View source for Correlation coefficient

A numerical characteristic of the joint distribution of two random variables, expressing a relationship between them. The correlation coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c0265301.png" /> for random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c0265302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c0265303.png" /> with mathematical expectations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c0265304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c0265305.png" /> and non-zero variances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c0265306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c0265307.png" /> is defined by

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The correlation coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c0265309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653010.png" /> is simply the [[Covariance|covariance]] of the normalized variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653012.png" />. The correlation coefficient is symmetric with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653014.png" /> and is invariant under change of the origin and scaling. In all cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653015.png" />. The importance of the correlation coefficient as one of the possible measures of dependence is determined by its following properties: 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653017.png" /> are independent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653018.png" /> (the converse is not necessarily true). Random variables for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653019.png" /> are said to be non-correlated. 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653020.png" /> if and only if the dependence between the random variables is linear:

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The difficulty of interpreting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653022.png" /> as a measure of dependence is that the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653023.png" /> may be valid for both independent and dependent random variables; in the general case, a necessary and sufficient condition for independence is that the [[Maximal correlation coefficient|maximal correlation coefficient]] equals zero. Thus, the correlation coefficient does not exhaust all types of dependence between random variables and it is a measure of linear dependence only. The degree of this linear dependence is characterized as follows: The random variable

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gives a linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653025.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026530/c02653026.png" /> which is best in the sense that

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see also [[Regression|Regression]]. As characteristic correlations between several random variables there are the [[Partial correlation coefficient|partial correlation coefficient]] and the [[Multiple-correlation coefficient|multiple-correlation coefficient]]. For methods for testing independence hypotheses and using correlation coefficients to study correlation, see [[Correlation (in statistics)|Correlation (in statistics)]].