# Pearson product-moment correlation coefficient

In statistics, the Pearson product-moment correlation coefficient (Template:IPAc-en) (sometimes referred to as the PPMCC or PCC or Pearson's r) is a measure of the linear correlation (dependence) between two variables X and Y, giving a value between +1 and −1 inclusive, where 1 is total positive correlation, 0 is no correlation, and −1 is total negative correlation. It is widely used in the sciences as a measure of the degree of linear dependence between two variables. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s.[1][2][3]

Examples of scatter diagrams with different values of correlation coefficient (ρ)
Several sets of (xy) points, with the correlation coefficient of x and y for each set. Note that the correlation reflects the non-linearity and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero.

## Definition

Pearson's correlation coefficient between two variables is defined as 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 about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.

### For a population

Pearson's correlation coefficient when applied to a 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. The formula for ρ is:

${\displaystyle \rho _{X,Y}={\mathrm {cov} (X,Y) \over \sigma _{X}\sigma _{Y}}={E[(X-\mu _{X})(Y-\mu _{Y})] \over \sigma _{X}\sigma _{Y}}}$

### For a sample

Pearson's correlation coefficient when applied to a sample is commonly represented by the letter r and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient. We can obtain a formula for r by substituting estimates of the covariances and variances based on a sample into the formula above. That formula for r is:

${\displaystyle r={\frac {\sum _{i=1}^{n}(X_{i}-{\bar {X}})(Y_{i}-{\bar {Y}})}{{\sqrt {\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}}}{\sqrt {\sum _{i=1}^{n}(Y_{i}-{\bar {Y}})^{2}}}}}}$

An equivalent expression gives the correlation coefficient as the mean of the products of the standard scores. Based on a sample of paired data (XiYi), the sample Pearson correlation coefficient is

${\displaystyle r={\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {X_{i}-{\bar {X}}}{s_{X}}}\right)\left({\frac {Y_{i}-{\bar {Y}}}{s_{Y}}}\right)}$

where

${\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i},{\text{ and }}s_{X}={\sqrt {{\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}}}}$

are the sample mean and sample standard deviation, respectively. Thus, the first parenthesized term in the previous summation is the standard score. (The terms for Y are similar.)

## Mathematical properties

The absolute values of both the sample and population Pearson correlation coefficients are less than or equal to 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 supported on a line (in the case of the population correlation). The Pearson correlation coefficient is symmetric: corr(X,Y) = corr(Y,X).

A key mathematical property of the Pearson correlation coefficient is that it is invariant to separate changes in location and scale in the two variables. That is, we may transform X to a + bX and transform Y to c + dY, where a, b, c, and d are constants with b, d > 0, without changing the correlation coefficient. (This fact holds for both the population and sample Pearson correlation coefficients.) Note that more general linear transformations do change the correlation: see a later section for an application of this.

The Pearson correlation can be expressed in terms of uncentered moments. Since μX = E(X), σX2 = E[(X − E(X))2] = E(X2) − E2(X) and likewise for Y, and since

${\displaystyle E[(X-E(X))(Y-E(Y))]=E(XY)-E(X)E(Y),\,}$

the correlation can also be written as

${\displaystyle \rho _{X,Y}={\frac {E(XY)-E(X)E(Y)}{{\sqrt {E(X^{2})-E(X)^{2}}}~{\sqrt {E(Y^{2})-E(Y)^{2}}}}}.}$

Alternative formulae for the sample Pearson correlation coefficient are also available:

${\displaystyle r_{xy}={\frac {\sum x_{i}y_{i}-n{\bar {x}}{\bar {y}}}{(n-1)s_{x}s_{y}}}={\frac {n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{{\sqrt {n\sum x_{i}^{2}-(\sum x_{i})^{2}}}~{\sqrt {n\sum y_{i}^{2}-(\sum y_{i})^{2}}}}}.}$

The above formula suggests a convenient single-pass algorithm for calculating sample correlations, but, depending on the numbers involved, it can sometimes be numerically unstable.

## Interpretation

The correlation coefficient ranges from −1 to 1. A value of 1 implies that a linear equation describes the relationship between X and Y perfectly, with all data points lying on a line for which Y increases as X increases. A value of −1 implies that all data points lie on a line for which Y decreases as X increases. A value of 0 implies that there is no linear correlation between the variables.

More generally, note that (Xi − X)(Yi − Y) is positive if and only if Xi and Yi lie on the same side of their respective means. Thus the correlation coefficient is positive if Xi and Yi tend to be simultaneously greater than, or simultaneously less than, their respective means. The correlation coefficient is negative if Xi and Yi tend to lie on opposite sides of their respective means.

### Geometric interpretation

Regression lines for y=gx(x) [red] and x=gy(y) [blue]

For uncentered data, it is possible to obtain a relation between correlation coefficient and the angle ${\displaystyle \varphi }$ between both possible regression lines y=gx(x) and x=gy(y). One can show[4] that r = sec(${\displaystyle \ \varphi }$) -tan(${\displaystyle \ \varphi }$).

For centered data (i.e., data which have been shifted by the sample mean so as to have an average of zero), the correlation coefficient can also be viewed as the cosine of the angle ${\displaystyle \ \theta }$ between the two vectors of samples drawn from the two random variables (see below).

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: x = (1, 2, 3, 5, 8) and y = (0.11, 0.12, 0.13, 0.15, 0.18).

By the usual procedure for finding the angle ${\displaystyle \ \theta }$ between two vectors (see dot product), the uncentered correlation coefficient is:

${\displaystyle \cos \theta ={\frac {{\mathbf {x}}\cdot {\mathbf {y}}}{\left\|{\mathbf {x}}\right\|\left\|{\mathbf {y}}\right\|}}={\frac {2.93}{{\sqrt {103}}{\sqrt {0.0983}}}}=0.920814711.}$

Note that the above data were deliberately chosen to be perfectly correlated: y = 0.10 + 0.01 x. The Pearson correlation coefficient must therefore be exactly one. Centering the data (shifting x by E(x) = 3.8 and y by E(y) = 0.138) yields x = (−2.8, −1.8, −0.8, 1.2, 4.2) and y = (−0.028, −0.018, −0.008, 0.012, 0.042), from which

${\displaystyle \cos \theta ={\frac {{\mathbf {x}}\cdot {\mathbf {y}}}{\left\|{\mathbf {x}}\right\|\left\|{\mathbf {y}}\right\|}}={\frac {0.308}{{\sqrt {30.8}}{\sqrt {0.00308}}}}=1=\rho _{xy},}$

as expected.

### Interpretation of the size of a correlation

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, ${\displaystyle 1-{\sqrt {1-\rho ^{2}}}}$ (plotted here as a function of ρ) is the factor by which a given prediction interval for Y may be reduced given the corresponding value of X. For example, if ρ = .5, then the 95% prediction interval of Y|X will be about 13% smaller than the 95% prediction interval of Y.

Several authors[5][6] have offered guidelines for the interpretation of a correlation coefficient. However, all such criteria are in some ways arbitrary and should not be observed too strictly.[6] 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.

### 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[7]

${\displaystyle d_{X,Y}=1-\rho _{X,Y}.}$

Considering that the Pearson correlation coefficient falls between [−1, 1], the Pearson distance lies in [0, 2].

## Inference

A graph showing the minimum value of Pearson's correlation coefficient that is significantly different from zero at the 0.05 level, for a given sample size.

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 construct a confidence interval around r that has a given probability of containing ρ.

We discuss methods of achieving one or both of these aims below.

### Use a permutation test

Permutation tests 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:

1. Using the original paired data (xiyi), randomly redefine the pairs to create a new data set (xiyi′), where the i′ are a permutation of the set {1,...,n}. The permutation i′ is selected randomly, with equal probabilities placed on all n! possible permutations. This is equivalent to drawing the i′ randomly "without replacement" from the set {1, ..., n}. A closely related and equally justified (bootstrapping) approach is to separately draw the i and the i′ "with replacement" from {1, ..., n};
2. Construct a correlation coefficient r from the randomized data.

To perform the permutation test, repeat steps (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 (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-sided or one-sided test is desired.

### Use a bootstrap

The bootstrap can be used to construct confidence intervals for Pearson's correlation coefficient. In the "non-parametric" bootstrap, n pairs (xiyi) 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.

### Testing using Student's t-distribution

For pairs from an uncorrelated bivariate normal distribution, the sampling distribution of Pearson's correlation coefficient follows Student's t-distribution with degrees of freedom n − 2. Specifically, if the underlying variables have a bivariate normal distribution, the variable

${\displaystyle t=r{\sqrt {\frac {n-2}{1-r^{2}}}}}$

has a Student's t-distribution in the null case (zero correlation).[8] This also holds approximately even if the observed values are non-normal, provided sample sizes are not very small.[9] For determining the critical values for r the inverse of this transformation is also needed:

${\displaystyle r={\frac {t}{\sqrt {n-2+t^{2}}}}.}$

Alternatively, large sample approaches can be used.

Early work on the distribution of the sample correlation coefficient was carried out by R. A. Fisher[10][11] and A. K. Gayen.[12] Another early paper[13] provides graphs and tables for general values of ρ, for small sample sizes, and discusses computational approaches.

### Use the exact distribution

For data that follows a bivariate normal distribution, the exact density function for the sample correlation of a normal bivariate is[14][15]

${\displaystyle f\left(r\right)={\frac {\left(n-2\right)\,\mathbf {\Gamma } \left(n-1\right)\left(1-\rho ^{2}\right)^{\frac {n-1}{2}}\left(1-r^{2}\right)^{\frac {n-4}{2}}}{{\sqrt {2\pi }}\,\mathbf {\Gamma } \left(n-{\frac {1}{2}}\right)\left(1-\rho r\right)^{n-{\frac {3}{2}}}}}\,\mathbf {_{2}F_{1}} \left({\frac {1}{2}},{\frac {1}{2}};{\frac {2n-1}{2}};{\frac {\rho r+1}{2}}\right)}$

where ${\displaystyle \mathbf {\Gamma } }$ is the gamma function, ${\displaystyle \,\mathbf {_{2}F_{1}} (a,b;c;z)}$ is the Gaussian hypergeometric function. In the special case when ${\displaystyle \,\rho =0}$, the density can be written as:

${\displaystyle f\left(r\right)={\frac {\left(1-r^{2}\right)^{\frac {n-4}{2}}}{\mathbf {B} \left({\frac {1}{2}},{\frac {n-2}{2}}\right)}},}$

where ${\displaystyle \mathbf {B} }$ is the beta function, which is one way of writing the density of a Student's t-distribution, as above.

Note that[16] ${\displaystyle E\left(r\right)=\rho -{\frac {\rho \left(1-\rho ^{2}\right)}{2\left(n-1\right)}}+\cdots }$, therefore r is a biased estimator of ${\displaystyle \,\rho }$. The unique minimum variance unbiased estimator is given by[17] ${\displaystyle {\hat {\rho }}=r\mathbf {_{2}F_{1}} \left({\frac {1}{2}},{\frac {1}{2}};{\frac {n-1}{2}};1-r^{2}\right)}$. An approximately unbiased estimator can be obtained by truncating the previously mentioned series for ${\displaystyle E\left(r\right)}$ and solving the equation ${\displaystyle r=E\left(r\right)=\rho -{\frac {\rho \left(1-\rho ^{2}\right)}{2\left(n-1\right)}}}$ for ${\displaystyle \,\rho }$. However, the solution, ${\displaystyle {\breve {\rho }}=r\left[1+{\frac {1-r^{2}}{2\left(n-1\right)}}\right]}$,{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} is suboptimal.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} An approximately unbiased estimator,{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} with minimum variance for large values of n, with a bias of order ${\displaystyle {\frac {1}{n-1}}}$, can be obtained by maximizing ${\displaystyle \log {f\left(r\right)}}$, i.e. ${\displaystyle {\hat {\rho }}=r\left[1-{\frac {1-r^{2}}{2\left(n-1\right)}}\right]}$.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

### Use the Fisher transformation

In practice, confidence intervals and hypothesis tests relating to ρ are usually carried out using the Fisher transformation:

${\displaystyle F(r)={1 \over 2}\ln {1+r \over 1-r}=\operatorname {arctanh} (r).}$

If F(r) is the Fisher transformation of r, and n is the sample size, then F(r) approximately follows a normal distribution with

${\displaystyle {\text{mean}}=F(\rho )=\operatorname {arctanh} (\rho )}$    and standard error    ${\displaystyle {\text{SE}}={\frac {1}{\sqrt {n-3}}}.}$

Thus, a z-score is

${\displaystyle z={\frac {x-{\text{mean}}}{\text{SE}}}=[F(r)-F(\rho _{0})]{\sqrt {n-3}}}$

under the null hypothesis of that ${\displaystyle \rho =\rho _{0}}$, 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 = 2.2 is observed and a two-sided p-value is desired to test the null hypothesis that ${\displaystyle \rho =0}$, the p-value is 2·Φ(−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(${\displaystyle \rho }$):

${\displaystyle 100(1-\alpha )\%{\text{CI}}:\operatorname {arctanh} (\rho )\in [\operatorname {arctanh} (r)\pm z_{\alpha /2}SE]}$

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

${\displaystyle 100(1-\alpha )\%{\text{CI}}:\rho \in [\operatorname {tanh} (\operatorname {arctanh} (r)-z_{\alpha /2}SE),\operatorname {tanh} (\operatorname {arctanh} (r)+z_{\alpha /2}SE)]}$

For example, suppose we observe r = 0.3 with a sample size of n=50, and we wish to obtain a 95% confidence interval for ρ. The transformed value is arctanh(r) = 0.30952, so the confidence interval on the transformed scale is 0.30952 ± 1.96/√47, or (0.023624, 0.595415). Converting back to the correlation scale yields (0.024, 0.534).

## Pearson's correlation and least squares regression analysis

The square of the sample correlation coefficient is typically denoted r2 and called the coefficient of determination, estimates the fraction of the variance in Y that is explained by X in a simple linear regression. As a starting point, the total variation in the Yi around their average value can be decomposed as follows

${\displaystyle \sum _{i}(Y_{i}-{\bar {Y}})^{2}=\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}+\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2},}$

where the ${\displaystyle {\hat {Y}}_{i}}$ are the fitted values from the regression analysis. This can be rearranged to give

${\displaystyle 1={\frac {\sum _{i}(Y_{i}-{\hat {Y}}_{i})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}+{\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}.}$

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 square regression models, that the sample covariance between ${\displaystyle {\hat {Y}}_{i}}$ and ${\displaystyle Y_{i}-{\hat {Y}}_{i}}$ is zero. Thus, the sample correlation coefficient between the observed and fitted response values in the regression can be written

${\displaystyle r(Y,{\hat {Y}})^{2}={\frac {\sum _{i}({\hat {Y}}_{i}-{\bar {Y}})^{2}}{\sum _{i}(Y_{i}-{\bar {Y}})^{2}}}}$