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Correlation

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Statistical concept

This article is about correlation and dependence in statistical data. For other uses, seeCorrelation (disambiguation).

It has been suggested thatCorrelation coefficient bemerged into this article. (Discuss) Proposed since February 2025.

Several sets of (x, y) points, with thePearson correlation coefficient ofx andy 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 ofY is zero.

Instatistics,correlation ordependence is any statistical relationship, whethercausal or not, between tworandom variables orbivariate 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 arelinearly related. Familiar examples of dependent phenomena include the correlation between theheight 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 thedemand 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 acausal relationship, becauseextreme 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 aredependent if they do not satisfy a mathematical property ofprobabilistic independence. In informal parlance,correlation is synonymous withdependence. However, when used in a technical sense, correlation refers to any of several specific types of mathematical relationship betweenthe 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 $E(Y|X=x)$ is related to $x {\displaystyle x}$ 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 severalcorrelation coefficients, often denoted $\rho$ or $r {\displaystyle r}$ , measuring the degree of correlation. The most common of these is thePearson 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 – such asSpearman's rank correlation coefficient – have been developed to be morerobust than Pearson's and to detect less structured relationships between variables.^[1]^[2]^[3]Mutual information can also be applied to measure dependence between two variables.

Pearson's product-moment coefficient

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Main article:Pearson product-moment correlation coefficient

Example scatterplots of various datasets with various correlation coefficients

The most familiar measure of dependence between two quantities is thePearson product-moment correlation coefficient (PPMCC), or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". It is obtained by taking the ratio of the covariance of the two variables in question of our numerical dataset, normalized to the square root of their variances. Mathematically, one simply divides thecovariance of the two variables by the product of theirstandard deviations.Karl Pearson developed the coefficient from a similar but slightly different idea byFrancis Galton.^[4]

A Pearson product-moment correlation coefficient attempts to establish a line of best fit through a dataset of two variables by essentially laying out the expected values and the resulting Pearson's correlation coefficient indicates how far away the actual dataset is from the expected values. Depending on the sign of our Pearson's correlation coefficient, we can end up with either a negative or positive correlation if there is any sort of relationship between the variables of our data set.^{[citation needed]}

The population correlation coefficient $\rho _{X,Y}$ between tworandom variables $X {\displaystyle X}$ and $Y {\displaystyle Y}$ withexpected values $\mu _{X}$ and $\mu _{Y}$ andstandard deviations $\sigma _{X}$ and $\sigma _{Y}$ is defined as:

$\rho _{X,Y}=\operatorname {corr} (X,Y)={\operatorname {cov} (X,Y) \over \sigma _{X}\sigma _{Y}}={\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})] \over \sigma _{X}\sigma _{Y}},\quad {\text{if}}\ \sigma _{X}\sigma _{Y}>0.$

where $\operatorname {E}$ is theexpected value operator, $\operatorname {cov}$ meanscovariance, and $\operatorname {corr}$ is a widely used alternative notation for the correlation coefficient. The Pearson correlation is defined only if both standard deviations are finite and positive. An alternative formula purely in terms ofmoments is:

$\rho _{X,Y}={\operatorname {E} (XY)-\operatorname {E} (X)\operatorname {E} (Y) \over {\sqrt {\operatorname {E} (X^{2})-\operatorname {E} (X)^{2}}}\cdot {\sqrt {\operatorname {E} (Y^{2})-\operatorname {E} (Y)^{2}}}}$

Correlation and independence

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It is a corollary of theCauchy–Schwarz inequality that theabsolute value of the Pearson correlation coefficient is not bigger than 1. Therefore, the value of a correlation coefficient ranges between −1 and +1. The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship (anti-correlation),^[5] and some value in theopen interval $(-1,1)$ in all other cases, indicating the degree oflinear dependence between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.

If the variables areindependent, Pearson's correlation coefficient is 0. However, because the correlation coefficient detects only linear dependencies between two variables, the converse is not necessarily true. A correlation coefficient of 0 does not imply that the variables are independent^{[citation needed]}.

${\begin{aligned}X,Y{\text{ independent}}\quad &\Rightarrow \quad \rho _{X,Y}=0\quad (X,Y{\text{ uncorrelated}})\\\rho _{X,Y}=0\quad (X,Y{\text{ uncorrelated}})\quad &\nRightarrow \quad X,Y{\text{ independent}}\end{aligned}}$

For example, suppose the random variable $X {\displaystyle X}$ is symmetrically distributed about zero, and $Y=X^{2}$ . Then $Y {\displaystyle Y}$ is completely determined by $X {\displaystyle X}$ , so that $X {\displaystyle X}$ and $Y {\displaystyle Y}$ are perfectly dependent, but their correlation is zero; they areuncorrelated. However, in the special case when $X {\displaystyle X}$ and $Y {\displaystyle Y}$ arejointly normal, uncorrelatedness is equivalent to independence.

Even though uncorrelated data does not necessarily imply independence, one can check if random variables are independent if theirmutual information is 0.

Sample correlation coefficient

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Given a series of $n {\displaystyle n}$ measurements of the pair $(X_{i},Y_{i})$ indexed by $i=1,\ldots ,n$ , thesample correlation coefficient can be used to estimate the population Pearson correlation $\rho _{X,Y}$ between $X {\displaystyle X}$ and $Y {\displaystyle Y}$ . The sample correlation coefficient is defined as

r_{xy}\quad {\overset {\underset {\mathrm {def} }{}}{=}}\quad {\frac {\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{(n-1)s_{x}s_{y}}}={\frac {\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sqrt {\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})^{2}\sum \limits _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}}},

where ${\overline {x}}$ and ${\overline {y}}$ are the samplemeans of $X {\displaystyle X}$ and $Y {\displaystyle Y}$ , and $s_{x}$ and $s_{y}$ are thecorrected sample standard deviations of $X {\displaystyle X}$ and $Y {\displaystyle Y}$ .

Equivalent expressions for $r_{xy}$ are

{\begin{aligned}r_{xy}&={\frac {\sum x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{ns'_{x}s'_{y}}}\\[5pt]&={\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{{\sqrt {n\sum x_{i}^{2}-(\sum x_{i})^{2}}}~{\sqrt {n\sum y_{i}^{2}-(\sum y_{i})^{2}}}}}.\end{aligned}}

where $s'_{x}$ and $s'_{y}$ are theuncorrected sample standard deviations of $X {\displaystyle X}$ and $Y {\displaystyle Y}$ .

If $x {\displaystyle x}$ and $y {\displaystyle y}$ are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range.^[6] For the case of a linear model with a single independent variable, thecoefficient of determination (R squared) is the square of $r_{xy}$ , Pearson's product-moment coefficient.

Example

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Consider thejoint probability distribution ofX andY given in the table below.

$\mathrm {P} (X=x,Y=y)$
y x	−1	0	1
0	0	⁠1/3⁠	0
1	⁠1/3⁠	0	⁠1/3⁠

For this joint distribution, themarginal distributions are:

\mathrm {P} (X=x)={\begin{cases}{\frac {1}{3}}&\quad {\text{for }}x=0\\{\frac {2}{3}}&\quad {\text{for }}x=1\end{cases}}

\mathrm {P} (Y=y)={\begin{cases}{\frac {1}{3}}&\quad {\text{for }}y=-1\\{\frac {1}{3}}&\quad {\text{for }}y=0\\{\frac {1}{3}}&\quad {\text{for }}y=1\end{cases}}

This yields the following expectations and variances:

\mu _{X}={\frac {2}{3}}

\mu _{Y}=0

\sigma _{X}^{2}={\frac {2}{9}}

\sigma _{Y}^{2}={\frac {2}{3}}

Therefore:

{\begin{aligned}\rho _{X,Y}&={\frac {1}{\sigma _{X}\sigma _{Y}}}\mathrm {E} [(X-\mu _{X})(Y-\mu _{Y})]\\[5pt]&={\frac {1}{\sigma _{X}\sigma _{Y}}}\sum _{x,y}{(x-\mu _{X})(y-\mu _{Y})\mathrm {P} (X=x,Y=y)}\\[5pt]&={\frac {3{\sqrt {3}}}{2}}\left(\left(1-{\frac {2}{3}}\right)(-1-0){\frac {1}{3}}+\left(0-{\frac {2}{3}}\right)(0-0){\frac {1}{3}}+\left(1-{\frac {2}{3}}\right)(1-0){\frac {1}{3}}\right)=0.\end{aligned}}

Rank correlation coefficients

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Main articles:Spearman's rank correlation coefficient andKendall tau rank correlation coefficient

Rank correlation coefficients, such asSpearman's rank correlation coefficient andKendall's rank correlation coefficient (τ) measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship. If, as the one variable increases, the otherdecreases, the rank correlation coefficients will be negative. It is common to regard these rank correlation coefficients as alternatives to Pearson's coefficient, used either to reduce the amount of calculation or to make the coefficient less sensitive to non-normality in distributions. However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than thePearson product-moment correlation coefficient, and are best seen as measures of a different type of association, rather than as an alternative measure of the population correlation coefficient.^[7]^[8]

To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers $(x,y)$ :

(0, 1), (10, 100), (101, 500), (102, 2000).

As we go from each pair to the next pair $x {\displaystyle x}$ increases, and so does $y {\displaystyle y}$ . This relationship is perfect, in the sense that an increase in $x {\displaystyle x}$ isalways accompanied by an increase in $y {\displaystyle y}$ . This means that we have a perfect rank correlation, and both Spearman's and Kendall's correlation coefficients are 1, whereas in this example Pearson product-moment correlation coefficient is 0.7544, indicating that the points are far from lying on a straight line. In the same way if $y {\displaystyle y}$ alwaysdecreases when $x {\displaystyle x}$ increases, the rank correlation coefficients will be −1, while the Pearson product-moment correlation coefficient may or may not be close to −1, depending on how close the points are to a straight line. Although in the extreme cases of perfect rank correlation the two coefficients are both equal (being both +1 or both −1), this is not generally the case, and so values of the two coefficients cannot meaningfully be compared.^[7] For example, for the three pairs (1, 1) (2, 3) (3, 2) Spearman's coefficient is 1/2, while Kendall's coefficient is 1/3.

Other measures of dependence among random variables

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The information given by a correlation coefficient is not enough to define the dependence structure between random variables. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is amultivariate normal distribution. (See diagram above.) In the case ofelliptical distributions it characterizes the (hyper-)ellipses of equal density; however, it does not completely characterize the dependence structure (for example, amultivariate t-distribution's degrees of freedom determine the level of tail dependence).

For continuous variables, multiple alternative measures of dependence were introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables (see^[9] and reference references therein for an overview). They all share the important property that a value of zero implies independence. This led some authors^[9]^[10] to recommend their routine usage, particularly ofDistance correlation.^[11]^[12] Another alternative measure is the Randomized Dependence Coefficient.^[13] The RDC is a computationally efficient,copula-based measure of dependence between multivariate random variables and is invariant with respect to non-linear scalings of random variables.

One important disadvantage of the alternative, more general measures is that, when used to test whether two variables are associated, they tend to have lower power compared to Pearson's correlation when the data follow a multivariate normal distribution.^[9] This is an implication of theNo free lunch theorem. To detect all kinds of relationships, these measures have to sacrifice power on other relationships, particularly for the important special case of a linear relationship with Gaussian marginals, for which Pearson's correlation is optimal. Another problem concerns interpretation. While Person's correlation can be interpreted for all values, the alternative measures can generally only be interpreted meaningfully at the extremes.^[14]

For twobinary variables, theodds ratio measures their dependence, and takes range non-negative numbers, possibly infinity:⁠ $[0,+\infty ]$ ⁠. Related statistics such asYule'sY andYule'sQ normalize this to the correlation-like range⁠ $[-1,1]$ ⁠. The odds ratio is generalized by thelogistic model to model cases where the dependent variables are discrete and there may be one or more independent variables.

Thecorrelation ratio,entropy-basedmutual information,total correlation,dual total correlation andpolychoric correlation are all also capable of detecting more general dependencies, as is consideration of thecopula between them, while thecoefficient of determination generalizes the correlation coefficient tomultiple regression.

Sensitivity to the data distribution

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Further information:Pearson product-moment correlation coefficient § Sensitivity to the data distribution

The degree of dependence between variablesX andY does not depend on the scale on which the variables are expressed. That is, if we are analyzing the relationship betweenX andY, most correlation measures are unaffected by transformingX toa +bX andY toc +dY, wherea,b,c, andd are constants (b andd being positive). This is true of some correlationstatistics as well as theirpopulation analogues. Some correlation statistics, such as the rank correlation coefficient, are also invariant tomonotone transformations of the marginal distributions ofX and/orY.

Pearson/Spearman correlation coefficients betweenX andY are shown when the two variables' ranges are unrestricted, and when the range ofX is restricted to the interval (0,1).

Most correlation measures are sensitive to the manner in whichX andY are sampled. Dependencies tend to be stronger if viewed over a wider range of values. Thus, if we consider the correlation coefficient between the heights of fathers and their sons over all adult males, and compare it to the same correlation coefficient calculated when the fathers are selected to be between 165 cm and 170 cm in height, the correlation will be weaker in the latter case. Several techniques have been developed that attempt to correct for range restriction in one or both variables, and are commonly used in meta-analysis; the most common are Thorndike's case II and case III equations.^[15]

Various correlation measures in use may be undefined for certain joint distributions ofX andY. For example, the Pearson correlation coefficient is defined in terms ofmoments, and hence will be undefined if the moments are undefined. Measures of dependence based onquantiles are always defined. Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as beingunbiased, orasymptotically consistent, based on the spatial structure of the population from which the data were sampled.

Sensitivity to the data distribution can be used to an advantage. For example,scaled correlation is designed to use the sensitivity to the range in order to pick out correlations between fast components oftime series.^[16] By reducing the range of values in a controlled manner, the correlations on long time scale are filtered out and only the correlations on short time scales are revealed.

Correlation matrices

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The correlation matrix of $n {\displaystyle n}$ random variables $X_{1},\ldots ,X_{n}$ is the $n\times n$ matrix $C {\displaystyle C}$ whose $(i,j)$ entry is

c_{ij}:=\operatorname {corr} (X_{i},X_{j})={\frac {\operatorname {cov} (X_{i},X_{j})}{\sigma _{X_{i}}\sigma _{X_{j}}}},\quad {\text{if}}\ \sigma _{X_{i}}\sigma _{X_{j}}>0.

Thus the diagonal entries are all identicallyone. If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as thecovariance matrix of thestandardized random variables $X_{i}/\sigma (X_{i})$ for $i=1,\dots ,n$ . This applies both to the matrix of population correlations (in which case $\sigma$ is the population standard deviation), and to the matrix of sample correlations (in which case $\sigma$ denotes the sample standard deviation). Consequently, each is necessarily apositive-semidefinite matrix. Moreover, the correlation matrix is strictlypositive definite if no variable can have all its values exactly generated as a linear function of the values of the others.

The correlation matrix is symmetric because the correlation between $X_{i}$ and $X_{j}$ is the same as the correlation between $X_{j}$ and $X_{i}$ .

A correlation matrix appears, for example, in one formula for thecoefficient of multiple determination, a measure of goodness of fit inmultiple regression.

Instatistical modelling, correlation matrices representing the relationships between variables are categorized into different correlation structures, which are distinguished by factors such as the number of parameters required to estimate them. For example, in anexchangeable correlation matrix, all pairs of variables are modeled as having the same correlation, so all non-diagonal elements of the matrix are equal to each other. On the other hand, anautoregressive matrix is often used when variables represent a time series, since correlations are likely to be greater when measurements are closer in time. Other examples include independent, unstructured, M-dependent, andToeplitz.

Inexploratory data analysis, theiconography of correlations consists in replacing a correlation matrix by a diagram where the "remarkable" correlations are represented by a solid line (positive correlation), or a dotted line (negative correlation).

Nearest valid correlation matrix

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In some applications (e.g., building data models from only partially observed data) one wants to find the "nearest" correlation matrix to an "approximate" correlation matrix (e.g., a matrix which typically lacks semi-definite positiveness due to the way it has been computed).

In 2002, Higham^[17] formalized the notion of nearness using theFrobenius norm and provided a method for computing the nearest correlation matrix using theDykstra's projection algorithm, of which an implementation is available as an online Web API.^[18]

This sparked interest in the subject, with new theoretical (e.g., computing the nearest correlation matrix with factor structure^[19]) and numerical (e.g. usage theNewton's method for computing the nearest correlation matrix^[20]) results obtained in the subsequent years.

Uncorrelatedness and independence of stochastic processes

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Similarly for two stochastic processes $\left\{X_{t}\right\}_{t\in {\mathcal {T}}}$ and $\left\{Y_{t}\right\}_{t\in {\mathcal {T}}}$ : If they are independent, then they are uncorrelated.^[21]^{: p. 151} The opposite of this statement might not be true. Even if two variables are uncorrelated, they might not be independent to each other.

Common misconceptions

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Correlation and causality

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Main article:Correlation does not imply causation

The conventional dictum that "correlation does not imply causation" means that correlation cannot be used by itself to infer a causal relationship between the variables.^[22] This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap withidentity relations (tautologies), where no causal process exists. Consequently, a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction).

A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health, or does good health lead to good mood, or both? Or does some other factor underlie both? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.

Simple linear correlations

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Anscombe's quartet: four sets of data with the same correlation of 0.816

The Pearson correlation coefficient indicates the strength of alinear relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if theconditional mean of $Y {\displaystyle Y}$ given $X {\displaystyle X}$ , denoted $\operatorname {E} (Y\mid X)$ , is not linear in $X {\displaystyle X}$ , the correlation coefficient will not fully determine the form of $\operatorname {E} (Y\mid X)$ .

The adjacent image showsscatter plots ofAnscombe's quartet, a set of four different pairs of variables created byFrancis Anscombe.^[23] The four $y {\displaystyle y}$ variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line ( ${\textstyle y=3+0.5x}$ ). However, as can be seen on the plots, the distribution of the variables is very different. The first one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear. In this case the Pearson correlation coefficient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for oneoutlier which exerts enough influence to lower the correlation coefficient from 1 to 0.816. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

These examples indicate that the correlation coefficient, as asummary statistic, cannot replace visual examination of the data. The examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow anormal distribution, but this is only partially correct.^[4] The Pearson correlation can be accurately calculated for any distribution that has a finitecovariance matrix, which includes most distributions encountered in practice. However, the Pearson correlation coefficient (taken together with the sample mean and variance) is only asufficient statistic if the data is drawn from amultivariate normal distribution. As a result, the Pearson correlation coefficient fully characterizes the relationship between variables if and only if the data are drawn from a multivariate normal distribution.

Bivariate normal distribution

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If a pair $\ (X,Y)\$ of random variables follows abivariate normal distribution, the conditional mean $\operatorname {\boldsymbol {\mathcal {E}}} (X\mid Y)$ is a linear function of $Y {\displaystyle Y}$ , and the conditional mean $\operatorname {\boldsymbol {\mathcal {E}}} (Y\mid X)$ is a linear function of $\ X~.$ The correlation coefficient $\ \rho _{X,Y}\$ between $\ X\$ and $\ Y\ ,$ and themarginal means and variances of $\ X\$ and $\ Y\$ determine this linear relationship:

\operatorname {\boldsymbol {\mathcal {E}}} (Y\mid X)=\operatorname {\boldsymbol {\mathcal {E}}} (Y)+\rho _{X,Y}\cdot \sigma _{Y}\cdot {\frac {\ X-\operatorname {\boldsymbol {\mathcal {E}}} (X)\ }{\sigma _{X}}}\ ,

where $\operatorname {\boldsymbol {\mathcal {E}}} (X)$ and $\operatorname {\boldsymbol {\mathcal {E}}} (Y)$ are the expected values of $\ X\$ and $\ Y\ ,$ respectively, and $\ \sigma _{X}\$ and $\ \sigma _{Y}\$ are the standard deviations of $\ X\$ and $\ Y\ ,$ respectively.

The empirical correlation $r {\displaystyle r}$ is anestimate of the correlation coefficient $\ \rho ~.$ A distribution estimate for $\ \rho \$ is given by

\pi (\rho \mid r)={\frac {\ \Gamma (N)\ }{\ {\sqrt {2\pi \ }}\cdot \Gamma (N-{\tfrac {\ 1\ }{2}})\ }}\cdot {\bigl (}1-r^{2}{\bigr )}^{\frac {\ N\ -2\ }{2}}\cdot {\bigl (}1-\rho ^{2}{\bigr )}^{\frac {\ N-3\ }{2}}\cdot {\bigl (}1-r\rho {\bigr )}^{-N+{\frac {\ 3\ }{2}}}\cdot F_{\mathsf {Hyp}}\left(\ {\tfrac {\ 3\ }{2}},-{\tfrac {\ 1\ }{2}};N-{\tfrac {\ 1\ }{2}};{\frac {\ 1+r\rho \ }{2}}\ \right)\

where $\ F_{\mathsf {Hyp}}\$ is theGaussian hypergeometric function.

This density is both a Bayesianposterior density and an exact optimalconfidence distribution density.^[24]^[25]

References

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^Croxton, Frederick Emory; Cowden, Dudley Johnstone; Klein, Sidney (1968)Applied General Statistics, Pitman.ISBN 9780273403159 (page 625)
^Dietrich, Cornelius Frank (1991)Uncertainty, Calibration and Probability: The Statistics of Scientific and Industrial Measurement 2nd Edition, A. Higler.ISBN 9780750300605 (Page 331)
^Aitken, Alexander Craig (1957)Statistical Mathematics 8th Edition. Oliver & Boyd.ISBN 9780050013007 (Page 95)
^^a ^bRodgers, J. L.; Nicewander, W. A. (1988). "Thirteen ways to look at the correlation coefficient".The American Statistician.42 (1):59–66.doi:10.1080/00031305.1988.10475524.JSTOR 2685263.
^Dowdy, S. and Wearden, S. (1983). "Statistics for Research", Wiley.ISBN 0-471-08602-9 pp 230
^Francis, DP; Coats AJ; Gibson D (1999). "How high can a correlation coefficient be?".Int J Cardiol.69 (2):185–199.doi:10.1016/S0167-5273(99)00028-5.PMID 10549842.
^^a ^bYule, G.U and Kendall, M.G. (1950), "An Introduction to the Theory of Statistics", 14th Edition (5th Impression 1968). Charles Griffin & Co. pp 258–270
^Kendall, M. G. (1955) "Rank Correlation Methods", Charles Griffin & Co.
^^a ^b ^cKarch, Julian D.; Perez-Alonso, Andres F.; Bergsma, Wicher P. (2024-08-04). "Beyond Pearson's Correlation: Modern Nonparametric Independence Tests for Psychological Research".Multivariate Behavioral Research.59 (5):957–977.doi:10.1080/00273171.2024.2347960.hdl:1887/4108931.PMID 39097830.
^Simon, Noah; Tibshirani, Robert (2014). "Comment on "Detecting Novel Associations In Large Data Sets" by Reshef Et Al, Science Dec 16, 2011". p. 3.arXiv:1401.7645 [stat.ME].
^Székely, G. J. Rizzo; Bakirov, N. K. (2007). "Measuring and testing independence by correlation of distances".Annals of Statistics.35 (6):2769–2794.arXiv:0803.4101.doi:10.1214/009053607000000505.S2CID 5661488.
^Székely, G. J.; Rizzo, M. L. (2009)."Brownian distance covariance".Annals of Applied Statistics.3 (4):1233–1303.arXiv:1010.0297.doi:10.1214/09-AOAS312.PMC 2889501.PMID 20574547.
^Lopez-Paz D. and Hennig P. and Schölkopf B. (2013). "The Randomized Dependence Coefficient", "Conference on Neural Information Processing Systems"Reprint
^Reimherr, Matthew; Nicolae, Dan L. (2013). "On Quantifying Dependence: A Framework for Developing Interpretable Measures".Statistical Science.28 (1):116–130.arXiv:1302.5233.doi:10.1214/12-STS405.
^Thorndike, Robert Ladd (1947).Research problems and techniques (Report No. 3). Washington DC: US Govt. print. off.
^Nikolić, D; Muresan, RC; Feng, W; Singer, W (2012). "Scaled correlation analysis: a better way to compute a cross-correlogram".European Journal of Neuroscience.35 (5):1–21.doi:10.1111/j.1460-9568.2011.07987.x.PMID 22324876.S2CID 4694570.
^Higham, Nicholas J. (2002). "Computing the nearest correlation matrix—a problem from finance".IMA Journal of Numerical Analysis.22 (3):329–343.CiteSeerX 10.1.1.661.2180.doi:10.1093/imanum/22.3.329.
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^Borsdorf, Rudiger; Higham, Nicholas J.; Raydan, Marcos (2010)."Computing a Nearest Correlation Matrix with Factor Structure"(PDF).SIAM J. Matrix Anal. Appl.31 (5):2603–2622.doi:10.1137/090776718.
^Qi, HOUDUO; Sun, DEFENG (2006). "A quadratically convergent Newton method for computing the nearest correlation matrix".SIAM J. Matrix Anal. Appl.28 (2):360–385.doi:10.1137/050624509.
^Park, Kun Il (2018).Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer.ISBN 978-3-319-68074-3.
^Aldrich, John (1995)."Correlations Genuine and Spurious in Pearson and Yule".Statistical Science.10 (4):364–376.doi:10.1214/ss/1177009870.JSTOR 2246135.
^Anscombe, Francis J. (1973). "Graphs in statistical analysis".The American Statistician.27 (1):17–21.doi:10.2307/2682899.JSTOR 2682899.
^Taraldsen, Gunnar (2021)."The confidence density for correlation".Sankhya A.85:600–616.doi:10.1007/s13171-021-00267-y.hdl:11250/3133125.ISSN 0976-8378.S2CID 244594067.
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External links

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Look upcorrelation ordependence in Wiktionary, the free dictionary.

Wikimedia Commons has media related toCorrelation.

Wikiversity has learning resources aboutCorrelation

MathWorld page on the (cross-)correlation coefficient/s of a sample
Compute significance between two correlations, for the comparison of two correlation values.
"A MATLAB Toolbox for computing Weighted Correlation Coefficients". Archived fromthe original on 24 April 2021.
Proof that the Sample Bivariate Correlation has limits plus or minus 1
Interactive Flash simulation on the correlation of two normally distributed variables by Juha Puranen.
Correlation analysis. Biomedical Statistics
R-PsychologistCorrelation visualization of correlation between two numeric variables

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Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in
Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife
Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons
Parametric tests	Likelihood-ratio Score/Lagrange multiplier Wald

Specific tests

Z-test(normal) Student'st-test F-test
Goodness of fit	Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality(Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC
Rank statistics	Sign Sample median Signed rank(Wilcoxon) Hodges–Lehmann estimator Rank sum(Mann–Whitney) Nonparametric anova 1-way(Kruskal–Wallis) 2-way(Friedman) Ordered alternative(Jonckheere–Terpstra) Van der Waerden test

Bayesian inference

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis	Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)
Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression
Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Homoscedasticity and Heteroscedasticity
Generalized linear model	Exponential families Logistic(Bernoulli) / Binomial / Poisson regressions
Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality
Specific tests	Dickey–Fuller Johansen Q-statistic(Ljung–Box) Durbin–Watson Breusch–Godfrey
Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model(Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)
Frequency domain	Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time
Hazard function	Nelson–Aalen estimator
Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics
Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification
Social statistics	Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging