View source for Normal distribution

{{Probability distribution||name=Normal|type=density|pdf_image=[[File:Normal distribution pdf.png|325px|Probability density function for the normal distribution]]<br /><small>The green line is the standard normal distribution</small>|cdf_image=[[File:Normal distribution cdf.png|325px|Cumulative distribution function for the normal distribution]]<br /><small>Colors match the image above</small>|parameters=<math>\mu</math> [[location parameter|location]] ([[real number|real]])<br/><math>\sigma^2>0</math> squared [[scale parameter|scale]] (real)|support=<math>x \in\mathbb{R}\!</math>|pdf=<math>\frac1{\sigma\sqrt{2\pi}}\; \exp\left(-\frac{\left(x-\mu\right)^2}{2\sigma^2} \right) \!</math>|cdf=<math>\frac12 \left(1 + \mathrm{erf}\,\frac{x-\mu}{\sigma\sqrt2}\right) \!</math>|mean=<math>\mu</math>|median=<math>\mu</math>|mode=<math>\mu</math>|variance=<math>\sigma^2</math>|skewness=0|kurtosis=0|entropy=<math>\ln\left(\sigma\sqrt{2\,\pi\,e}\right)\!</math>|mgf=<math>M_X(t)= \exp\left(\mu\,t+\frac{\sigma^2 t^2}{2}\right)</math>|char=<math>\chi_X(t)=\exp\left(\mu\,i\,t-\frac{\sigma^2 t^2}{2}\right)</math>}}[[File:Standard deviation diagram.svg|thumb|350px|For the normal distribution, the values less than one standard deviation away 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%.]]

The '''normal distribution''' is a [[probability distribution]] used in [[probability theory]] and [[statistics]]. It is also called '''Gaussian distribution''' because it was first discovered by [[Carl Friedrich Gauss]]. The normal distribution is very important in many fields because many things take this form.<ref>{{Cite web|date=2021-11-26|title=Normal Distribution {{!}} Data Basecamp|url=https://databasecamp.de/en/statistics/normal-distribution|access-date=2022-07-15|language=en-US}}</ref> A [[random variable]] that takes this form is '''normally distributed''', and can be called a '''normal deviate'''. The normal distribution is often called the ''bell curve'', because the graph of its [[probability density function|probability density]] looks like a [[bell]]. The '''standard normal distribution''' (also known as the '''Z distribution''') is a normal distribution that has a [[mean]] of zero and a [[standard deviation]] of one.<ref name=":0">{{Cite web|date=2020-04-26|title=List of Probability and Statistics Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/probability-statistics-symbols/|access-date=2020-08-15|website=Math Vault|language=en-US}}</ref> 

The form of the distribution is  

<math>
f(x) = \frac{1}{\sigma \sqrt{2\pi} } e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}
</math> 

In a normal distribution, the [[parameter]] <math>\mu</math> is the [[mean]] ("average"). The [[standard deviation]] ("variability") is <math>\sigma</math>.<ref name=":0" /> The [[variance]] of the distribution is <math>\sigma^2</math>.

The normal distribution is important because it can represent real-life examples. It is used in [[Natural sciences|natural]] and [[social sciences]].<ref>{{Cite web|date=2021-11-26|title=Normal Distribution - easily explained! {{!}} Data Basecamp|url=https://databasecamp.de/en/statistics/normal-distribution|access-date=2023-05-29|language=en-US}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Normal Distribution|url=https://mathworld.wolfram.com/NormalDistribution.html|access-date=2020-08-15|website=mathworld.wolfram.com|language=en}}</ref> Some examples include:
*Height
*Test scores
*Measurement errors
*[[Light]] intensity (as in [[laser]] light)
*Insurance companies use normal distributions to model certain [[average]] cases.<ref>{{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>
The [[central limit theorem]] can be used to describe real-life data as a normal distribution.<ref>{{Cite web|last=Kwak|first=Sang Gyu|last2=Kim|first2=Jong Hae|date=2017-02-21|title=Central limit theorem: the cornerstone of modern statistics|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5370305/|access-date=2024-05-30|website=National Library of Medicine}}</ref>
== Origin ==
Its origin goes back to the 18th century to [[De Moivre's formula|De Moivre]] and in the early 19th century with [[Carl Friedrich Gauss|Gauss]].
== Use ==
It is the most widely used piece of statistics by far. It was famously used in [[World War I]] by the [[United States Army]] to decide when men were so poor mentally that they could not be used by the Army in any job.

== Normal Distribution Media ==
<gallery widths='160px' heights='100%' mode='traditional' caption=''>
File:Normal Distribution PDF.svg|A selection of Normal Distribution Probability Density Functions (PDFs). Both the mean, μ, and variance, σ², are varied. The key is given on the graph.

File:Normal Distribution CDF.svg|A selection of Normal Distribution Cumulative Density Functions (CDFs). Both the mean, μ, and variance, σ², are varied. The key is given on the graph.

File:De moivre-laplace.gif|As the number of discrete events increases, the function begins to resemble a normal distribution.

File:Dice sum central limit theorem.svg|Comparison of probability density functions, {{math|''p''(''k'')}} for the sum of {{mvar|n}} fair 6-sided dice to show their convergence to a normal distribution with increasing {{mvar|na}}, 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).

File:QHarmonicOscillator.png|The ground state of a [[quantum harmonic oscillator]] has the Gaussian distribution.

File:Fisher iris versicolor sepalwidth.svg|Histogram of sepal widths for ''Iris versicolor'' from Fisher's [[Iris flower data set]], with superimposed best-fitting normal distribution

File:FitNormDistr.tif|Fitted cumulative normal distribution to October rainfalls, see [[distribution fitting]]
</gallery>
== Related pages ==
* [[Frequency distribution]]
* [[Least squares]]
* [[Student's t-distribution]]
== References ==
{{reflist}}
== Other websites ==
*[https://www.danielsoper.com/statcalc/calculator.aspx?id=2 Cumulative Area Under the Standard Normal Curve Calculator]  from Daniel Soper's ''Free Statistics Calculators'' website. Computes the cumulative area under the normal curve (i.e., the cumulative probability), given a z-score.
*[https://ztable.io Interactive Standard Normal Distribution]
*[http://www.gnu.org/software/gsl/manual/html_node/Random-Number-Distributions.html GNU Scientific Library &ndash; Reference Manual &ndash; The Gaussian Distribution]
*[https://www.ztable.net Normal Distribution Table]
*[http://www.waterlog.info/normdis.htm Download free two-way normal distribution calculator]
*[http://www.waterlog.info/cumfreq.htm Download free normal distribution fitting software]

[[Category:Probability distributions]]