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| [[Image:Homoscedasticity.png|thumb|right|Plot with random data showing homoscedasticity.]]
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| In [[statistics]], a [[sequence]] or a vector of [[random variable]]s is '''homoscedastic''' {{IPAc-en|ˌ|h|oʊ|m|oʊ|s|k|ə|ˈ|d|æ|s|t|ɪ|k}} if all random variables in the sequence or vector have the same [[finite set|finite]] [[variance]]. This is also known as '''homogeneity of variance'''. The complementary notion is called [[heteroscedasticity]]. The spellings ''homos'''k'''edasticity'' and ''heteros'''k'''edasticity'' are also frequently used.<ref>For the Greek etymology of the term, see J. Huston McCulloch (1985), [https://docs.google.com/viewer?a=v&q=cache:n9WwLxZX9LsJ:www.ime.usp.br/~abe/lista/pdfZAptC9KazU.pdf+On+Heteros*edasticity&hl=en&gl=uk&pid=bl&srcid=ADGEEShKto-i0r96p6JIcxnvPB9w4abkVWuKcpSwR8IUT_hDcvabhUEPS6KW0Hs9lGBESqZOutTYpLCcbe-4vj_ldq4nmkUlYTx6ZiL-c2f4ZFMJpauBi-dMPrJSRMVbQSianWK-FPct&sig=AHIEtbQHgBYoPXlrwLXby_joAnqpVKQ2PQ "On Heteros*edasticity"], ''[[Econometrica]]'', 53(2), p. 483.</ref>
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| The assumption of homoscedasticity simplifies mathematical and computational treatment. <!-- and usually leads to adequate estimation results (e.g. in [[data mining]]) even if the assumption is not true.--> Serious violations in homoscedasticity (assuming a distribution of data is homoscedastic when in actuality it is heteroscedastic {{IPAc-en|ˌ|h|ɛ|t|ər|oʊ|s|k|ə|ˈ|d|æ|s|t|ɪ|k}}) may result in overestimating the goodness of fit as measured by the [[Pearson product-moment correlation coefficient|Pearson coefficient]].
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| ==Assumptions of a regression model==
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| As used in describing [[simple linear regression]] analysis, one assumption of the fitted model (to ensure that the least-squares estimators are each a [[best linear unbiased estimator]] of the respective population parameters, by the [[Gauss–Markov theorem]]) is that the standard deviations of the error terms are constant and do not depend on the ''x''-value. Consequently, each probability distribution for ''y'' (response variable) has the same standard deviation regardless of the ''x''-value (predictor). In short, this assumption is homoscedasticity. Homoscedasticity is not required for the estimates to be unbiased, consistent, and asymptotically normal {{Citation needed|date=August 2013}}
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| ==Testing==
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| {{expand section|date=November 2013}}
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| Residuals can be tested for homoscedasticity using the [[Breusch–Pagan test]], which regresses square residuals to independent variables. Since the Breusch–Pagan test is sensitive to normality, the Koenker–Basset or 'generalized Breusch–Pagan' test is used for general purposes{{cn|date=November 2013}}. Testing for groupwise heteroscedasticity requires the [[Goldfeld–Quandt test]].
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| ==Homoscedastic distributions==
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| Two or more [[normal distribution]]s, <math>N(\mu_i,\Sigma_i)</math>, are homoscedastic if they share a common [[covariance matrix|covariance]] (or [[correlation matrix|correlation]]) matrix, <math>\Sigma_i = \Sigma_j,\ \forall i,j</math>. Homoscedastic distributions are especially useful to derive statistical [[pattern recognition]] and [[machine learning]] algorithms. One popular example is Fisher's [[linear discriminant analysis]].
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| The concept of homoscedasticity can be applied to distributions on spheres.<ref>Hamsici, Onur C.; Martinez, Aleix M. (2007) [http://jmlr.csail.mit.edu/papers/volume8/hamsici07a/hamsici07a.pdf "Spherical-Homoscedastic Distributions: The Equivalency of Spherical and Normal Distributions in Classification"], ''Journal of Machine Learning Research'', 8, 1583-1623</ref>
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| ==See also==
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| *[[Bartlett's test]]
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| *[[Homogeneity (statistics)]]
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| *[[Heterogeneous#Statistics|Heterogeneity]]
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| {{refimprove|date=October 2011}}
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| ==References==
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| {{reflist}}
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| {{statistics}}
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| [[Category:Statistical deviation and dispersion]]
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| [[de:Homoskedastizität]]
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