View source for Pearson correlation coefficient

{{Short description|Measure of linear correlation}}
{{Distinguish|Coefficient of determination}}
{{Use dmy dates|date=August 2022}}
[[File:Correlation examples2.svg|thumb|400px|right|Several sets of (''x'',&nbsp;''y'') points, with the correlation coefficient of ''x'' and ''y'' for each set. The correlation reflects the strength and direction of a linear relationship (top row), but not the slope of that relationship (middle row), nor many aspects of nonlinear relationships (bottom row). 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]], the '''Pearson correlation coefficient''' ('''PCC'''){{efn|Also known as '''Pearson's ''r''''', the '''Pearson product-moment correlation coefficient''' ('''PPMCC'''), the '''bivariate correlation''',<ref>{{cite web |url=http://libguides.library.kent.edu/SPSS/PearsonCorr |title=SPSS Tutorials: Pearson Correlation}}</ref> or simply the unqualified '''correlation coefficient'''<ref>{{cite web|url=https://www.statisticshowto.com/probability-and-statistics/correlation-coefficient-formula/|title=Correlation Coefficient: Simple Definition, Formula, Easy Steps|website=Statistics How To}}</ref>}} is a [[correlation coefficient]] that measures [[linear]] correlation between two sets of data. It is the ratio between the [[covariance]] of two variables and the product of their [[standard deviation]]s; thus, it is essentially a normalized measurement of the covariance, such that the result always has a value between −1 and 1.<ref>{{Cite web |title=4.3. Covariance and correlation coefficient — TU Delft textbook |url=https://mude.citg.tudelft.nl/book/2025/multivariate/Correlation_and_Covariance.html |access-date=2025-10-30 |website=mude.citg.tudelft.nl}}</ref> A key difference is that unlike covariance, this correlation coefficient does not have [[Unit of measurement|units]], allowing comparison of the strength of the joint association between different pairs of random variables that do not necessarily have the same units.<ref>{{Cite web |title=4.3. Covariance and correlation coefficient — TU Delft textbook |url=https://mude.citg.tudelft.nl/book/2025/multivariate/Correlation_and_Covariance.html |access-date=2025-10-30 |website=mude.citg.tudelft.nl}}</ref> As with covariance itself, the measure can only reflect a linear [[correlation]] of variables, and ignores many other types of relationships or correlations. As a simple example, one would expect the age and height of a sample of children from a school to have a Pearson correlation coefficient significantly greater than 0, but less than 1 (as 1 would represent an unrealistically perfect correlation).

==Naming and history==
It was developed by [[Karl Pearson]] from a related idea introduced by [[Francis Galton]] in the 1880s, and for which the mathematical formula was derived and published by [[Auguste Bravais]] in 1844.{{efn|As early as 1877, Galton was using the term "reversion" and the symbol "''r''" for what would become "regression".<ref>{{cite journal |first=F. |last=Galton |date=5–19 April 1877 |title=Typical laws of heredity |journal=Nature |volume=15 |issue=388, 389, 390 |pages=492–495; 512–514; 532–533 |doi=10.1038/015492a0 |bibcode=1877Natur..15..492. |s2cid=4136393 |url=https://books.google.com/books?id=eskKAAAAYAAJ&pg=PA512|doi-access=free }} In the "Appendix" on page 532, Galton uses the term "reversion" and the symbol ''r''.</ref><ref>{{cite journal |first=F. |last=Galton |date=24 September 1885 |title=The British Association: Section II, Anthropology: Opening address by Francis Galton, F.R.S., etc., President of the Anthropological Institute, President of the Section |journal=Nature |volume=32 |issue=830 |pages=507–510 |url=https://books.google.com/books?id=lN3RjXLUuWsC&pg=PA499}}</ref><ref>{{cite journal |first=F. |last=Galton |year=1886 |title=Regression towards mediocrity in hereditary stature |journal=Journal of the Anthropological Institute of Great Britain and Ireland |volume=15 |pages=246–263 |doi=10.2307/2841583 |jstor=2841583 |url=https://books.google.com/books?id=JPcRAAAAYAAJ&pg=PA246|doi-access=free }}</ref>}}<ref>{{cite journal |first=Karl |last=Pearson |date=20 June 1895 |title=Notes on regression and inheritance in the case of two parents |journal=Proceedings of the Royal Society of London |volume=58 |pages=240–242 |bibcode=1895RSPS...58..240P |url=https://books.google.com/books?id=60aL0zlT-90C&pg=PA240}}</ref><ref>{{cite journal |last=Stigler |first=Stephen M. |year=1989 |title=Francis Galton's account of the invention of correlation |journal=Statistical Science |volume=4 |issue=2 |pages=73–79 |jstor=2245329 |doi=10.1214/ss/1177012580 |doi-access=free}}</ref><ref>{{cite journal |title=Analyse mathematique sur les probabilités des erreurs de situation d'un point |journal=Mem. Acad. Roy. Sci. Inst. France |series=Sci. Math, et Phys. |volume=9 |pages=255–332 |year=1844 |url=https://books.google.com/books?id=y3s_AAAAcAAJ&q=Sur%20Les%20Probabilit%C3%A9s%20des%20Erreurs%20de%20Situation%20d'un%20Point&pg=PA1 |via=Google Books |language=fr}}</ref><ref>{{cite journal |last1=Wright |first1=S. |year=1921 |title=Correlation and causation |url=https://archive.org/details/sim_journal-of-agricultural-research_1921-01-03_20_7/page/n51 |journal=Journal of Agricultural Research |volume=20 |issue=7 |pages=557–585}}</ref> The naming of the coefficient is thus an example of [[Stigler's Law]].

==Intuitive explanation==
The correlation coefficient can be derived by considering the cosine of the angle between two points representing the two sets of x and y co-ordinate data.<ref>{{Cite web |title=How was the correlation coefficient formula derived? |url=https://stats.stackexchange.com/questions/635637/how-was-the-correlation-coefficient-formula-derived |access-date=2024-10-26 |website=Cross Validated |language=en}}</ref> This expression is therefore a number between -1 and 1 and is equal to unity when all the points lie on a straight line.

==Definition==
Pearson's correlation coefficient is the [[covariance]] of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first [[Moment (mathematics)|moment]] about the origin) of the product of the mean-adjusted random variables; hence the modifier ''product-moment'' in the name.{{verify source|date=February 2024}}

===For a population===
Pearson's correlation coefficient, when applied to a [[statistical population|population]], is commonly represented by the Greek letter ''ρ'' (rho) and may be referred to as the ''population correlation coefficient'' or the ''population Pearson correlation coefficient''. Given a pair of random variables <math>(X,Y)</math> (for example, Height and Weight), the formula for ''ρ''<ref name="RealCorBasic">Real Statistics Using Excel, "[http://www.real-statistics.com/correlation/basic-concepts-correlation/ Basic Concepts of Correlation]", retrieved 22 February 2015.</ref> is<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Statistical Correlation|url=https://mathworld.wolfram.com/StatisticalCorrelation.html|access-date=2020-08-22|website=Wolfram MathWorld|language=en}}</ref>

<math display=block> \rho_{X,Y}= \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y}</math>

where
*<math> \operatorname{cov} </math> is the [[covariance]]
*<math> \sigma_X </math> is the [[standard deviation]] of  <math> X </math>
*<math> \sigma_Y </math> is the standard deviation of  <math> Y </math>.

The formula for <math>\operatorname{cov}(X,Y)</math> can be expressed in terms of [[mean]] and [[Expected Value|expectation]]. Since<ref name="RealCorBasic"/>

:<math>\operatorname{cov}(X,Y) = \operatorname\mathbb{E}[(X-\mu_X)(Y-\mu_Y)],</math>

the formula for <math>\rho</math> can also be written as

<math display=block> \rho_{X,Y} = \frac{\operatorname\mathbb{E}[(X - \mu_X)(Y - \mu_Y)]}{\sigma_X\sigma_Y}</math>

where
*<math> \sigma_Y </math> and <math> \sigma_X </math> are defined as above
*<math> \mu_X </math> is the mean of <math> X </math>
*<math> \mu_Y </math> is the mean of <math> Y </math>
*<math> \operatorname\mathbb{E} </math> is the expectation.

The formula for <math>\rho</math> can be expressed in terms of uncentered moments.  Since

:<math>\begin{align}
       \mu_X ={} &\operatorname\mathbb{E}[X] \\
       \mu_Y ={} &\operatorname\mathbb{E}[Y] \\
  \sigma_X^2 ={} &\operatorname\mathbb{E}\left[\left(X - \operatorname\mathbb{E}[X]\right)^2\right] = \operatorname\mathbb{E}\left[X^2\right] - \left(\operatorname\mathbb{E}[X]\right)^2 \\
   \sigma_Y^2 ={} &\operatorname\mathbb{E}\left[\left(Y - \operatorname\mathbb{E}[Y]\right)^2\right] = \operatorname\mathbb{E}\left[Y^2\right] - \left(\operatorname\mathbb{E}[Y]\right)^2 \\
\operatorname{cov}(X,Y) ={} &\operatorname\mathbb{E}[\left(X - \mu_X\right)\left(Y - \mu_Y\right)] = \operatorname\mathbb{E}[\left(X - \operatorname\mathbb{E}[X]\right)\left(Y - \operatorname\mathbb{E}[Y]\right)] = \operatorname\mathbb{E}[XY] - \operatorname\mathbb{E}[X]\operatorname\mathbb{E}[Y] ,
\end{align}</math>

the formula for <math>\rho</math> can also be written as
<math display="block">\rho_{X,Y} =
  \frac{\operatorname\mathbb{E}[XY] - \operatorname\mathbb{E}[X]\operatorname\mathbb{E}[Y]}{\sqrt{\operatorname\mathbb{E}\left[X^2\right] - \left(\operatorname\mathbb{E}[X] \right)^2} ~ \sqrt{\operatorname\mathbb{E}\left[Y^2\right] - \left(\operatorname\mathbb{E}[Y] \right)^2}}.</math>

===For a sample===
Pearson's correlation coefficient, when applied to a [[sample (statistics)|sample]], is commonly represented by <math>r_{xy}</math> and may be referred to as the ''sample correlation coefficient'' or the ''sample Pearson correlation coefficient''. We can obtain a formula for <math>r_{xy}</math> by substituting estimates of the covariances and variances based on a sample into the formula above. Given paired data <math>\left\{ (x_1,y_1),\ldots,(x_n,y_n) \right\}</math> consisting of <math>n</math> pairs, <math>r_{xy}</math> is defined as

<math display=block>r_{xy} =\frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^n _{i=1}(x_i - \bar{x})^2} \sqrt{\sum ^n _{i=1}(y_i - \bar{y})^2}}</math>

where
*<math>n</math> is sample size
*<math>x_i, y_i</math> are the individual sample points indexed with ''i''
*<math display="inline">\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i</math> (the sample mean); and analogously for <math>\bar{y}</math>.

Rearranging gives us this<ref name="RealCorBasic"/> formula for <math>r_{xy}</math>:

:<math>r_{xy} = \frac{\sum_i x_i y_i-n\bar{x}\bar{y}}
{\sqrt{\sum_i x_i^2-n\bar{x}^2}~\sqrt{\sum_i y_i^2-n\bar{y}^2}},</math>

where <math>n, x_i, y_i, \bar{x}, \bar{y}</math> are defined as above.

Rearranging again gives us this formula for <math>r_{xy}</math>:

:<math>r_{xy} = \frac{n\sum x_i y_i - \sum x_i\sum y_i}
{\sqrt{n\sum x_i^2-\left(\sum x_i\right)^2}~\sqrt{n\sum y_i^2-\left(\sum y_i\right)^2}},</math>

where <math>n, x_i, y_i</math> are defined as above.

This formula suggests a convenient single-pass algorithm for calculating sample correlations, though depending on the numbers involved, it can sometimes be [[numerical stability|numerically unstable]].

An equivalent expression gives the formula for <math>r_{xy}</math> as the mean of the products of the [[standard score]]s as follows:

:<math>r_{xy} = \frac{1}{n-1} \sum ^n _{i=1} \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right)</math>

where
*<math>n, x_i, y_i, \bar{x}, \bar{y}</math> are defined as above, and <math>s_x, s_y</math> are defined below
*<math display="inline">\left( \frac{x_i - \bar{x}}{s_x} \right)</math> is the standard score (and analogously for the standard score of <math>y</math>).

Alternative formulae for <math>r_{xy}</math> are also available. For example, one can use the following formula for <math>r_{xy}</math>:

:<math>r_{xy} =\frac{\sum x_iy_i-n \bar{x} \bar{y}}{(n-1) s_x s_y}</math>
where
*<math>n, x_i, y_i, \bar{x}, \bar{y}</math> are defined as above and:
*<math display="inline">s_x = \sqrt{\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2}</math> (the [[sample standard deviation]]); and analogously for <math>s_y</math>.

=== For jointly Gaussian distributions ===
If <math>(X, Y)</math> is [[Joint probability distribution|jointly]] [[Gaussian distribution|gaussian]], with mean zero and [[variance]] <math>\Sigma</math>, then <math>\Sigma = \begin{bmatrix}
\sigma_X^2 & \rho_{X,Y}\sigma_X\sigma_Y \\
\rho_{X,Y}\sigma_X\sigma_Y & \sigma_Y^2 \\
\end{bmatrix}</math>.

===Practical issues===
Under heavy noise conditions, extracting the correlation coefficient between two sets of [[Random variables|stochastic variables]] is nontrivial, in particular where [[Canonical Correlation Analysis]] reports degraded correlation values due to the heavy noise contributions. A generalization of the approach is given elsewhere.<ref>{{cite book |first= N. |last=Moriya |year=2008 |contribution=Noise-related multivariate optimal joint-analysis in longitudinal stochastic processes  |pages=[https://books.google.com/books?id=4XvRgF0QfqkC&pg=PA223 223–260] |editor=Yang, Fengshan  |title=[[Progress in Applied Mathematical Modeling]] |publisher=[[Nova Science Publishers, Inc.]] |isbn=978-1-60021-976-4 }}</ref>

In case of missing data, Garren derived the [[maximum likelihood]] estimator.<ref>{{cite journal |last=Garren |first=Steven T. |date=15 June 1998 |title=Maximum likelihood estimation of the correlation coefficient in a bivariate normal model, with missing data |journal=Statistics & Probability Letters |volume=38 |issue=3 |pages=281–288 |doi=10.1016/S0167-7152(98)00035-2 }}</ref>

Some distributions (e.g., [[stable distribution]]s other than a [[normal distribution]]) do not have a defined variance.

==Mathematical properties==
The values of both the sample and population Pearson correlation coefficients are on or between −1 and 1. Correlations equal to +1 or −1 correspond to data points lying exactly on a line (in the case of the sample correlation), or to a bivariate distribution entirely [[Support (measure theory)|supported]] on a line (in the case of the population correlation). The Pearson correlation coefficient is symmetric: corr(''X'',''Y'')&nbsp;=&nbsp;corr(''Y'',''X'').

A key mathematical property of the Pearson correlation coefficient is that it is [[invariant estimator|invariant]] under separate changes in location and scale in the two variables. That is, we may transform ''X'' to {{math|''a'' + ''bX''}} and transform ''Y'' to {{math|''c'' + ''dY''}}, where ''a'', ''b'', ''c'', and ''d'' are constants with {{math|''b'', ''d'' > 0}}, without changing the correlation coefficient. (This holds for both the population and sample Pearson correlation coefficients.) More general linear transformations do change the correlation: see ''{{section link||Decorrelation of n random variables}}'' for an application of this. In particular, it might be useful to notice that corr(''-X'',''Y'')&nbsp;=&nbsp;-corr(''X'',''Y'')

==Interpretation==
The correlation coefficient ranges from −1 to 1. An absolute value of exactly 1 implies that a linear equation describes the relationship between ''X'' and ''Y'' perfectly, with all data points lying on a [[line (mathematics)|line]]. The correlation sign is determined by the [[regression slope]]: a value of +1 implies that all data points lie on a line for which ''Y'' increases as ''X'' increases, whereas a value of -1 implies a line where ''Y'' increases while ''X'' decreases.<ref name="STAT 462">{{cite web | title=2.6 - (Pearson) Correlation Coefficient r | website=STAT 462 | url=https://online.stat.psu.edu/stat462/node/96/ | access-date=2021-07-10}}</ref> A value of 0 implies that there is no linear dependency between the variables.<ref>{{Cite web|title=Introductory Business Statistics: The Correlation Coefficient r|url=https://opentextbc.ca/introbusinessstatopenstax/chapter/the-correlation-coefficient-r/|access-date=21 August 2020|website=opentextbc.ca}}</ref>

More generally, {{math|(''X''<sub>''i''</sub> − {{overline|''X''}})(''Y''<sub>''i''</sub> − {{overline|''Y''}})}} is positive if and only if ''X''<sub>''i''</sub> and ''Y''<sub>''i''</sub> lie on the same side of their respective means. Thus the correlation coefficient is positive if ''X''<sub>''i''</sub> and ''Y''<sub>''i''</sub> tend to be simultaneously greater than, or simultaneously less than, their respective means. The correlation coefficient is negative ([[anti-correlation]]) if ''X''<sub>''i''</sub> and ''Y''<sub>''i''</sub> tend to lie on opposite sides of their respective means. Moreover, the stronger either tendency is, the larger is the [[absolute value]] of the correlation coefficient.

Rodgers and Nicewander<ref>{{cite journal |author1=Rodgers |author2=Nicewander |year=1988  |title=Thirteen ways to look at the correlation coefficient |journal=The American Statistician |volume=42 |issue=1 |pages=59–66 |url=https://www.stat.berkeley.edu/~rabbee/correlation.pdf |doi=10.2307/2685263 |jstor=2685263}}</ref> cataloged thirteen ways of interpreting correlation or simple functions of it:
* Function of raw scores and means
* Standardized covariance
* Standardized slope of the regression line
* Geometric mean of the two regression slopes
* Square root of the ratio of two variances
* Mean cross-product of standardized variables
* Function of the angle between two standardized regression lines
* Function of the angle between two variable vectors
* Rescaled variance of the difference between standardized scores
* Estimated from the balloon rule
* Related to the bivariate ellipses of isoconcentration
* Function of test statistics from designed experiments
* Ratio of two means

===Geometric interpretation===
[[File:Regression lines.png|thumb|upright=1.5|Regression lines for {{math|1=''y'' = ''g''<sub>''X''</sub>(''x'')}} [{{color|red|red}}] and {{math|1=''x'' = ''g''<sub>''Y''</sub>(''y'')}} [{{color|blue|blue}}]]]

For uncentered data, there is a relation between the correlation coefficient and the angle ''φ'' between the two regression lines, {{nowrap|1=''y'' = ''g''<sub>''X''</sub>(''x'')}} and {{nowrap|1=''x'' = ''g''<sub>''Y''</sub>(''y'')}}, obtained by regressing ''y'' on ''x'' and ''x'' on ''y'' respectively. (Here, ''φ'' is measured counterclockwise within the first quadrant formed around the lines' intersection point if {{math|''r'' > 0}}, or counterclockwise from the fourth to the second quadrant if {{nowrap|''r'' < 0}}.) One can show<ref>{{cite journal |last=Schmid |first=John Jr. |title=The relationship between the coefficient of correlation and the angle included between regression lines |journal=The Journal of Educational Research |date=December 1947 |volume=41 |issue=4 |pages=311–313 |jstor=27528906 |doi=10.1080/00220671.1947.10881608}}</ref> that if the standard deviations are equal, then {{nowrap|1=''r'' = sec ''φ'' − tan ''φ''}}, where sec and tan are [[trigonometric functions]].

For centered data (i.e., data which have been shifted by the sample means of their respective variables so as to have an average of zero for each variable), the correlation coefficient can also be viewed as the [[cosine]] of the [[angle]] ''θ'' between the two observed [[Vector (geometry)|vectors]] in ''N''-dimensional space (for ''N'' observations of each variable).<ref>{{cite web |last=Rummel |first=R.J. |title=Understanding Correlation |year=1976 |url=http://www.hawaii.edu/powerkills/UC.HTM |at=ch. 5 (as illustrated for a special case in the next paragraph)}}</ref>

Both the uncentered (non-Pearson-compliant) and centered correlation coefficients can be determined for a dataset. As an example, suppose five countries are found to have gross national products of 1, 2, 3, 5, and 8 billion dollars, respectively. Suppose these same five countries (in the same order) are found to have 11%, 12%, 13%, 15%, and 18% poverty. Then let '''x''' and '''y''' be ordered 5-element vectors containing the above data: {{nowrap|1='''x''' = (1, 2, 3, 5, 8)}} and {{nowrap|1='''y''' = (0.11, 0.12, 0.13, 0.15, 0.18)}}.

By the usual procedure for finding the angle ''θ'' between two vectors (see [[dot product]]), the ''uncentered'' correlation coefficient is

:<math> \cos \theta = \frac { \mathbf{x} \cdot \mathbf{y} } { \left\| \mathbf{x} \right\| \left\| \mathbf{y} \right\|} = \frac {2.93} { \sqrt{103} \sqrt{0.0983} } = 0.920814711. </math>

This uncentered correlation coefficient is identical with the [[cosine similarity]]. The above data were deliberately chosen to be perfectly correlated: {{math|1=''y'' = 0.10 + 0.01 ''x''}}. The Pearson correlation coefficient must therefore be exactly one. Centering the data (shifting '''x''' by {{math|1=ℰ('''x''') = 3.8}} and '''y''' by {{math|1=ℰ('''y''') = 0.138}}) yields {{math|1='''x''' = (−2.8, −1.8, −0.8, 1.2, 4.2)}} and {{math|1='''y''' = (−0.028, −0.018, −0.008, 0.012, 0.042)}}, from which

:<math> \cos \theta = \frac{\mathbf{x} \cdot \mathbf{y}} {\left\| \mathbf{x} \right\| \left\| \mathbf{y} \right\|} = \frac {0.308}{\sqrt{30.8}\sqrt{0.00308}} = 1 = \rho_{xy}, </math>
as expected.

===Interpretation of the size of a correlation===
[[File:Pearson correlation and prediction intervals.svg|thumb|200px|right|This figure gives a sense of how the usefulness of a Pearson correlation for predicting values varies with its magnitude. Given jointly normal ''X'', ''Y'' with correlation ''ρ'', <math>1 - \sqrt{1 - \rho^2}</math> (plotted here as a function of ''ρ'') is the factor by which a given [[w:Prediction interval|prediction interval]] for ''Y'' may be reduced given the corresponding value of ''X''. For example, if ''ρ'' = 0.5, then the 95% prediction interval of ''Y''{{pipe}}''X'' will be about 13% smaller than the 95% prediction interval of ''Y''.]]

Several authors have offered guidelines for the interpretation of a correlation coefficient.<ref name="Buda">{{cite book |last1=Buda |first1=Andrzej |last2=Jarynowski |first2=Andrzej |title=Life Time of Correlations and its Applications |date=December 2010 |publisher=Wydawnictwo Niezależne |isbn=978-83-915272-9-0 |pages=5–21}}</ref><ref name="Cohen88"/> However, all such criteria are in some ways arbitrary.<ref name="Cohen88">{{cite book |last=Cohen |first=J. |year=1988 |title=Statistical Power Analysis for the Behavioral Sciences |edition=2nd}}</ref> The interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0.8 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences, where there may be a greater contribution from complicating factors.

==Inference==
Statistical inference based on Pearson's correlation coefficient often focuses on one of the following two aims:
* One aim is to test the [[null hypothesis]] that the true correlation coefficient ''ρ'' is equal to 0, based on the value of the sample correlation coefficient ''r''.
* The other aim is to derive a [[confidence interval]] that, on repeated sampling, has a given probability of containing ''ρ''.
Methods of achieving one or both of these aims are discussed below.

===Using a permutation test===
[[Permutation test]]s provide a direct approach to performing hypothesis tests and constructing confidence intervals. A permutation test for Pearson's correlation coefficient involves the following two steps: 
# Using the original paired data (''x''<sub>''i''</sub>,&nbsp;''y''<sub>''i''</sub>), randomly redefine the pairs to create a new data set (''x''<sub>''i''</sub>,&nbsp;''y''<sub>''{{prime|i}}''</sub>), where the ''{{prime|i}}'' are a [[permutation]] of the set {1,...,''n''}.  The permutation ''{{prime|i}}'' is selected randomly, with equal probabilities placed on all ''n''! possible permutations.  This is equivalent to drawing the ''{{prime|i}}'' randomly without replacement from the set {1, ..., ''n''}.  In [[Bootstrapping (statistics)|bootstrapping]], a closely related approach, the ''i'' and the ''{{prime|i}}'' are equal and drawn with replacement from {1, ..., ''n''};
# Construct a correlation coefficient ''r'' from the randomized data.
To perform the permutation test, repeat steps&nbsp;(1) and (2) a large number of times.  The [[p-value]] for the permutation test is the proportion of the ''r'' values generated in step&nbsp;(2) that are larger than the Pearson correlation coefficient that was calculated from the original data.  Here "larger" can mean either that the value is larger in magnitude, or larger in signed value, depending on whether a [[two-tailed test|two-sided]] or [[one-sided test|one-sided]] test is desired.

===Using a bootstrap===
The [[bootstrapping (statistics)|bootstrap]] can be used to construct confidence intervals for Pearson's correlation coefficient.  In the "non-parametric" bootstrap, ''n'' pairs (''x''<sub>''i''</sub>,&nbsp;''y''<sub>''i''</sub>) are resampled "with replacement" from the observed set of ''n'' pairs, and the correlation coefficient ''r'' is calculated based on the resampled data.  This process is repeated a large number of times, and the empirical distribution of the resampled ''r'' values are used to approximate the [[sampling distribution]] of the statistic.  A 95% [[confidence interval]] for ''ρ'' can be defined as the interval spanning from the 2.5th to the 97.5th [[percentile]] of the resampled ''r'' values.

=== Standard error ===
If <math>x</math> and <math>y</math> are random variables, with a simple linear relationship between them with an additive normal noise (i.e., y= a + bx + e), then a [[standard error]] associated to the correlation is

:<math>\sigma_r \approx \frac{1-r^2}{\sqrt{n}}</math>

where <math>r</math> is the correlation and <math>n</math> the sample size.<ref>{{Cite journal|last=Bowley|first=A. L.|date=1928|title=The Standard Deviation of the Correlation Coefficient|journal=Journal of the American Statistical Association|volume=23|issue=161|pages=31–34|doi=10.2307/2277400|jstor=2277400|issn=0162-1459}}</ref><ref>{{Cite web|title=Derivation of the standard error for Pearson's correlation coefficient|url=https://stats.stackexchange.com/q/226380 |access-date=2021-07-30|website=Cross Validated}}</ref>

===Testing using Student's ''t''-distribution===
[[File:Critical correlation vs. sample size.svg|thumb|324x324px|Critical values of Pearson's correlation coefficient that must be exceeded to be considered significantly nonzero at the 0.05 level]]For pairs from an uncorrelated [[bivariate normal distribution]], the [[sampling distribution]] of the [[studentized]] Pearson's correlation coefficient follows [[Student's t-distribution|Student's ''t''-distribution]] with degrees of freedom ''n''&nbsp;−&nbsp;2.  Specifically, if the underlying variables have a bivariate normal distribution, the variable

:<math>t = \frac{r}{\sigma_r} = r\sqrt{\frac{n-2}{1 - r^2}}</math>

has a student's ''t''-distribution in the null case (zero correlation).<ref>Rahman, N. A. (1968) ''A Course in Theoretical Statistics'', Charles Griffin and Company, 1968</ref> This holds approximately in case of non-normal observed values if sample sizes are large enough.<ref>Kendall, M. G., Stuart, A. (1973) ''The Advanced Theory of Statistics, Volume 2: Inference and Relationship'', Griffin. {{isbn|0-85264-215-6}} (Section 31.19)</ref> For determining the critical values for ''r'' the inverse function is needed:

:<math>r = \frac{t}{\sqrt{n - 2 + t^2}}.</math>

Alternatively, large sample, asymptotic approaches can be used.

Another early paper<ref>{{cite journal |last1=Soper |first1=H.E. |author-link=H. E. Soper |last2=Young |first2=A.W. |last3=Cave |first3=B.M. |last4=Lee |first4=A. |last5=Pearson |first5=K. |year=1917 |title=On the distribution of the correlation coefficient in small samples. Appendix II to the papers of "Student" and R.A. Fisher. A co-operative study |url=https://zenodo.org/record/1431587 |journal=[[Biometrika]] |volume=11 |issue=4 |pages=328–413 |doi=10.1093/biomet/11.4.328}}</ref> provides graphs and tables for general values of ''ρ'', for small sample sizes, and discusses computational approaches.

In the case where the underlying variables are not normal, the sampling distribution of Pearson's correlation coefficient follows a Student's ''t''-distribution, but the degrees of freedom are reduced.<ref>{{cite journal |last1=Davey |first1=Catherine E. |last2=Grayden |first2=David B. |last3=Egan |first3=Gary F. |last4=Johnston |first4=Leigh A. |title=Filtering induces correlation in fMRI resting state data |journal=NeuroImage |date=January 2013 |volume=64 |pages=728–740 |doi=10.1016/j.neuroimage.2012.08.022 |pmid=22939874 |hdl=11343/44035 |s2cid=207184701 |hdl-access=free }}</ref>

===Using the exact distribution===
For data that follow a [[bivariate normal distribution]], the exact density function ''f''(''r'') for the sample correlation coefficient  ''r'' of a normal bivariate is<ref>{{cite journal |last1=Hotelling |first1=Harold |title=New Light on the Correlation Coefficient and its Transforms |journal=Journal of the Royal Statistical Society |series=Series B (Methodological) |date=1953 |volume=15 |issue=2 |pages=193–232 |jstor=2983768 |doi=10.1111/j.2517-6161.1953.tb00135.x }}</ref><ref>{{cite book |author1=Kenney, J.F. |author2=Keeping, E.S. |title=Mathematics of Statistics |volume=Part 2 |edition=2nd |place=Princeton, NJ |publisher=Van Nostrand |year=1951}}</ref><ref>{{cite web |url=http://mathworld.wolfram.com/CorrelationCoefficientBivariateNormalDistribution.html |title=Correlation Coefficient—Bivariate Normal Distribution |first=Eric W. |last=Weisstein |website=Wolfram MathWorld}}</ref>

:<math>f(r) = \frac{(n - 2)\, \mathrm{\Gamma}(n - 1) \left(1 - \rho^2\right)^{\frac{n - 1}{2}} \left(1 - r^2\right)^{\frac{n - 4}{2}}}{\sqrt{2\pi}\, \operatorname{\Gamma}\mathord\left(n - \tfrac{1}{2}\right) (1 - \rho r)^{n - \frac{3}{2}}} {}_{2}\mathrm{F}_{1}\mathord\left(\tfrac{1}{2}, \tfrac{1}{2}; \tfrac{1}{2}(2n - 1); \tfrac{1}{2}(\rho r + 1)\right)</math>

where <math>\Gamma</math> is the [[gamma function]] and <math>{}_{2}\mathrm{F}_{1}(a,b;c;z)</math> is the [[hypergeometric function|Gaussian hypergeometric function]].

In the special case when <math>\rho = 0</math> (zero population correlation), the exact density function ''f''(''r'') can be written as

:<math>f(r) = \frac{\left( 1-r^2 \right)^{\frac{n - 4}{2}}}{\operatorname{\Beta}\mathord\left(\tfrac{1}{2}, \tfrac{n - 2}{2}\right)},</math>

where <math>\Beta</math> is the [[beta function]], which is one way of writing the density of a Student's t-distribution for a [[studentized]] sample correlation coefficient, as above.

===Using the Fisher transformation===
{{main|Fisher transformation}}

In practice, [[confidence intervals]] and [[hypothesis test]]s relating to ''ρ'' are usually carried out using the, [[Variance-stabilizing transformation]], [[Fisher transformation]], <math>F</math>'':

:<math>F(r) \equiv \tfrac{1}{2} \, \ln \left(\frac{1 + r}{1 - r}\right) = \operatorname{artanh}(r)</math>

''F''(''r'') approximately follows a [[normal distribution]] with

:<math>\text{mean} = F(\rho) = \operatorname{artanh}(\rho)</math>{{spaces|4}}and [[standard error]] <math>=\text{SE} = \frac{1}{\sqrt{n - 3}},</math>

where ''n'' is the sample size. The approximation error is lowest for a large sample size <math>n</math> and small <math>r</math> and <math>\rho_0</math> and increases otherwise.

Using the approximation, a [[standard score|z-score]] is

:<math>z = \frac{x - \text{mean}}{\text{SE}} = [F(r) - F(\rho_0)]\sqrt{n - 3}</math>

under the [[null hypothesis]] that <math>\rho = \rho_0</math>, given the assumption that the sample pairs are [[independent and identically distributed]] and follow a [[bivariate normal distribution]].  Thus an approximate [[p-value]] can be obtained from a normal probability table.  For example, if ''z''&nbsp;=&nbsp;2.2 is observed and a two-sided p-value is desired to test the null hypothesis that <math>\rho = 0</math>, the p-value is {{nowrap|1=2Φ(−2.2) = 0.028}}, where Φ is the standard normal [[cumulative distribution function]].

To obtain a confidence interval for ρ, we first compute a confidence interval for ''F''(''<math>\rho</math>''):

:<math>100(1 - \alpha)\%\text{CI}: \operatorname{artanh}(\rho) \in [\operatorname{artanh}(r) \pm z_{\alpha/2}\text{SE}]</math>

The inverse Fisher transformation brings the interval back to the correlation scale.

:<math>100(1 - \alpha)\%\text{CI}: \rho \in [\tanh(\operatorname{artanh}(r) - z_{\alpha/2}\text{SE}), \tanh(\operatorname{artanh}(r) + z_{\alpha/2}\text{SE})]</math>

For example, suppose we observe ''r''&nbsp;=&nbsp;0.7 with a sample size of ''n''=50, and we wish to obtain a 95% confidence interval for&nbsp;''ρ''. The transformed value is <math display="inline">\operatorname{arctanh} \left ( r \right ) = 0.8673</math>, so the confidence interval on the transformed scale is <math>0.8673 \pm \frac{1.96}{\sqrt{47}} </math>, or (0.5814,&nbsp;1.1532). Converting back to the correlation scale yields (0.5237,&nbsp;0.8188).

==In least squares regression analysis==
{{For|more general, non-linear dependency|Coefficient of determination#In a multiple linear model}}
The square of the sample correlation coefficient is typically denoted ''r''<sup>2</sup> and is a special case of the [[coefficient of determination]]. In this case, it estimates the fraction of the variance in ''Y'' that is explained by ''X'' in a [[simple linear regression]]. So if we have the observed dataset <math>Y_1, \dots , Y_n</math> and the fitted dataset <math>\hat Y_1, \dots , \hat Y_n</math> then as a starting point the total variation in the ''Y''<sub>''i''</sub> around their average value can be decomposed as follows

:<math>\sum_i (Y_i - \bar{Y})^2 = \sum_i (Y_i-\hat{Y}_i)^2 + \sum_i (\hat{Y}_i-\bar{Y})^2,</math>

where the <math>\hat{Y}_i</math> are the fitted values from the regression analysis.  This can be rearranged to give

:<math>1 = \frac{\sum_i (Y_i-\hat{Y}_i)^2}{\sum_i (Y_i - \bar{Y})^2} + \frac{\sum_i (\hat{Y}_i-\bar{Y})^2}{\sum_i (Y_i - \bar{Y})^2}.</math>

The two summands above are the fraction of variance in ''Y'' that is explained by ''X'' (right) and that is unexplained by ''X'' (left).

Next, we apply a property of [[least squares]] regression models, that the sample covariance between <math>\hat{Y}_i</math> and <math>Y_i-\hat{Y}_i</math> is zero.  Thus, the sample correlation coefficient between the observed and fitted response values in the regression can be written (calculation is under expectation, assumes Gaussian statistics)

:<math>
\begin{align}
r(Y,\hat{Y}) &= \frac{\sum_i(Y_i-\bar{Y})(\hat{Y}_i-\bar{Y})}{\sqrt{\sum_i(Y_i-\bar{Y})^2\cdot \sum_i(\hat{Y}_i-\bar{Y})^2}}\\[6pt]
&= \frac{\sum_i(Y_i-\hat{Y}_i+\hat{Y}_i-\bar{Y})(\hat{Y}_i-\bar{Y})}{\sqrt{\sum_i(Y_i-\bar{Y})^2\cdot \sum_i(\hat{Y}_i-\bar{Y})^2}}\\[6pt]
&= \frac{ \sum_i [(Y_i-\hat{Y}_i)(\hat{Y}_i-\bar{Y}) +(\hat{Y}_i-\bar{Y})^2 ]}{\sqrt{\sum_i(Y_i-\bar{Y})^2\cdot \sum_i(\hat{Y}_i-\bar{Y})^2}}\\[6pt]
&= \frac{ \sum_i (\hat{Y}_i-\bar{Y})^2 }{\sqrt{\sum_i(Y_i-\bar{Y})^2\cdot \sum_i(\hat{Y}_i-\bar{Y})^2}}\\[6pt]
&= \sqrt{\frac{\sum_i(\hat{Y}_i-\bar{Y})^2}{\sum_i(Y_i-\bar{Y})^2}}.
\end{align}
</math>

Thus

:<math>r(Y,\hat{Y})^2 = \frac{\sum_i(\hat{Y}_i-\bar{Y})^2}{\sum_i(Y_i-\bar{Y})^2}</math>

where <math>r(Y,\hat{Y})^2</math> is the proportion of variance in ''Y'' explained by a linear function of ''X''.

In the derivation above, the fact that
:<math>\sum_i (Y_i-\hat{Y}_i)(\hat{Y}_i-\bar{Y}) = 0</math>
can be proved by noticing that the partial derivatives of the [[residual sum of squares]] ({{math|RSS}}) over ''β''<sub>0</sub> and ''β''<sub>1</sub> are equal to 0 in the least squares model, where
:<math>\text{RSS} = \sum_i (Y_i - \hat{Y}_i)^2</math>.

In the end, the equation can be written as

:<math>r(Y,\hat{Y})^2 = \frac{\text{SS}_\text{reg}}{\text{SS}_\text{tot}}</math>
where
*<math>\text{SS}_\text{reg} = \sum_i (\hat{Y}_i-\bar{Y})^2</math>
*<math>\text{SS}_\text{tot} = \sum_i (Y_i-\bar{Y})^2</math>.
The symbol <math>\text{SS}_\text{reg}</math> is called the regression sum of squares, also called the [[explained sum of squares]], and <math>\text{SS}_\text{tot}</math> is the [[total sum of squares]] (proportional to the [[variance]] of the data).

==Sensitivity to the data distribution==
{{Further|Correlation and dependence#Sensitivity to the data distribution}}

===Existence===
The population Pearson correlation coefficient is defined in terms of [[moment (mathematics)|moments]], and therefore exists for any bivariate [[probability distribution]] for which the [[statistical population|population]] [[covariance]] is defined and the [[marginal distribution|marginal]] [[population variance]]s are defined and are non-zero. Some probability distributions, such as the [[Cauchy distribution]], have undefined variance and hence ρ is not defined if ''X'' or ''Y'' follows such a distribution. In some practical applications, such as those involving data suspected to follow a [[heavy-tailed distribution]], this is an important consideration.  However, the existence of the correlation coefficient is usually not a concern; for instance, if the range of the distribution is bounded, ρ is always defined.

===Sample size===
*If the sample size is moderate or large and the population is normal, then, in the case of the bivariate [[normal distribution]], the sample correlation coefficient is the [[maximum likelihood estimate]] of the population correlation coefficient, and is [[asymptotic distribution|asymptotically]] [[bias of an estimator|unbiased]] and [[efficiency (statistics)|efficient]], which roughly means that it is impossible to construct a more accurate estimate than the sample correlation coefficient. 
*If the sample size is large and the population is not normal, then the sample correlation coefficient remains approximately unbiased, but may not be efficient.
*If the sample size is large, then the sample correlation coefficient is a [[consistent estimator]] of the population correlation coefficient as long as the sample means, variances, and covariance are consistent (which is guaranteed when the [[law of large numbers]] can be applied).
*If the sample size is small, then the sample correlation coefficient ''r'' is not an unbiased estimate of ''ρ''.<ref name="RealCorBasic"/> The adjusted correlation coefficient must be used instead: see elsewhere in this article for the definition.
*Correlations can be different for imbalanced [[dichotomous variable|dichotomous]] data when there is variance error in sample.<ref>{{cite journal |last1=Lai |first1=Chun Sing |last2=Tao |first2=Yingshan |last3=Xu |first3=Fangyuan |last4=Ng |first4=Wing W.Y. |last5=Jia |first5=Youwei |last6=Yuan |first6=Haoliang |last7=Huang |first7=Chao |last8=Lai |first8=Loi Lei |last9=Xu |first9=Zhao |last10=Locatelli |first10=Giorgio |title=A robust correlation analysis framework for imbalanced and dichotomous data with uncertainty |journal=Information Sciences |date=January 2019 |volume=470 |pages=58–77 |doi=10.1016/j.ins.2018.08.017 |s2cid=52878443 |url=http://eprints.whiterose.ac.uk/134706/2/ELSEVI_3.pdf }}</ref>

===Robustness===
Like many commonly used statistics, the sample [[statistic]] ''r'' is not [[robust statistics|robust]],<ref name="wilcox">{{Cite book| title=Introduction to robust estimation and hypothesis testing | last = Wilcox | first = Rand R. | publisher= Academic Press | year=2005}}</ref> so its value can be misleading if [[outlier]]s are present.<ref>{{Cite journal |title=Robust estimation and outlier detection with correlation coefficients |author1=Devlin, Susan J. |author1-link=Susan J. Devlin |author2=Gnanadesikan, R. |author3=Kettenring J.R. |journal=Biometrika |volume=62 |issue=3 |year=1975 |pages=531–545 |doi=10.1093/biomet/62.3.531 |jstor=2335508}}</ref><ref>{{Cite book| title=Robust Statistics | last = Huber | first = Peter. J.| publisher= Wiley | year=2004}}{{Page needed|date=September 2010}}</ref> Specifically, the PMCC is neither distributionally robust,<ref>{{Cite book |last=Vaart |first=A. W. van der |title=Asymptotic Statistics |date=1998-10-13 |publisher=Cambridge University Press |doi=10.1017/cbo9780511802256 |isbn=978-0-511-80225-6}}</ref> nor outlier resistant<ref name="wilcox"/> (see ''{{section link|Robust statistics#Definition}}''). Inspection of the [[scatterplot]] between ''X'' and ''Y'' will typically reveal a situation where lack of robustness might be an issue, and in such cases it may be advisable to use a robust measure of association. Note however that while most robust estimators of association measure [[statistical dependence]] in some way, they are generally not interpretable on the same scale as the Pearson correlation coefficient.

Statistical inference for Pearson's correlation coefficient is sensitive to the data distribution.  Exact tests, and asymptotic tests based on the [[Fisher transformation]] can be applied if the data are approximately normally distributed, but may be misleading otherwise.  In some situations, the [[bootstrapping (statistics)|bootstrap]] can be applied to construct confidence intervals, and [[permutation test]]s can be applied to carry out hypothesis tests.  These [[non-parametric statistics|non-parametric]] approaches may give more meaningful results in some situations where bivariate normality does not hold.  However the standard versions of these approaches rely on [[exchangeable random variables|exchangeability]] of the data, meaning that there is no ordering or grouping of the data pairs being analyzed that might affect the behavior of the correlation estimate.

A stratified analysis is one way to either accommodate a lack of bivariate normality, or to isolate the correlation resulting from one factor while controlling for another.  If ''W'' represents cluster membership or another factor that it is desirable to control, we can [[Stratified sampling|stratify]] the data based on the value of ''W'', then calculate a correlation coefficient within each stratum.  The stratum-level estimates can then be combined to estimate the overall correlation while controlling for ''W''.<ref>Katz., Mitchell H. (2006) ''Multivariable Analysis – A Practical Guide for Clinicians''. 2nd Edition.  Cambridge University Press. {{isbn|978-0-521-54985-1}}. {{isbn|0-521-54985-X}}</ref>

==Variants==
{{See also|Correlation and dependence#Other measures of dependence among random variables}}

Variations of the correlation coefficient can be calculated for different purposes. Here are some examples.

===Adjusted correlation coefficient===
The sample correlation coefficient {{mvar|r}} is not an unbiased estimate of {{mvar|ρ}}. For data that follows a [[bivariate normal distribution]], the expectation {{math|E[''r'']}} for the sample correlation coefficient  {{mvar|r}} of a normal bivariate is<ref>{{Cite journal | first = H. | last = Hotelling | year = 1953 | title = New Light on the Correlation Coefficient and its Transforms | journal = Journal of the Royal Statistical Society. Series B (Methodological) | volume = 15 | issue = 2 | pages = 193–232 | jstor = 2983768| doi = 10.1111/j.2517-6161.1953.tb00135.x }}</ref>

:<math>\operatorname\mathbb{E}\left[r\right] = \rho - \frac{\rho \left(1 - \rho^2\right)}{2n} + \cdots, \quad</math> therefore {{mvar|r}} is a biased estimator of <math>\rho.</math>

The unique minimum variance unbiased estimator {{math|''r''<sub>adj</sub>}} is given by<ref>{{Cite journal | first=Ingram | last=Olkin |author2=Pratt, John W.  |date=March 1958  | title=Unbiased Estimation of Certain Correlation Coefficients | journal=The Annals of Mathematical Statistics | volume=29| issue=1 | pages=201–211 | jstor=2237306 | doi=10.1214/aoms/1177706717| doi-access=free }}.</ref>

{{NumBlk|:|<math> r_\text{adj} = r \, \mathbf{_2F_1}\left(\frac{1}{2}, \frac{1}{2}; \frac{n - 1}{2}; 1 - r^2\right),</math>|{{EquationRef|1}}}}

where:
*<math>r, n</math> are defined as above,
*<math>\mathbf{_2 F_1}(a, b; c; z)</math> is the [[hypergeometric function|Gaussian hypergeometric function]].

An approximately unbiased estimator {{math|''r''<sub>adj</sub>}} can be obtained{{citation needed|date=April 2012}} by truncating {{math|E[''r'']}} and solving this truncated equation:

{{NumBlk|:|<math> r = \operatorname\mathbb{E}[r] \approx r_\text{adj} - \frac{r_\text{adj} \left(1 - r_\text{adj}^2\right)}{2n}.</math>|{{EquationRef|2}}}}

An approximate solution{{citation needed|date=April 2012}} to equation ({{EquationNote|2}}) is

{{NumBlk|:|<math> r_\text{adj} \approx r \left[1 + \frac{1 - r^2}{2n}\right],</math>|{{EquationRef|3}}}}

where in ({{EquationNote|3}})
*<math>r, n</math> are defined as above,
*{{math|''r''<sub>adj</sub>}} is a suboptimal estimator,{{citation needed|date=April 2012}}{{clarify|date=February 2015| reason=suboptimal in what sense?}}
*{{math|''r''<sub>adj</sub>}} can also be obtained by maximizing log(''f''(''r'')),
*{{math|''r''<sub>adj</sub>}} has minimum variance for large values of {{mvar|n}},
*{{math|''r''<sub>adj</sub>}} has a bias of order {{math|{{frac|1|(''n'' − 1)}}}}.

Another proposed<ref name="RealCorBasic"/> adjusted correlation coefficient is{{citation needed|date=February 2015|reason=is this in a published article?}}

:<math>r_\text{adj}=\sqrt{1-\frac{(1-r^2)(n-1)}{(n-2)}}.</math>

{{math|''r''<sub>adj</sub> ≈ ''r''}} for large values of&nbsp;{{mvar|n}}.

===Weighted correlation coefficient===
Suppose observations to be correlated have differing degrees of importance that can be expressed with a weight vector ''w''. To calculate the correlation between vectors ''x'' and ''y'' with the weight vector ''w'' (all of length&nbsp;''n''),<ref>{{cite web|url=http://sci.tech-archive.net/Archive/sci.stat.math/2006-02/msg00171.html|title=Re: Compute a weighted correlation|website=sci.tech-archive.net}}</ref><ref>{{cite web|url=http://www.mathworks.com/matlabcentral/fileexchange/20846-weighted-correlation-matrix|title=Weighted Correlation Matrix – File Exchange – MATLAB Central|access-date=18 November 2017|archive-date=15 May 2021|archive-url=https://web.archive.org/web/20210515121719/https://www.mathworks.com/matlabcentral/fileexchange/20846-weighted-correlation-matrix}}</ref>

* Weighted mean: <math display="block">\operatorname{m}(x; w) = \frac{\sum_i w_i x_i}{\sum_i w_i}.</math>
* Weighted covariance <math display="block">\operatorname{cov}(x,y;w) = \frac{\sum_i w_i \cdot (x_i - \operatorname{m}(x; w)) (y_i - \operatorname{m}(y; w))}{\sum_i w_i }.</math>
* Weighted correlation <math display="block">\operatorname{corr}(x,y;w) = \frac{\operatorname{cov}(x,y;w)}{\sqrt{\operatorname{cov}(x,x;w) \operatorname{cov}(y,y;w)}}.</math>

===Reflective correlation coefficient===
The reflective correlation is a variant of Pearson's correlation in which the data are not centered around their mean values.{{citation needed|date=January 2011}} The population reflective correlation is

:<math>\operatorname{corr}_r(X,Y) = \frac{\operatorname\mathbb{E}[\,X\,Y\,]}{\sqrt{\operatorname\mathbb{E}[\,X^2\,]\cdot \operatorname\mathbb{E}[\,Y^2\,]}}.</math>

The reflective correlation is symmetric, but it is not invariant under translation:

:<math>\operatorname{corr}_r(X, Y) = \operatorname{corr}_r(Y, X) = \operatorname{corr}_r(X, bY) \neq \operatorname{corr}_r(X, a + b Y), \quad a \neq 0, b > 0.</math>

The sample reflective correlation is equivalent to [[cosine similarity]]:

:<math>rr_{xy} = \frac{\sum x_i y_i}{\sqrt{(\sum x_i^2)(\sum y_i^2)}}.</math>

The weighted version of the sample reflective correlation is

:<math>rr_{xy, w} = \frac{\sum w_i x_i y_i}{\sqrt{(\sum w_i x_i^2)(\sum w_i y_i^2)}}.</math>

===Scaled correlation coefficient===
{{Main|Scaled correlation}}

Scaled correlation is a variant of Pearson's correlation in which the range of the data is restricted intentionally and in a controlled manner to reveal correlations between fast components in [[time series]].<ref name = "Nikolicetal">{{cite journal | last1 = Nikolić | first1 = D | last2 = Muresan | first2 = RC | last3 = Feng | first3 = W | last4 = Singer | first4 = W | year = 2012 | title = Scaled correlation analysis: a better way to compute a cross-correlogram | url = http://www.danko-nikolic.com/wp-content/uploads/2012/03/Scaled-correlation-analysis.pdf | journal = European Journal of Neuroscience | volume =  35| issue = 5| pages = 1–21 | doi = 10.1111/j.1460-9568.2011.07987.x | pmid = 22324876 | s2cid = 4694570 }}</ref> Scaled correlation is defined as average correlation across short segments of data.

Let <math>K</math> be the number of segments that can fit into the total length of the signal <math>T</math> for a given scale <math>s</math>:

:<math>K = \operatorname{round}\left(\frac{T}{s}\right).</math>

The scaled correlation across the entire signals <math>\bar{r}_s</math> is then computed as

:<math>\bar{r}_s = \frac{1}{K} \sum\limits_{k=1}^K r_k,</math>

where <math>r_k</math> is Pearson's coefficient of correlation for segment <math>k</math>.

By choosing the parameter <math>s</math>, the range of values is reduced and the correlations on long time scale are filtered out, only the correlations on short time scales being revealed. Thus, the contributions of slow components are removed and those of fast components are retained.

===Pearson's distance===
A distance metric for two variables ''X'' and ''Y'' known as ''Pearson's distance'' can be defined from their correlation coefficient as<ref>Fulekar (Ed.), M.H. (2009) ''Bioinformatics: Applications in Life and Environmental Sciences'', Springer (pp. 110) {{isbn|1-4020-8879-5}}</ref>
:<math>d_{X,Y}=1-\rho_{X,Y}.</math>
Considering that the Pearson correlation coefficient falls between [−1, +1], the Pearson distance lies in [0, 2]. The Pearson distance has been used in [[cluster analysis]] and data detection for communications and storage with unknown gain and offset.<ref>{{cite journal
 |author1=Immink, K. Schouhamer
 |author2=Weber, J. 
 |title=Minimum Pearson distance detection for multilevel channels with gain and / or offset mismatch 
 |date=October 2010
 |journal=IEEE Transactions on Information Theory 
 |volume=60 |issue=10 |pages=5966–5974
 |doi=10.1109/tit.2014.2342744 |citeseerx=10.1.1.642.9971
 |s2cid=1027502 
 |url=https://www.researchgate.net/publication/265604603
 |access-date=11 February 2018}}</ref>

The Pearson "distance" defined this way assigns distance greater than 1 to negative correlations. In reality, both strong positive correlation and negative correlations are meaningful, so care must be taken when Pearson "distance" is used for nearest neighbor algorithm as such algorithm will only include neighbors with positive correlation and exclude neighbors with negative correlation. Alternatively, an absolute valued distance, <math>d_{X,Y}=1-|\rho_{X,Y}|</math>, can be applied, which will take both positive and negative correlations into consideration. The information on positive and negative association can be extracted separately, later.

===Circular correlation coefficient{{anchor|Circular}}===
{{further|Circular statistics}}

For variables ''X'' = {''x''<sub>1</sub>,...,''x''<sub>''n''</sub>} and ''Y'' = {''y''<sub>1</sub>,...,''y''<sub>''n''</sub>} that are defined on the unit circle {{Not a typo|{{closed-open|0, 2π}}}}, it is possible to define a circular analog of Pearson's coefficient.<ref name="SRJ">{{cite book |title=Topics in circular statistics |last1=Jammalamadaka |first1=S. Rao |last2=SenGupta |first2=A. |year=2001 |publisher=World Scientific |location=New Jersey |isbn=978-981-02-3778-3 |page=176 |url=https://books.google.com/books?id=sKqWMGqQXQkC&q=Jammalamadaka+Topics+in+circular |access-date=21 September 2016}}</ref> This is done by transforming data points in ''X'' and ''Y'' with a [[sine]] function such that the correlation coefficient is given as:

:<math>r_\text{circular} = \frac{\sum ^n _{i=1}\sin(x_i - \bar{x}) \sin(y_i - \bar{y})}{\sqrt{\sum^n_{i=1} \sin(x_i - \bar{x})^2} \sqrt{\sum ^n_{i=1} \sin(y_i - \bar{y})^2}}</math>

where <math>\bar{x}</math> and <math>\bar{y}</math> are the [[Mean of circular quantities|circular means]] of ''X'' and&nbsp;''Y''. This measure can be useful in fields like meteorology where the angular direction of data is important.

===Partial correlation===
{{Main|Partial correlation}}
If a population or data-set is characterized by more than two variables, a [[partial correlation]] coefficient measures the strength of dependence between a pair of variables that is not accounted for by the way in which they both change in response to variations in a selected subset of the other variables.

===Pearson correlation coefficient in quantum systems===
For two observables, <math>X</math> and <math>Y</math>, in a bipartite quantum system Pearson correlation coefficient is defined as <ref>{{cite journal |first=M. D. |last=Reid |date=1 July 1989 |title=Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification |journal=Physical Review A |volume=40 |issue=2 |pages=913–923 |doi=10.1103/PhysRevA.40.913 |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.40.913}}</ref><ref>{{cite journal |author1=Maccone, L. |author2= Dagmar, B. |author3= Macchiavello, C. |date=1 April 2015 |title=Complementarity and Correlations |journal=Physical Review Letters |volume=114 |issue=13 |article-number=130401 |doi= 10.1103/PhysRevLett.114.130401 |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.130401|arxiv=1408.6851 }}</ref>

:<math>\mathbb{Cor}(X,Y) = \frac{\mathbb{E}[X \otimes Y] - \mathbb{E}[X] \cdot \mathbb{E}[Y]}{\sqrt{\mathbb{V}[X] \cdot \mathbb{V}[Y]}} \,,</math>

where
*<math> \mathbb{E}[X] </math> is the expectation value of the observable <math> X </math>,
*<math> \mathbb{E}[Y] </math> is the expectation value of the observable <math> Y </math>,
*<math> \mathbb{E}[X \otimes Y] </math> is the expectation value of the observable <math> X \otimes Y </math>,
*<math> \mathbb{V}[X] </math> is the variance of the observable <math> X </math>, and
*<math> \mathbb{V}[Y] </math> is the variance of the observable <math> Y </math>.

<math>\mathbb{Cor}(X,Y)</math> is symmetric, i.e., <math>\mathbb{Cor}(X,Y)= \mathbb{Cor}(Y, X)</math>, and its absolute value is invariant under affine transformations.

==Decorrelation of ''n'' random variables==
{{main|Decorrelation}}

It is always possible to remove the correlations between all pairs of an arbitrary number of random variables by using a data transformation, even if the relationship between the variables is nonlinear. A presentation of this result for population distributions is given by Cox & Hinkley.<ref>{{cite book |author1=Cox, D.R. |author2=Hinkley, D.V. |year=1974 |title=Theoretical Statistics |publisher=Chapman & Hall |at=Appendix 3 |isbn=0-412-12420-3}}</ref>

A corresponding result exists for reducing the sample correlations to zero. Suppose a vector of ''n'' random variables is observed ''m'' times. Let ''X'' be a matrix where <math>X_{i,j}</math> is the ''j''th variable of observation ''i''. Let <math>Z_{m,m}</math> be an ''m'' by ''m'' square matrix with every element 1. Then ''D'' is the data transformed so every random variable has zero mean, and ''T'' is the data transformed so all variables have zero mean and zero correlation with all other variables – the sample [[correlation matrix]] of ''T'' will be the identity matrix. This has to be further divided by the standard deviation to get unit variance. The transformed variables will be uncorrelated, even though they may not be [[Statistical independence|independent]].

:<math>D = X -\frac{1}{m} Z_{m,m} X</math>
:<math>T = D (D^{\mathsf{T}} D)^{-\frac{1}{2}},</math>

where an exponent of {{frac|−|1|2}} represents the [[matrix square root]] of the [[matrix inverse|inverse]] of a matrix.  The correlation matrix of ''T'' will be the identity matrix.  If a new data observation ''x'' is a row vector of ''n'' elements, then the same transform can be applied to ''x'' to get the transformed vectors ''d'' and ''t'':

:<math>d = x - \frac{1}{m} Z_{1,m} X,</math>
:<math>t = d (D^{\mathsf{T}} D)^{-\frac{1}{2}}.</math>
This decorrelation is related to [[principal components analysis]] for multivariate data.

==Software implementations==
* [[R (programming language)|R]]'s statistics base-package implements the correlation coefficient with <code>cor(x, y)</code>, or (with the p-value also) with [http://stat.ethz.ch/R-manual/R-patched/library/stats/html/cor.test.html <code>cor.test(x, y)</code>].
* The [[SciPy]] [[Python (programming language)|Python]] library via [https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.pearsonr.html <code>pearsonr(x, y)</code>].
* The [[Pandas (software)|Pandas]] and Polars Python libraries implement the Pearson correlation coefficient calculation as the default option for the methods [https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.corr.html <code>pandas.DataFrame.corr</code>] and [https://docs.pola.rs/api/python/stable/reference/expressions/api/polars.corr.html <code>polars.corr</code>], respectively.
* [[Wolfram Mathematica]] via the [https://reference.wolfram.com/language/ref/Correlation.html <code>Correlation</code>] function, or (with the p-value) with [https://reference.wolfram.com/language/ref/CorrelationTest.html <code>CorrelationTest</code>].
* The [[Boost (C++ libraries)|Boost]] [[C++]] library via the [https://www.boost.org/doc/libs/1_76_0/libs/math/doc/html/math_toolkit/bivariate_statistics.html <code>correlation_coefficient</code>] function.
* [[Excel]] has an in-built [https://support.microsoft.com/en-us/office/correl-function-995dcef7-0c0a-4bed-a3fb-239d7b68ca92 <code>correl(array1, array2)</code>] function for calculating the Pearson's correlation coefficient.

==See also==
{{Portal|Mathematics}}
* [[Anscombe's quartet]]
* [[Association (statistics)]]
* [[Coefficient of colligation]]
** [[Yule's Q]]
** [[Yule's Y]]
* [[Coefficient of multiple correlation]]
* [[Concordance correlation coefficient]]
* [[Correlation and dependence]]
* [[Correlation ratio]]
* [[Disattenuation]]
* [[Distance correlation]]
* [[Maximal information coefficient]]
* [[Multiple correlation]]
* [[Normally distributed and uncorrelated does not imply independent]]
* [[Odds ratio]]
* [[Partial correlation]]
* [[Polychoric correlation]]
* [[Quadrant count ratio]]
* [[RV coefficient]]
* [[Spearman's rank correlation coefficient]]
* [[Kendall rank correlation coefficient]]

==Footnotes==
{{Notelist|1}}

==References==
{{Reflist}}

==External links==
{{Wikiversity|Linear correlation}}
* {{cite web |url=http://comparingcorrelations.org |title=cocor |website=comparingcorrelations.org}} – A free web interface and R package for the statistical comparison of two dependent or independent correlations with overlapping or non-overlapping variables.
* {{cite web |url=http://nagysandor.eu/AsimovTeka/correlation_en/index.html |title=Correlation |language=en |website=nagysandor.eu |access-date=30 January 2013 |archive-date=17 May 2021 |archive-url=https://web.archive.org/web/20210517153242/http://nagysandor.eu/AsimovTeka/correlation_en/index.html }} – an interactive Flash simulation on the correlation of two normally distributed variables.
* {{cite web |url=https://www.hackmath.net/en/calculator/linear-regression |title=Correlation coefficient calculator |series=Linear regression |website=hackmath.net}}
* {{cite web |url=http://frank.mtsu.edu/~dkfuller/tables/correlationtable.pdf |title=Critical values for Pearson's correlation coefficient |website=frank.mtsu.edu/~dkfuller}} – large table.
* {{cite web |url=http://guessthecorrelation.com |title=Guess the Correlation }} –  A game where players guess how correlated two variables in a scatter plot are, in order to gain a better understanding of the concept of correlation.

{{Statistics|descriptive}}
{{Machine learning evaluation metrics}}
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{{DEFAULTSORT:Pearson product-moment correlation coefficient}}
[[Category:Correlation indicators]]
[[Category:Parametric statistics]]
[[Category:Statistical ratios]]