Correlation

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

Several sets of (x,y) points, with thePearson correlation coefficientofxandyfor 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 ofYis zero.

Instatistics,correlationordependenceis any statistical relationship, whethercausalor not, between tworandom variablesorbivariate 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 arelinearlyrelated. Familiar examples of dependent phenomena include the correlation between theheightof 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 so-calleddemand 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 weathercauses 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 aredependentif they do not satisfy a mathematical property ofprobabilistic independence.In informal parlance,correlationis 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$x$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$r$,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– have been developed to be morerobustthan Pearson's, that is, more sensitive to nonlinear relationships.Mutual informationcan also be applied to measure dependence between two variables.

Pearson's product-moment coefficient

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 thecovarianceof the two variables by the product of theirstandard deviations.Karl Pearsondeveloped the coefficient from a similar but slightly different idea byFrancis Galton.

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.

The population correlation coefficient$\rho _{X,Y}$between tworandom variables$X$and$Y$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 valueoperator,$\operatorname {cov}$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 ofmomentsis:

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

Correlation and independence

It is a corollary of theCauchy–Schwarz inequalitythat theabsolute valueof 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),and some value in theopen interval$(-1,1)$in all other cases, indicating the degree oflinear dependencebetween 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.

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

For example, suppose the random variable$X$is symmetrically distributed about zero, and$Y=X^{2}$.Then$Y$is completely determined by$X$,so that$X$and$Y$are perfectly dependent, but their correlation is zero; they areuncorrelated.However, in the special case when$X$and$Y$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 informationis 0.

Sample correlation coefficient

Given a series of$n$measurements of the pair$(X_{i},Y_{i})$indexed by$i=1,\ldots,n$,thesample correlation coefficientcan be used to estimate the population Pearson correlation$\rho _{X,Y}$between$X$and$Y$.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 samplemeansof$X$and$Y$,and$s_{x}$and$s_{y}$are thecorrected sample standard deviationsof$X$and$Y$.

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 theuncorrectedsample standard deviationsof$X$and$Y$.

If$x$and$y$are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range.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

Consider thejoint probability distributionofXandYgiven 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 distributionsare:

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

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

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

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

Rank correlationcoefficients, such asSpearman's rank correlation coefficientandKendall'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.

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$x$increases, and so does$y$.This relationship is perfect, in the sense that an increase in$x$isalwaysaccompanied by an increase in$y$.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$y$alwaysdecreaseswhen$x$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.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

See also:Pearson product-moment correlation coefficient § Variants

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 distributionsit 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 (seeand reference references therein for an overview). They all share the important property that a value of zero implies independence. This led some authorsto recommend their routine usage, particularly ofDistance correlation.Another alternative measure is the Randomized Dependence Coefficient.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.This is an implication of theNo free lunch theoremtheorem. 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 meaningfull at the extremes.

For twobinary variables,theodds ratiomeasures their dependence, and takes range non-negative numbers, possibly infinity:⁠$[0,+\infty ]$⁠.Related statistics such asYule'sYandYule'sQnormalize this to the correlation-like range⁠$[-1,1]$⁠.The odds ratio is generalized by thelogistic modelto 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 correlationandpolychoric correlationare all also capable of detecting more general dependencies, as is consideration of thecopulabetween them, while thecoefficient of determinationgeneralizes the correlation coefficient tomultiple regression.

Sensitivity to the data distribution

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

The degree of dependence between variablesXandYdoes not depend on the scale on which the variables are expressed. That is, if we are analyzing the relationship betweenXandY,most correlation measures are unaffected by transformingXtoa+bXandYtoc+dY,wherea,b,c,anddare constants (banddbeing positive). This is true of some correlationstatisticsas well as theirpopulationanalogues. Some correlation statistics, such as the rank correlation coefficient, are also invariant tomonotone transformationsof the marginal distributions ofXand/orY.

Pearson/Spearmancorrelation coefficients betweenXandYare shown when the two variables' ranges are unrestricted, and when the range ofXis restricted to the interval (0,1).

Most correlation measures are sensitive to the manner in whichXandYare 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.

Various correlation measures in use may be undefined for certain joint distributions ofXandY.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 onquantilesare 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 correlationis designed to use the sensitivity to the range in order to pick out correlations between fast components oftime series.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

The correlation matrix of$n$random variables$X_{1},\ldots,X_{n}$is the$n\times n$matrix$C$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 matrixof 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 definiteif 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 anexchangeablecorrelation 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, anautoregressivematrix 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 correlationsconsists 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

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, Highamformalized the notion of nearness using theFrobenius normand 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.

This sparked interest in the subject, with new theoretical (e.g., computing the nearest correlation matrix with factor structure) and numerical (e.g. usage theNewton's methodfor computing the nearest correlation matrix) results obtained in the subsequent years.

Uncorrelatedness and independence of stochastic processes

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.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

Correlation and causality

Main article:Correlation does not imply causation

See also:Normally distributed and uncorrelated does not imply independent

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.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 withidentityrelations (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

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

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

The adjacent image showsscatter plotsofAnscombe's quartet,a set of four different pairs of variables created byFrancis Anscombe.The four$y$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 oneoutlierwhich 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.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 statisticif 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

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$,and the conditional mean$\operatorname {\boldsymbol {\mathcal {E}}} (Y\mid X)$is a linear function of$\ X~.$The correlation coefficient$\ \rho _{X,Y}\$between$\ X\$and$\ Y\,$and themarginalmeans and variances of$\ 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$r$is anestimateof 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 Bayesianposteriordensity and an exact optimalconfidence distributiondensity.

See also

Further information:Correlation (disambiguation)

• Autocorrelation
• Canonical correlation
• Coefficient of determination
• Cointegration
• Concordance correlation coefficient
• Cophenetic correlation
• Correlation disattenuation
• Correlation function
• Correlation gap
• Covariance
• Covariance and correlation
• Cross-correlation
• Ecological correlation
• Fraction of variance unexplained
• Genetic correlation
• Goodman and Kruskal's lambda
• Iconography of correlations
• Illusory correlation
• Interclass correlation
• Intraclass correlation
• Lift (data mining)
• Mean dependence
• Modifiable areal unit problem
• Multiple correlation
• Point-biserial correlation coefficient
• Quadrant count ratio
• Spurious correlation
• Statistical correlation ratio
• Subindependence

Further reading

• Cohen, J.; Cohen P.; West, S.G. &Aiken, L.S.(2002).Applied multiple regression/correlation analysis for the behavioral sciences(3rd ed.). Psychology Press.ISBN978-0-8058-2223-6.
• "Correlation (in statistics)",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Oestreicher, J. & D. R. (February 26, 2015).Plague of Equals: A science thriller of international disease, politics and drug discovery.California: Omega Cat Press. p. 408.ISBN978-0963175540.

External links

• 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 originalon 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 variablesby Juha Puranen.
• Correlation analysis. Biomedical Statistics
• R-PsychologistCorrelationvisualization of correlation between two numeric variables