Editing Correlation (section)

{{Short description|Statistical concept}}
{{About|correlation and dependence in statistical data||Correlation (disambiguation)}}
{{Merge from|Correlation coefficient|discuss=Talk:Correlation#Proposed merge of Correlation coefficient into Correlation|date=February 2025}}
[[File:Correlation examples2.svg|thumb|400px|right|Several sets of (''x'',&nbsp;''y'') points, with the [[Pearson correlation coefficient]] of ''x'' and ''y'' for each set. The correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case, the correlation coefficient is undefined because the variance of ''Y'' is zero.]]

In [[statistics]], '''correlation ''' or '''dependence ''' is any statistical relationship, whether [[causality|causal]] or not, between two [[random variable]]s or [[bivariate data]]. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are ''[[line (geometry)|linearly]]'' related.  
Familiar examples of dependent phenomena include the correlation between the [[human height|height]] of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the [[demand curve]].

Correlations are useful because they can indicate a predictive relationship that can be exploited in practice.  For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a [[causality|causal relationship]], because [[extreme weather]] causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., [[correlation does not imply causation]]).

Formally, random variables are ''dependent'' if they do not satisfy a mathematical property of [[independence (probability theory)|probabilistic independence]].  In informal parlance, ''correlation'' is synonymous with ''dependence''. However, when used in a technical sense, correlation refers to any of several specific types of mathematical relationship between [[Conditional expectation|the conditional expectation of one variable given the other is not constant as the conditioning variable changes]]; broadly correlation in this specific sense is used when <math>E(Y|X=x)</math> is related to <math>x</math> in some manner (such as linearly, monotonically, or perhaps according to some particular functional form such as logarithmic). Essentially, correlation is the measure of how two or more variables are related to one another.  There are several [[correlation coefficient]]s, often denoted <math>\rho</math> or <math>r</math>, measuring the degree of correlation.  The most common of these is the ''[[Pearson product-moment correlation coefficient|Pearson correlation coefficient]]'', which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other).  Other correlation coefficients&nbsp;– such as ''[[Spearman's rank correlation coefficient]]'' –&nbsp;have been developed to be more [[robust statistics|robust]] than Pearson's and to detect less structured relationships between variables.<ref>Croxton, Frederick Emory; Cowden, Dudley Johnstone; Klein, Sidney (1968) ''Applied General Statistics'', Pitman. {{ISBN|9780273403159}} (page 625)</ref><ref>Dietrich, Cornelius Frank (1991) ''Uncertainty, Calibration and Probability: The Statistics of Scientific and Industrial Measurement'' 2nd Edition, A. Higler. {{ISBN|9780750300605}} (Page 331)</ref><ref>Aitken, Alexander Craig (1957) ''Statistical Mathematics'' 8th Edition. Oliver & Boyd. {{ISBN|9780050013007}}  (Page 95)</ref> [[Mutual information]] can also be applied to measure dependence between two variables.