Gaussian distribution

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[[File:Standard deviation diagram.svg|right|300px|thumb|Gaussian density plot illustrating the probability of being n standard deviations away from the mean.|alt=A graph of the Gaussian distribution probability density function.]] {{mathematics}}
A '''Gaussian distribution''' or '''Gauss distribution''' is a [[Statistics|statistical]] model that describes a particular way that data can be distributed around the [[mean]], the average. It is also sometimes called the '''normal distribution''', the '''Laplace–Gauss distribution''', or the '''bell curve''' (because its graph looks like a bell).{{efn|Though this isn't terribly useful since the graph for many other distributions look like bells, such as the {{wpl|Student's t-distribution}} and some versions of the {{wpl|logistic distribution}} and the {{wpl|Weibull distribution||.}} While they're in a different class of distributions, the {{wpl|Poisson distribution}} and {{wpl|binomial distribution}} look like bells too. In fairness, however, as the parameters increase, those other distributions ''become'' the Gaussian distribution.}} It's incredibly useful for analyzing certain kinds of data in [[physics]], [[medicine]], [[anthropology]], and many other types of [[science]]. At the same time, it is often misinterpreted and misused, especially by [[Racialism|racialists]]. 

==Definition==

A standard Gaussian distribution is such that the [[mean]] is zero, the variance is one (unit variance), and the most likely something can occur is 1 (100%). As such, the probability density function of the standard Gaussian distribution may be written as: <math>\varphi\!\left(x\right)=\frac{1}{\sqrt{2\pi}}e^{-\tfrac{1}{2}x^2}</math>.  This makes the distribution technically in the exponential family of distributions. The Gaussian distribution is a specific kind of the more general {{wpl|Gaussian function}}.

In general, the Gaussian distribution with mean μ and [[standard deviation]] σ is equivalent to the standard Gaussian multiplied by σ, then added to μ.

In multiple dimensions, a random vector may be said to follow a multivariate Gaussian distribution or multinormal distribution if and only if every linear combination of its components follows a univariate Gaussian distribution. The distribution of a multinormal random variable is uniquely determined by its mean and the variances and covariances of its components.<ref>[https://online.stat.psu.edu/stat505/book/export/html/636 Multivariate Normal Distribution], ''Applied Multivariate Statistical Analysis'', Penn State University.</ref><ref group=note>This means that if two multinormal random vectors have the same mean, and their components have the same variances and covariances, then they must have the same distribution.</ref>

==Occurrences in Statistics==
According to the [[Central limit theorem|central limit theorem]], the average and sum of n identically distributed random variables with finite variance approaches a Gaussian distribution as n becomes large.<ref>Weisstein, Eric W., [https://mathworld.wolfram.com/CentralLimitTheorem.html Central Limit Theorem], ''Wolfram MathWorld''.</ref>

The Maxwell-Herschel Theorem states that the multivariate standard normal distribution<ref group=note>This is defined as a vector composed of multiple independent standard univariate normal distributions</ref> multiplied by a constant is the only rotationally invariant<ref group=note>This means that applying an orthonormal (i.e. distance-preserving and origin-preserving) transformation to the vector does not change its distribution</ref> distribution where the coordinates are independent.<ref>Kolditz, Tobias, [https://web.archive.org/web/20210307110938/https://proceed-to-decode.com/posts/herschel-maxwell-derivation-of-the-normal-distribution/ Herschel-Maxwell Derivation of the Normal Distribution]. October 26, 2020.</ref><ref>Jaynes, E. T., [http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/JaynesProbabilityTheoryChapter7.pdf The central, Gaussian or normal distribution], ''Probability Theory''. 2002.</ref> {{wpl|James Clerk Maxwell}} used this fact to derive a description of the distribution of velocities of particles in an ideal gas, since he assumed that particle velocities would follow a distribution with these properties.<ref>Gyenis, Balázs, [https://arxiv.org/abs/1702.01411 Maxwell and the normal distribution: A colored story of probability, independence, and tendency toward equilibrium], ''History and Philosophy of Physics''. February 5, 2017.</ref>

Gauss used it to model errors in astronomical observations<ref name=gauss>Sedgewick, Robert; Wayne, Kevin; and Dondero, Robert. [https://introcs.cs.princeton.edu/python/appendix_gaussian/ Gaussian Distribution], ''Introduction to Programming in Python''. June 2015.</ref> on the basis that if a series of independent observations are assumed to be identically distributed around some mean value which happens to be equal to the sample mean, then the probability of the sample is maximized if the distribution of the observations is assumed to be Gaussian.

The "continuity correction" is a relatively simple fix that stats students everwhere trip over.  The Gaussian Distribution is ''continuous'' by Baire's theorem, so discrete outcomes need to be approximated by intervals with bounds equal to the discrete outcome ± 0.5.<ref>
Campbell, Russell Bruce, [https://www.cs.uni.edu/~campbell/stat/prob9.html Continuity correction factor].</ref><ref group=note>Technically it'll be slightly different from .5 depending on where you are in the curve, but the actual differences are so miniscule and no human alive could do the math in a reasonable amount of time, so it can be safely ignored</ref>  The need is best demonstrated through an absurd case.  Let's say you want to know the odds of getting exactly 50 heads out of 100 flips of a fair coin. Using a binomial distribution to find the exact odds will give you (100 choose 50)*.5^100, or about 7.96%.  If you apply the normal approximation without the continuity correction, you would get Φ(0) - Φ(0) = 0.5 - 0.5 = 0<ref group=note>Φ denotes the cumulative distribution function of the standard normal distribution.</ref> chance of this happening.<ref group=note>The same will happen when computing the odds of getting any specific number of heads, for that matter.</ref>  Obviously the chance of getting 50 heads is greater than 0%. Instead, you should calculate the probability that X~N(μ=50, σ=5)<ref>Glen, Stephanie. [https://www.statisticshowto.com/probability-and-statistics/binomial-theorem/normal-approximation-to-the-binomial/ Normal Approximation to the Binomial], ''Statistics How To''.</ref> is between 49.5 and 50.5, which will give you 7.97%, which is a pretty good estimate of the actual probability.

==Where it applies==
Some examples of things that follow a Gaussian distribution:
*[[IQ]] - IQ scores are by definition fit or normed so that they follow a Gaussian curve.<ref>Spielman, et al., [https://web.archive.org/web/20220422115736/https://opentextbc.ca/psychologyopenstax/chapter/measures-of-intelligence/ Measures of Intelligence], ''Psychology: OpenStax''. August 7, 2019.</ref> When [[High IQ society|high IQ societies]] talk about being "3 standard deviations above the norm" or allowing only the "top 1%", this is what they're going on about. 
*Manufacturing and other process errors.  This is where "six sigma" gets its name, the idea of the chance of an error occurring being more than six standard deviations from the mean
*Distance between stars and planets - In fact, Gauss encountered the normal distribution by studying the orbits of the planets.<ref name=gauss/> {{wpl|Pierre-Simon de Laplace}} used Gaussian distribution to show that the motion of planets remained invariant over time and so, contrary to [[Isaac Newton]], there was no need for [[Goddidit|some sort of deity]] to come in and realign the Solar System.<ref>Pannekoek, Antonie, [https://adsabs.harvard.edu/full/1948PA.....56..300P  The planetary theory of Laplace], ''Popular Astronomy'' (56). 1948.</ref>
*Accuracy of projectiles - In ballistics, the {{wpl|circular error probable}} describes the chances of a missile or other projectile landing within a certain distance of a target. It follows Gaussian probabilities and often appears in analyses regarding nuclear missiles and [[nuclear war]].<ref>Wellenstein, A. (n.d.) ''[https://nuclearsecrecy.com/missilemap/faq/ MISSILEMAP Frequently Asked Questions (FAQ)]''. Restricted Data. </ref>
* Errors in scientific [[Observation|observations]] and measurements - observational errors are often an aggregate of many smaller sources, and thus per the Central Limit Theorem will be approximately normal. In fact, an earlier name for the Gaussian distribution was the "law of errors".<ref name=gauss/>

==Where it doesn't apply==
* As the Gaussian distribution is a continuous distribution, it will not be a good model for discrete data unless the discrete data is sufficiently granular (although it may still be a good model for the average of many observations of a discrete outcome.) Discrete data usually includes counts, such as amount of votes a candidate got, number of [[goat]]s per square mile, or amount of times you got heads after flipping a coin five times.
* It is not a good model for the lifespan of, for example, a product or an [[organism]]; usually different distributions are used instead.<ref>Tian, Lu, [https://web.stanford.edu/~lutian/coursepdf/unit1.pdf Survival Distributions, Hazard Functions, Cumulative Hazards], BIO 224, Stanford University.</ref><ref>Rodríguez, Germán, [https://data.princeton.edu/wws509/notes/c7.pdf Survival Models], ''Generalized Linear Models''. September 2010.</ref> This is especially the case if there is a chance that some units might be defective, as they will drive a high early failure rate, while wear over time leads to a higher failure rate of non-defective units, resulting in a "bathtub-shaped" curve.<ref>Wilkins, Dennis J., [https://www.weibull.com/hotwire/issue21/hottopics21.htm The Bathtub Curve and Product Failure Behavior], ''Reliability HotWire'', November 2002.</ref>

==Use by Eugenicists==
Adolphe Quetelet sought to prove that various physical attributes of humans followed a Gaussian distribution, and sought to prove this by measuring physical attributes in many subjects.<ref name=aq>Eknoyan, Garabed, [https://academic.oup.com/ndt/article/23/1/47/1923176 Adolphe Quetelet (1796–1874)—the average man and indices of obesity], ''Nephrology Dialysis Transplantation'' (23)1. January 2008.</ref> He believed that individual humans represent deviations from an "average man" representative of all of humanity.<ref>Rose, Todd, [https://www.theatlantic.com/business/archive/2016/02/the-invention-of-the-normal-person/463365/ How the Idea of a ‘Normal’ Person Got Invented], ''The Atlantic''. February 18, 2016.</ref> Quetelet's ideas would influence the [[eugenics]] movement and its supporters, such as [[Francis Galton]].<ref name=aq/>

Galton assumed that Quetelet's demonstration that various biometric traits followed a Gaussian distribution should also hold for traits of ability such as [[intelligence]].<ref name=galton>Krashniak, Adam; Lamm, Ehud, [https://www.sciencedirect.com/science/article/pii/S0039368120302090 Francis Galton’s regression towards mediocrity and the stability of types], ''Studies in History and Philosophy of Science'' (86). April 2021.</ref> Francis Galton derived the concept of regression to determine how genetic traits were inherited.<ref name=galton/> He observed that tall parents tended to have tall children, but that those children were not on average as tall as their parents, a phenomenon he described as "regression to mediocrity".<ref name=galton/><ref group=note>This can be thought of as a predecessor to the modern concept of [[Regression to the mean|regression to the mean]].</ref> Galton saw this as an improvement on and explanation for Quetelet's observations.<ref name=galton/>

While IQ tests originally expressed scores as a quotient between a person's "mental age" by their actual age,<ref group=note>Hence the term "Intelligence ''Quotient''".</ref><ref>[https://web.archive.org/web/20170722194028/http://www.ncme.org/ncme/NCME/Resource_Center/Glossary/NCME/Resource_Center/Glossary1.aspx?hkey=4bb87415-44dc-4088-9ed9-e8515326a061#anchorI Intelligence Quotient (IQ), ''Glossary of Important Assessment and Measurement Terms''], ''National Council on Measurement in Education''.</ref> this was only meaningful when testing children. IQ tests began using "deviation IQs" based on relative comparisons of performance between a test-taker against a sample used to norm the test in order to be able to measure adult intelligence, especially David Wechsler's 1939 test.<ref>[https://www.mentalhelp.net/psychological-testing/intelligence-quotient/ Psychological Testing: Intelligence Quotient], ''MentalHelp.net''.</ref> These IQ test scores were fit to a Gaussian distribution because intelligence was assumed to follow such a distribution.<ref>Brock, Stephen E., [https://www.csus.edu/indiv/b/brocks/courses/eds%20245/Handouts/Week%2010/Descrptive%20Statistics%20and%20the%20Normal%20Curve.pdf Descriptive Statistics and Psychological Testing], ''Psychology in the Schools'', 2020.</ref><ref name=galton/> Intelligence testing has historically been of interest to eugenicists as a way of identifying who is exceptional and who is defective.

In practice, intelligence in humans is... weird.  An IQ test can identify someone who is ''exceptionally'' dumb or intelligent, but a person who scores a few points higher or lower is not necessarily "smarter" or "dumber", just able to solve whatever pattern the IQ test is testing; at best the IQ test is the map, not the territory.  Richard Feynman scored an IQ of "only" 124<ref>[[Scott Alexander|Alexander, Scott]]. [https://www.vox.com/the-big-idea/2017/10/23/16516516/iq-tests-high-low-achievement-sat-anxiety-determinism So you’ve learned you’ve got a “pitifully” low IQ. How worried should you be?], ''Vox''. October 23, 2017.</ref>, which would put him in the "significantly above average" category but much less than the "genius level" one would expect of a Nobel-prize winning Physicist, or any random physicist for that matter; the average IQ of Physicists is 128.<ref>McLain, Sylvia. [https://www.theguardian.com/science/occams-corner/2014/feb/27/physicists-intelligence-social-scientists-religious-dumb-bad-science If physicists are smarter than social scientists, are religious people dumb?], ''The Guardian''. February 27, 2014.</ref>   He would not be allowed to join Mensa with that IQ, but he was probably smart enough to notice that anyone bragging about going to such a meeting was advertising the schedule of when their wife would be all alone.  Meanwhile, plenty of "geniuses" fail to achieve success and plenty of people with below average intelligence manage to do quite well and establish businesses.

Furthermore, as anyone who has done studies on intelligence can assure you, a penguin scores consistently as highly as non-English speakers.<ref>[https://www.dailymotion.com/video/x2nd67f Monty Python, Penguin Intelligence], Dailymotion.</ref>

==See also==
*[[Francis Galton]]
*[[Probability]]
*[[Statistics]]

==Notes==
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==References==
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[[Category:Statistics]]

Categories: Statistics