}</math> and precision <math display=inline>n\tau + \tau_0</math>, i.e.

<math display=block>p(\mu\mid\mathbf{X}) \sim \mathcal{N}\left(\frac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}, \frac{1}{n\tau + \tau_0}\right)</math>

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

<math display=block>\begin{align}
\tau_0' &= \tau_0 + n\tau \\[5pt]
\mu_0' &= \frac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0} \\[5pt]
\bar{x} &= \frac{1}{n}\sum_{i=1}^n x_i
\end{align}</math>

That is, to combine ''n'' data points with total precision of ''nτ'' (or equivalently, total variance of ''n''/''σ''<sup>2</sup>) and mean of values <math display=inline>\bar{x}</math>, derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a ''precision-weighted average'', i.e. a [[weighted average]] of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do [[Bayesian analysis]] of [[conjugate prior]]s for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

<math display=block>\begin{align}
{\sigma^2_0}' &= \frac{1}{\frac{n}{\sigma^2} + \frac{1}{\sigma_0^2}} \\[5pt]
\mu_0' &= \frac{\frac{n\bar{x}}{\sigma^2} + \frac{\mu_0}{\sigma_0^2}}{\frac{n}{\sigma^2} + \frac{1}{\sigma_0^2}} \\[5pt]
\bar{x} &= \frac{1}{n}\sum_{i=1}^n x_i
\end{align}</math>

==== With known mean ====
For a set of [[i.i.d.]] normally distributed data points '''X''' of size ''n'' where each individual point ''x'' follows <math display=inline>x \sim \mathcal{N}(\mu, \sigma^2)</math> with known mean μ, the [[conjugate prior]] of the [[variance]] has an [[inverse gamma distribution]] or a [[scaled inverse chi-squared distribution]]. The two are equivalent except for having different [[parameter]]izations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ<sup>2</sup> is as follows:

<math display=block>p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac{(\sigma_0^2\frac{\nu_0}{2})^{\nu_0/2}}{\Gamma\left(\frac{\nu_0}{2} \right)}~\frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\frac{\nu_0}{2}}} \propto \frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\frac{\nu_0}{2}}}</math>

The [[likelihood function]] from above, written in terms of the variance, is:

<math display=block>\begin{align}
p(\mathbf{X}\mid\mu,\sigma^2) &= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{1}{2\sigma^2} \sum_{i=1}^n (x_i-\mu)^2\right] \\
&= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{S}{2\sigma^2}\right]
\end{align}</math>

where

<math display=block>S = \sum_{i=1}^n (x_i-\mu)^2.</math>

Then:

<math display=block>\begin{align}
p(\sigma^2\mid\mathbf{X}) &\propto p(\mathbf{X}\mid\sigma^2) p(\sigma^2) \\
&= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{S}{2\sigma^2}\right] \frac{(\sigma_0^2\frac{\nu_0}{2})^{\frac{\nu_0}{2}}}{\Gamma\left(\frac{\nu_0}{2} \right)}~\frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\frac{\nu_0}{2}}} \\
&\propto \left(\frac{1}{\sigma^2}\right)^{n/2} \frac{1}{(\sigma^2)^{1+\frac{\nu_0}{2}}} \exp\left[-\frac{S}{2\sigma^2} + \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right] \\
&= \frac{1}{(\sigma^2)^{1+\frac{\nu_0+n}{2}}} \exp\left[-\frac{\nu_0 \sigma_0^2 + S}{2\sigma^2}\right]
\end{align}</math>

The above is also a scaled inverse chi-squared distribution where

<math display=block>\begin{align}
\nu_0' &= \nu_0 + n \\
\nu_0'{\sigma_0^2}' &= \nu_0 \sigma_0^2 + \sum_{i=1}^n (x_i-\mu)^2
\end{align}</math>

or equivalently

<math display=block>\begin{align}
\nu_0' &= \nu_0 + n \\
{\sigma_0^2}' &= \frac{\nu_0 \sigma_0^2 + \sum_{i=1}^n (x_i-\mu)^2}{\nu_0+n}
\end{align}</math>

Reparameterizing in terms of an [[inverse gamma distribution]], the result is:

<math display=block>\begin{align}
\alpha' &= \alpha + \frac{n}{2} \\
\beta' &= \beta + \frac{\sum_{i=1}^n (x_i-\mu)^2}{2}
\end{align}</math>

==== With unknown mean and unknown variance ====
For a set of [[i.i.d.]] normally distributed data points '''X''' of size ''n'' where each individual point ''x'' follows <math display=inline>x \sim \mathcal{N}(\mu, \sigma^2)</math> with unknown mean μ and unknown [[variance]] σ<sup>2</sup>, a combined (multivariate) [[conjugate prior]] is placed over the mean and variance, consisting of a [[normal-inverse-gamma distribution]].
Logically, this originates as follows:
# From the analysis of the case with unknown mean but known variance, we see that the update equations involve [[sufficient statistic]]s computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
# From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and [[sum of squared deviations]].
# Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
# To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
# This suggests that we create a ''conditional prior'' of the mean on the unknown variance, with a hyperparameter specifying the mean of the [[pseudo-observation]]s associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
# This leads immediately to the [[normal-inverse-gamma distribution]], which is the product of the two distributions just defined, with [[conjugate prior]]s used (an [[inverse gamma distribution]] over the variance, and a normal distribution over the mean, ''conditional'' on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

<math display=block>\begin{align}
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal{N}(\mu_0,\sigma^2/n_0) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2)
\end{align}</math>
<!-- \\
 & =\frac{(\sigma_0^2\nu_0/2)^{\nu_0/2}}{\Gamma(\nu_0/2)}~\frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\nu_0/2}} \propto \frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\nu_0/2}}
-->

The update equations can be derived, and look as follows:

<math display=block>\begin{align}
\bar{x} &= \frac 1 n \sum_{i=1}^n x_i \\
\mu_0' &= \frac{n_0\mu_0 + n\bar{x}}{n_0 + n} \\
n_0' &= n_0 + n \\
\nu_0' &= \nu_0 + n \\
\nu_0'{\sigma_0^2}' &= \nu_0 \sigma_0^2 + \sum_{i=1}^n (x_i-\bar{x})^2 + \frac{n_0 n}{n_0 + n}(\mu_0 - \bar{x})^2
\end{align}</math>

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for <math display=inline>\nu_0'{\sigma_0^2}'</math> is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

{{math proof | proof =
The prior distributions are
<math display=block>\begin{align}
p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal{N}(\mu_0,\sigma^2/n_0) = \frac{1}{\sqrt{2\pi\frac{\sigma^2}{n_0}}} \exp\left(-\frac{n_0}{2\sigma^2}(\mu-\mu_0)^2\right) \\
&\propto (\sigma^2)^{-1/2} \exp\left(-\frac{n_0}{2\sigma^2}(\mu-\mu_0)^2\right) \\
p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2) \\
&= \frac{(\sigma_0^2\nu_0/2)^{\nu_0/2}}{\Gamma(\nu_0/2)}~\frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\nu_0/2}} \\
&\propto {(\sigma^2)^{-(1+\nu_0/2)}} \exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right].
\end{align}</math>

Therefore, the joint prior is

<math display=block>\begin{align}
p(\mu,\sigma^2; \mu_0, n_0, \nu_0,\sigma_0^2) &= p(\mu\mid\sigma^2; \mu_0, n_0)\,p(\sigma^2; \nu_0,\sigma_0^2) \\
&\propto (\sigma^2)^{-(\nu_0+3)/2} \exp\left[-\frac 1 {2\sigma^2}\left(\nu_0\sigma_0^2 + n_0(\mu-\mu_0)^2\right)\right].
\end{align}</math>

The [[likelihood function]] from the section above with known variance is:

<math display=block>\begin{align}
p(\mathbf{X}\mid\mu,\sigma^2) &= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{1}{2\sigma^2} \left(\sum_{i=1}^n(x_i -\mu)^2\right)\right]
\end{align}</math>

Writing it in terms of variance rather than precision, we get:
<math display=block>\begin{align}
p(\mathbf{X}\mid\mu,\sigma^2) &= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{1}{2\sigma^2} \left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right)\right] \\
&\propto {\sigma^2}^{-n/2} \exp\left[-\frac{1}{2\sigma^2} \left(S + n(\bar{x} -\mu)^2\right)\right]
\end{align}</math>
where <math display=inline>S = \sum_{i=1}^n(x_i-\bar{x})^2.</math>

Therefore, the posterior is (dropping the hyperparameters as conditioning factors):
<math display=block>\begin{align}
p(\mu,\sigma^2\mid\mathbf{X}) & \propto p(\mu,\sigma^2) \, p(\mathbf{X}\mid\mu,\sigma^2) \\
& \propto (\sigma^2)^{-(\nu_0+3)/2} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + n_0(\mu-\mu_0)^2\right)\right] {\sigma^2}^{-n/2} \exp\left[-\frac{1}{2\sigma^2} \left(S + n(\bar{x} -\mu)^2\right)\right] \\
&= (\sigma^2)^{-(\nu_0+n+3)/2} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + S + n_0(\mu-\mu_0)^2 + n(\bar{x} -\mu)^2\right)\right] \\
&= (\sigma^2)^{-(\nu_0+n+3)/2} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + S + \frac{n_0 n}{n_0+n}(\mu_0-\bar{x})^2 + (n_0+n)\left(\mu-\frac{n_0\mu_0 + n\bar{x}}{n_0 + n}\right)^2\right)\right] \\
& \propto (\sigma^2)^{-1/2} \exp\left[-\frac{n_0+n}{2\sigma^2}\left(\mu-\frac{n_0\mu_0 + n\bar{x}}{n_0 + n}\right)^2\right] \\
& \quad\times (\sigma^2)^{-(\nu_0/2+n/2+1)} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + S + \frac{n_0 n}{n_0+n}(\mu_0-\bar{x})^2\right)\right] \\
& = \mathcal{N}_{\mu\mid\sigma^2}\left(\frac{n_0\mu_0 + n\bar{x}}{n_0 + n}, \frac{\sigma^2}{n_0+n}\right) \cdot {\rm IG}_{\sigma^2}\left(\frac12(\nu_0+n), \frac12\left(\nu_0\sigma_0^2 + S + \frac{n_0 n}{n_0+n}(\mu_0-\bar{x})^2\right)\right).
\end{align}</math>

In other words, the posterior distribution has the form of a product of a normal distribution over <math display=inline>p(\mu|\sigma^2)</math> times an inverse gamma distribution over <math display=inline>p(\sigma^2)</math>, with parameters that are the same as the update equations above.
}}

==Occurrence and applications==
The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the [[central limit theorem]]; and
# Distributions modeled as normal – the normal distribution being the distribution with [[Principle of maximum entropy|maximum entropy]] for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

=== Exact normality ===
[[File:QHarmonicOscillator.png|thumb|The ground state of a [[quantum harmonic oscillator]] has the Gaussian distribution.]]
Certain quantities in [[physics]] are distributed normally, as was first demonstrated by [[James Clerk Maxwell]]. Examples of such quantities are:
* Probability density function of a ground state in a [[quantum harmonic oscillator]].
* The position of a particle that experiences [[diffusion]]. If initially the particle is located at a specific point (that is its probability distribution is the [[Dirac delta function]]), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the [[diffusion equation]]&nbsp;<math display=inline>\frac{\partial}{\partial t} f(x,t) = \frac{1}{2} \frac{\partial^2}{\partial x^2} f(x,t)</math>. If the initial location is given by a certain density function <math display=inline>g(x)</math>, then the density at time ''t'' is the [[convolution]] of ''g'' and the normal probability density function.

=== Approximate normality ===
''Approximately'' normal distributions occur in many situations, as explained by the [[central limit theorem]]. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where [[infinitely divisible]] and [[Indecomposable distribution|decomposable]] distributions are involved, such as
** [[binomial distribution|Binomial random variables]], associated with binary response variables;
** [[Poisson random variables]], associated with rare events;
* [[Thermal radiation]] has a [[Bose–Einstein statistics|Bose–Einstein]] distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

=== Assumed normality ===
[[File:Fisher iris versicolor sepalwidth.svg|thumb|right|Histogram of sepal widths for ''Iris versicolor'' from Fisher's [[Iris flower data set]], with superimposed best-fitting normal distribution]]
{{Blockquote|I can only recognize the occurrence of the normal curve – the Laplacian curve of errors – as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations.|{{harvtxt |Pearson |1901 }}}}
There are statistical methods to empirically test that assumption; see the above [[#Normality tests|Normality tests]] section.
* In [[biology]], the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a [[log-normal distribution]] (after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);<ref>{{harvtxt |Huxley |1932 }}</ref>
** The ''length'' of ''inert'' appendages (hair, claws, nails, teeth) of biological specimens, ''in the direction of growth''; presumably the thickness of tree bark also falls under this category;
** Certain physiological measurements, such as blood pressure of adult humans.
* In finance, in particular the [[Black–Scholes model]], changes in the ''logarithm'' of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like [[compound interest]], not like simple interest, and so are multiplicative). Some mathematicians such as [[Benoit Mandelbrot]] have argued that [[Levy skew alpha-stable distribution|log-Levy distributions]], which possesses [[heavy tails]] would be a more appropriate model, in particular for the analysis for [[stock market crash]]es. The use of the assumption of normal distribution occurring in financial models has also been criticized by [[Nassim Nicholas Taleb]] in his works.
* [[Propagation of uncertainty|Measurement errors]] in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.<ref>{{cite book|last=Jaynes|first=Edwin T.|year=2003|title=Probability Theory: The Logic of Science|publisher=Cambridge University Press|pages=592–593|url=https://books.google.com/books?id=tTN4HuUNXjgC&pg=PA592|isbn=9780521592710}}</ref>
* In [[Standardized testing (statistics)|standardized testing]], results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the [[Intelligence quotient|IQ test]]) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the [[SAT]]'s traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.
[[File:FitNormDistr.tif|thumb|220px|Fitted cumulative normal distribution to October rainfalls, see [[distribution fitting]] ]]
* Many scores are derived from the normal distribution, including [[percentile rank]]s (percentiles or quantiles), [[normal curve equivalent]]s, [[stanine]]s, [[z-scores]], and T-scores. Additionally, some [[Psychological statistics|behavioral statistical]] procedures assume that scores are normally distributed; for example, [[t-tests]] and [[Analysis of variance|ANOVAs]]. [[Bell curve grading]] assigns relative grades based on a normal distribution of scores.
* In [[hydrology]] the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the [[central limit theorem]].<ref>{{cite book|last=Oosterbaan|first=Roland J. | editor-last=Ritzema |editor-first=Henk P.|chapter=Chapter 6: Frequency and Regression Analysis of Hydrologic Data|year=1994 | edition=second revised|title=Drainage Principles and Applications, Publication 16|publisher=International Institute for Land Reclamation and Improvement (ILRI)|location=Wageningen, The Netherlands|pages=175–224|chapter-url=http://www.waterlog.info/pdf/freqtxt.pdf|isbn=978-90-70754-33-4}}</ref> The blue picture, made with [[CumFreq]], illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].

=== Methodological problems and peer review ===
[[John Ioannidis]] [[Why Most Published Research Findings Are False|argued]] that using normally distributed standard deviations as standards for validating research findings leave [[falsifiability|falsifiable predictions]] about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.<ref>Why Most Published Research Findings Are False, John P. A. Ioannidis, 2005</ref>

== Computational methods ==
=== Generating values from normal distribution ===
[[File:Planche de Galton.jpg|thumb|250px|right|The [[bean machine]], a device invented by [[Francis Galton]], can be called the first generator of normal random variables. This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.]]

In computer simulations, especially in applications of the [[Monte-Carlo method]], it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a {{math|''N''(''μ'', ''σ''<sup>2</sup>)}} can be generated as {{math|1=''X'' = ''μ'' + ''σZ''}}, where ''Z'' is standard normal. All these algorithms rely on the availability of a [[random number generator]] ''U'' capable of producing [[Uniform distribution (continuous)|uniform]] random variates.
* The most straightforward method is based on the [[probability integral transform]] property: if ''U'' is distributed uniformly on (0,1), then Φ<sup>−1</sup>(''U'') will have the standard normal distribution. The drawback of this method is that it relies on calculation of the [[probit function]] Φ<sup>−1</sup>, which cannot be done analytically. Some approximate methods are described in {{harvtxt |Hart |1968 }} and in the [[error function|erf]] article. Wichura gives a fast algorithm for computing this function to 16 decimal places,<ref>{{cite journal|last=Wichura |first=Michael J.|year=1988|title=Algorithm AS241: The Percentage Points of the Normal Distribution|journal=Applied Statistics |volume=37|pages=477–84|doi=10.2307/2347330|jstor=2347330|issue=3}}</ref> which is used by [[R programming language|R]] to compute random variates of the normal distribution.
* [[Irwin–Hall distribution#Approximating a Normal distribution|An easy-to-program approximate approach]] that relies on the [[central limit theorem]] is as follows: generate 12 uniform ''U''(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be [[Irwin–Hall distribution|Irwin–Hall]], which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6,&nbsp;6).<ref>{{harvtxt |Johnson |Kotz |Balakrishnan |1995 |loc=Equation (26.48) }}</ref> Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
* The [[Box–Muller method]] uses two independent random numbers ''U'' and ''V'' distributed [[uniform distribution (continuous)|uniformly]] on (0,1). Then the two random variables ''X'' and ''Y'' <math display=block>
    X = \sqrt{- 2 \ln U} \, \cos(2 \pi V) , \qquad
    Y = \sqrt{- 2 \ln U} \, \sin(2 \pi V) .
</math> will both have the standard normal distribution, and will be [[independence (probability theory)|independent]]. This formulation arises because for a [[bivariate normal]] random vector (''X'', ''Y'') the squared norm {{math|''X''<sup>2</sup> + ''Y''<sup>2</sup>}} will have the [[chi-squared distribution]] with two degrees of freedom, which is an easily generated [[exponential random variable]] corresponding to the quantity −2&nbsp;ln(''U'') in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable ''V''.
* The [[Marsaglia polar method]] is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, ''U'' and ''V'' are drawn from the uniform (−1,1) distribution, and then {{math|1=''S'' = ''U''<sup>2</sup> + ''V''<sup>2</sup>}} is computed. If ''S'' is greater or equal to 1, then the method starts over, otherwise the two quantities <math display=block>X = U\sqrt{\frac{-2\ln S}{S}}, \qquad Y = V\sqrt{\frac{-2\ln S}{S}}</math> are returned. Again, ''X'' and ''Y'' are independent, standard normal random variables.
* The Ratio method<ref>{{harvtxt |Kinderman |Monahan |1977 }}</ref> is a rejection method. The algorithm proceeds as follows:
** Generate two independent uniform deviates ''U'' and ''V'';
** Compute ''X'' = {{sqrt|8/''e''}} (''V'' − 0.5)/''U'';
** Optional: if ''X''<sup>2</sup> ≤ 5 − 4''e''<sup>1/4</sup>''U'' then accept ''X'' and terminate algorithm;
** Optional: if ''X''<sup>2</sup> ≥ 4''e''<sup>−1.35</sup>/''U'' + 1.4 then reject ''X'' and start over from step 1;
** If ''X''<sup>2</sup> ≤ −4 ln''U'' then accept ''X'', otherwise start over the algorithm.
*: The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved<ref>{{harvtxt|Leva|1992}}</ref> so that the logarithm is rarely evaluated.
* The [[ziggurat algorithm]]<ref>{{harvtxt |Marsaglia |Tsang |2000 }}</ref> is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
* Integer arithmetic can be used to sample from the standard normal distribution.<ref>{{harvtxt|Karney|2016}}</ref><ref>{{harvtxt|Du|Fan|Wei|2022}}</ref> This method is exact in the sense that it satisfies the conditions of ''ideal approximation'';<ref>{{harvtxt|Monahan|1985|loc=section 2}}</ref> i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
* There is also some investigation<ref>{{harvtxt |Wallace |1996}}</ref> into the connection between the fast [[Hadamard transform]] and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

=== Numerical approximations for the normal cumulative distribution function and normal quantile function ===
The standard normal [[cumulative distribution function]] is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as [[numerical integration]], [[Taylor series]], [[asymptotic series]] and [[Gauss's continued fraction#Of Kummer's confluent hypergeometric function|continued fractions]]. Different approximations are used depending on the desired level of accuracy.
* {{harvtxt |Zelen |Severo |1964 }} give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error {{math|{{abs|''ε''(''x'')}} < 7.5·10<sup>−8</sup>}} (algorithm [https://secure.math.ubc.ca/~cbm/aands/page_932.htm 26.2.17]): <math display=block>
    \Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac{1}{1+b_0x},
</math> where ''ϕ''(''x'') is the standard normal probability density function, and ''b''<sub>0</sub> = 0.2316419, ''b''<sub>1</sub> = 0.319381530, ''b''<sub>2</sub> = −0.356563782, ''b''<sub>3</sub> = 1.781477937, ''b''<sub>4</sub> = −1.821255978, ''b''<sub>5</sub> = 1.330274429.
* {{harvtxt |Hart |1968 }} lists some dozens of approximations – by means of rational functions, with or without exponentials – for the {{mono|erfc()}} function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by {{harvtxt |West |2009 }} combines Hart's algorithm 5666 with a [[continued fraction]] approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* {{harvtxt |Cody |1969 }} after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via [[rational function|Rational Chebyshev Approximation]].
* {{harvtxt |Marsaglia |2004 }} suggested a simple algorithm{{NoteTag|For example, this algorithm is given in the article [[Bc programming language#A translated C function|Bc programming language]].}} based on the Taylor series expansion <math display=block>
    \Phi(x) = \frac12 + \varphi(x)\left( x + \frac{x^3} 3 + \frac{x^5}{3 \cdot 5} + \frac{x^7}{3 \cdot 5 \cdot 7} + \frac{x^9}{3 \cdot 5 \cdot 7 \cdot 9} + \cdots \right)
</math> for calculating {{math|Φ(''x'')}} with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when {{math|1=''x'' = 10}}).
* The [[GNU Scientific Library]] calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with [[Chebyshev polynomial]]s.
* {{harvtxt |Dia|2023 }} proposes the following approximation of <math display=inline>1-\Phi</math> with a maximum relative error less than <math display=inline>2^{-53}</math> <math display=inline>
\left(\approx 1.1 \times 10^{-16}\right)
</math> in absolute value: for <math display=inline>x \ge 0</math><math display=inline>
\begin{aligned}
    1-\Phi\left(x\right) & = \left(\frac{0.39894228040143268}{x+2.92678600515804815}\right)
    \left(\frac{x^2+8.42742300458043240 x+18.38871225773938487}{x^2+5.81582518933527391 x+8.97280659046817350} \right) \\
    & \left(\frac{x^2+7.30756258553673541 x+18.25323235347346525}{x^2+5.70347935898051437 x+10.27157061171363079}\right)
    \left(\frac{x^2+5.66479518878470765 x+18.61193318971775795}{x^2+5.51862483025707963 x+12.72323261907760928}\right) \\
    & \left( \frac{x^2+4.91396098895240075 x+24.14804072812762821}{x^2+5.26184239579604207 x+16.88639562007936908}\right)
    \left( \frac{x^2+3.83362947800146179 x+11.61511226260603247}{x^2+4.92081346632882033 x+24.12333774572479110}\right) e^{-\frac{x^2}{2}}
\end{aligned}
</math> and for <math display=inline>
    x<0
</math>,
<math display=block>
    1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)
</math>

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting {{math|1=''p'' = Φ(''z'')}}, the simplest approximation for the quantile function is:
<math display=block>z = \Phi^{-1}(p)=5.5556\left[1- \left( \frac{1-p} p \right)^{0.1186}\right],\qquad p\ge 1/2</math>

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for {{math|0.5 ≤ ''p'' ≤ 0.9999}}, corresponding to {{math|0 ≤ ''z'' ≤ 3.719}}). For {{math|''p'' < 1/2}} replace ''p'' by {{math|1 − ''p''}} and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
<math display=block> z=-0.4115\left\{ \frac{1-p} p + \log \left[ \frac{1-p} p \right] - 1 \right\}, \qquad p\ge 1/2</math>

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
<math display=block>\begin{align}
L(z) & =\int_z^\infty (u-z)\varphi(u) \, du=\int_z^\infty [1-\Phi (u)] \, du \\[5pt]
L(z) & \approx \begin{cases}
   0.4115\left(\dfrac p {1-p} \right) - z, & p<1/2, \\ \\
   0.4115\left( \dfrac {1-p} p \right), & p\ge 1/2.
\end{cases} \\[5pt]
\text{or, equivalently,} \\
L(z) & \approx \begin{cases}
   0.4115\left\{ 1-\log \left[ \frac p {1-p} \right] \right\}, & p < 1/2, \\ \\
   0.4115 \dfrac{1-p} p, & p\ge 1/2.
\end{cases}
\end{align}</math>

This approximation is particularly accurate for the right far-tail (maximum error of 10<sup>−3</sup> for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on [[Response Modeling Methodology]] (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: [[Error function#Approximation with elementary functions]]. In particular, small ''relative'' error on the whole domain for the cumulative distribution function {{tmath|\Phi}} and the quantile function <math display=inline>\Phi^{-1}</math> as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

== History ==

=== Development ===
Some authors<ref>{{harvtxt |Johnson |Kotz |Balakrishnan |1994 |p=85 }}</ref><ref>{{harvtxt |Le Cam | Lo Yang |2000 |p=74 }}</ref> attribute the discovery of the normal distribution to [[de Moivre]], who in 1738{{NoteTag|De Moivre first published his findings in 1733, in a pamphlet ''Approximatio ad Summam Terminorum Binomii {{math|(''a'' + ''b'')<sup>''n''</sup>}} in Seriem Expansi'' that was designated for private circulation only. But it was not until the year 1738 that he made his results publicly available. The original pamphlet was reprinted several times, see for example {{harvtxt |Walker |1985 }}.}} published in the second edition of his ''[[The Doctrine of Chances]]'' the study of the coefficients in the [[binomial expansion]] of {{math|(''a'' + ''b'')<sup>''n''</sup>}}. De Moivre proved that the middle term in this expansion has the approximate magnitude of <math display=inline>2^n/\sqrt{2\pi n}</math>, and that "If ''m'' or {{sfrac|1|2}}''n'' be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ''ℓ'', has to the middle Term, is <math display=inline>-\frac{2\ell\ell}{n}</math>."<ref>De Moivre, Abraham (1733), Corollary I – see {{harvtxt |Walker |1985 |p=77 }}</ref> Although this theorem can be interpreted as the first obscure expression for the normal probability law, [[Stephen Stigler|Stigler]] points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.<ref>{{harvtxt |Stigler |1986 |p=76 }}</ref>

[[File:Carl Friedrich Gauss.jpg|thumb|180px|left|[[Carl Friedrich Gauss]] discovered the normal distribution in 1809 as a way to rationalize the [[method of least squares]].]]

In 1823 [[Gauss]] published his monograph <span title="Theory of the Combination of Observations Least Subject to Errors">"''Theoria combinationis observationum erroribus minimis obnoxiae''"</span> where among other things he introduces several important statistical concepts, such as the [[method of least squares]], the [[method of maximum likelihood]], and the ''normal distribution''. Gauss used ''M'', {{nobr|''M''′}}, {{nobr|''M''′′, ...}} to denote the measurements of some unknown quantity&nbsp;''V'', and sought the most probable estimator of that quantity: the one that maximizes the probability {{math|''φ''(''M'' − ''V'') · ''φ''(''M′'' − ''V'') · ''φ''(''M''′′ − ''V'') · ...}} of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function ''φ'' is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values.{{NoteTag|"It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it." — {{harvtxt |Gauss |1809 |loc=section 177 }} }} Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:<ref>{{harvtxt |Gauss |1809 |loc=section 177 }}</ref>
<math display=block>
    \varphi\mathit{\Delta} = \frac h {\surd\pi} \, e^{-\mathrm{hh}\Delta\Delta},
</math> <!-- please do not modify this formula; its spacing and style follow the original as closely as possible -->
where ''h'' is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the [[non-linear least squares|non-linear]] [[weighted least squares]] method.<ref>{{harvtxt |Gauss |1809 |loc=section 179 }}</ref>

[[File:Pierre-Simon Laplace.jpg|thumb|180px|right| [[Pierre-Simon Laplace]] proved the [[central limit theorem]] in 1810, consolidating the importance of the normal distribution in statistics.]]

Although Gauss was the first to suggest the normal distribution law, [[Laplace]] made significant contributions.{{NoteTag|"My custom of terming the curve the Gauss–Laplacian or ''normal'' curve saves us from proportioning the merit of discovery between the two great astronomer mathematicians." quote from {{harvtxt |Pearson |1905 |p=189 }} }} It was Laplace who first posed the problem of aggregating several observations in 1774,<ref>{{harvtxt |Laplace |1774 |loc = Problem III }}</ref> although his own solution led to the [[Laplacian distribution]]. It was Laplace who first calculated the value of the [[Gaussian integral|integral {{math|∫ ''e''<sup>−''t''<sup>2</sup></sup>&nbsp;''dt'' {{=}} {{sqrt|{{pi}}}}}}]] in 1782, providing the normalization constant for the normal distribution.<ref>{{harvtxt |Pearson |1905 |p=189 }}</ref> For this accomplishment, Gauss acknowledged the priority of Laplace.<ref>{{harvtxt |Gauss |1809 |loc=section 177 }}</ref> Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental [[central limit theorem]], which emphasized the theoretical importance of the normal distribution.<ref>{{harvtxt |Stigler |1986 |p=144 }}</ref>

It is of interest to note that in 1809 an Irish-American mathematician [[Robert Adrain]] published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss.<ref>{{harvtxt |Stigler |1978 |p=243 }}</ref> His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by [[Cleveland Abbe|Abbe]].<ref>{{harvtxt |Stigler |1978 |p=244 }}</ref>

In the middle of the 19th century [[James Clerk Maxwell|Maxwell]] demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena:<ref>{{harvtxt |Maxwell |1860 |p=23 }}</ref> The number of particles whose velocity, resolved in a certain direction, lies between ''x'' and ''x''&nbsp;+&nbsp;''dx'' is
<math display=block>
    \operatorname{N} \frac{1}{\alpha\;\sqrt\pi}\; e^{-\frac{x^2}{\alpha^2}} \, dx
</math> <!-- please do not modify this formula; its spacing and style follow the original as close as possible -->

=== Naming ===
Today, the concept is usually known in English as the '''normal distribution''' or '''Gaussian distribution'''.  Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual.<ref>Jaynes, Edwin J.; ''Probability Theory: The Logic of Science'', [http://www-biba.inrialpes.fr/Jaynes/cc07s.pdf Ch. 7].</ref> However, by the end of the 19th century some authors{{NoteTag|Besides those specifically referenced here, such use is encountered in the works of [[Charles Sanders Peirce|Peirce]], [[Galton]] ({{harvtxt |Galton |1889 |loc = chapter V }}) and [[Wilhelm Lexis|Lexis]] ({{harvtxt | Lexis |1878 }}, {{harvtxt |Rohrbasser |Véron |2003 }}) c. 1875.{{Citation needed |date=June 2011 }} }} had started using the name ''normal distribution'', where the word "normal" was used as an adjective&nbsp;– the term now being seen as a reflection of the fact that this distribution was seen as typical, common&nbsp;– and thus normal. [[Charles Sanders Peirce|Peirce]] (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what ''would'', in the long run, occur under certain circumstances."<ref>Peirce, Charles S. (c. 1909 MS), ''[[Charles Sanders Peirce bibliography#CP|Collected Papers]]'' v. 6, paragraph 327.</ref> Around the turn of the 20th century [[Karl Pearson|Pearson]] popularized the term ''normal'' as a designation for this distribution.<ref>{{harvtxt |Kruskal |Stigler |1997 }}.</ref>
{{Blockquote|Many years ago I called the Laplace–Gaussian curve the ''normal'' curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'. |{{harvtxt |Pearson |1920 }}}}

Also, it was Pearson who first wrote the distribution in terms of the standard deviation ''σ'' as in modern notation. Soon after this, in year 1915, [[Ronald Fisher|Fisher]] added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
<math display=block> df = \frac{1}{\sqrt{2\sigma^2\pi}} e^{-(x - m)^2/(2\sigma^2)} \, dx.</math>

The term "standard normal", which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P.&nbsp;G. Hoel (1947) ''Introduction to Mathematical Statistics'' and A.&nbsp;M. Mood (1950) ''Introduction to the Theory of Statistics''.<ref>{{cite web|title=Earliest Uses... (Entry Standard Normal Curve)|url=http://jeff560.tripod.com/s.html}}</ref>

== See also ==
{{Portal|Mathematics}}
* [[Bates distribution]] – similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
* [[Behrens–Fisher problem]] – the long-standing problem of testing whether two normal samples with different variances have same means;
* [[Bhattacharyya distance]] – method used to separate mixtures of normal distributions
* [[Erdős–Kac theorem]] – on the occurrence of the normal distribution in [[number theory]]
* [[Full width at half maximum]]
* [[Gaussian blur]] – [[convolution]], which uses the normal distribution as a kernel
* [[Gaussian function]]
* [[Modified half-normal distribution]]<ref name="Sun-2021">{{cite journal |last1=Sun |first1=Jingchao |last2=Kong |first2=Maiying |last3=Pal |first3=Subhadip |title=The Modified-Half-Normal distribution: Properties and an efficient sampling scheme |journal=Communications in Statistics – Theory and Methods |date=22 June 2021 |volume=52 |issue=5 |pages=1591–1613 |doi=10.1080/03610926.2021.1934700 |s2cid=237919587 |url=https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20 |issn=0361-0926}}</ref> with the pdf on <math display=inline>(0, \infty)</math> is given as <math display=inline>f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}</math>, where <math display=inline>\Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\(1,0)\end{matrix};z \right)</math> denotes the [[Fox–Wright Psi function]].
* [[Normally distributed and uncorrelated does not imply independent]]
* [[Ratio normal distribution]]
* [[Reciprocal normal distribution]]
* [[Standard normal table]]
* [[Stein's lemma]]
* [[Sub-Gaussian distribution]]
* [[Sum of normally distributed random variables]]
* [[Tweedie distribution]] – The normal distribution is a member of the family of Tweedie [[exponential dispersion model]]s.
* [[Wrapped normal distribution]] – the Normal distribution applied to a circular domain
* [[Z-test]] – using the normal distribution

== Notes ==
{{NoteFoot}}

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{Refbegin}}
{{divcol}}
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{{divcol end}}
{{Refend}}

== External links ==
{{Commons category|Normal distribution}}
* {{springer|title=Normal distribution|id=p/n067460}}
* [https://www.hackmath.net/en/calculator/normal-distribution Normal distribution calculator]

{{ProbDistributions|continuous-infinite}}
{{Authority control}}

[[Category:Normal distribution| ]]
[[Category:Continuous distributions]]
[[Category:Conjugate prior distributions]]
[[Category:Exponential family distributions]]
[[Category:Stable distributions]]
[[Category:Location-scale family probability distributions]]