Correlation
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- This article is about the correlation coefficient between two random variables. The term correlation can also mean the cross-correlation of two functions or electron correlation in molecular systems.

In probability theory and statistics, correlation, also called correlation coefficient, indicates the strength and direction of a linear relationship between two random variables. In general statistical usage, correlation or co-relation refers to the departure of two variables from independence, although correlation does not imply causality. In this broad sense there are several coefficients, measuring the degree of correlation, adapted to the nature of data.
A number of different coefficients are used for different situations. The best known is the Pearson product-moment correlation coefficient, which is obtained by dividing the covariance of the two variables by the product of their standard deviations. Despite its name, it was first introduced by Francis Galton.
[edit] Pearson's product-moment coefficient
[edit] Mathematical properties
The correlation ρX, Y between two random variables X and Y with expected values μX and μY and standard deviations σX and σY is defined as:
where E is the expected value of the variable and cov means covariance. Since μX = E(X), σX2 = E(X2) − E2(X) and likewise for Y, we may also write
The correlation is defined only if both of the standard deviations are finite and both of them are nonzero. It is a corollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in absolute value.
The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables. The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.
If the variables are independent then the correlation is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. Here is an example: Suppose the random variable X is uniformly distributed on the interval from −1 to 1, and Y = X2. Then Y is completely determined by X, so that X and Y are dependent, but their correlation is zero; they are uncorrelated. However, in the special case when X and Y are jointly normal, independence is equivalent to uncorrelatedness.
A correlation between two variables is diluted in the presence of measurement error around estimates of one or both variables, in which case disattenuation provides a more accurate coefficient.
[edit] The sample correlation
If we have a series of n measurements of X and Y written as xi and yi where i = 1, 2, ..., n, then the Pearson product-moment correlation coefficient can be used to estimate the correlation of X and Y . The Pearson coefficient is also known as the "sample correlation coefficient". It is especially important if X and Y are both normally distributed. The Pearson correlation coefficient is then the best estimate of the correlation of X and Y . The Pearson correlation coefficient is written:
where and
are the sample means of xi and yi , sx and sy are the sample standard deviations of xi and yi and the sum is from i = 1 to n. As with the population correlation, we may rewrite this as
Again, as is true with the population correlation, the absolute value of the sample correlation must be less than or equal to 1. Though the above formula conveniently suggests a single-pass algorithm for calculating sample correlations, it is notorious for its numerical instability (see below for something more accurate).
The square of the sample correlation coefficient, which is also known as the coefficient of determination, is the fraction of the variance in yi that is accounted for by a linear fit of xi to yi . This is written
where σy|x2 is the square of the error of a linear regression of xi on yi by the equation y = a + bx.
and σy2 is just the variance of y
Note that since the sample correlation coefficient is symmetric in xi and yi , we will get the same value for a fit of xi to yi :
This equation also gives an intuitive idea of the correlation coefficient for higher dimensions. Just as the above described sample correlation coefficient is the fraction of variance accounted for by the fit of a 1-dimensional linear submanifold to a set of 2-dimensional vectors (xi , yi ), so we can define a correlation coefficient for a fit of an m-dimensional linear submanifold to a set of n-dimensional vectors. For example, if we fit a plane z = a + bx + cy to a set of data (xi , yi , zi ) then the correlation coefficient of z to x and y is
[edit] Geometric Interpretation of correlation
The correlation coefficient can also be viewed as the cosine of the angle between the two vectors of samples drawn from the two random variables.
Caution: This method only works with centered data, i.e., data which have been shifted by the sample mean so as to have an average of zero. Some practitioners prefer an uncentered (non-Pearson-compliant) correlation coefficient. See the example below for a comparison.
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 between two vectors (see dot product), the uncentered correlation coefficient is:
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
as expected.
[edit] Interpretation of the size of a correlation
Several authors have offered guidelines for the interpretation of a correlation coefficient. Cohen (1988), for example, has suggested the following interpretations for correlations in psychological research:
Correlation | Negative | Positive |
Small | −0.29 to −0.10 | 0.10 to 0.29 |
Medium | −0.49 to −0.30 | 0.30 to 0.49 |
Large | −1.00 to −0.50 | 0.50 to 1.00 |
As Cohen himself has observed, however, all such criteria are in some ways arbitrary and should not be observed too strictly. This is because the interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0.9 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.
[edit] Non-parametric correlation coefficients
Pearson's correlation coefficient is a parametric statistic, and it may be less useful if the underlying assumption of normality is violated. Non-parametric correlation methods, such as Spearman's ρ and Kendall's τ may be useful when distributions are not normal; they are a little less powerful than parametric methods if the assumptions underlying the latter are met, but are less likely to give distorted results when the assumptions fail.
[edit] Other measures of dependence among random variables
To get a measure for more general dependencies in the data (also nonlinear) it is better to use the correlation ratio which is able to detect almost any functional dependency, or mutual information/total correlation which detect even more general dependencies.
[edit] Copulas and correlation
The information given by a correlation coefficient is not enough to define the dependence structure between random variables; to fully capture it we must consider the copula between them. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the cumulative distribution functions are elliptic (as with, for example, the multivariate normal distribution).
[edit] Correlation matrices
The correlation matrix of n random variables X1, ..., Xn is the n × n matrix whose i,j entry is corr(Xi, Xj). If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables Xi /SD(Xi) for i = 1, ..., n. Consequently it is necessarily a non-negative definite matrix.
The correlation matrix is symmetrical (the correlation between Xi and Xj is the same as the correlation between Xj and Xi).
[edit] Common misconceptions about correlation
[edit] Correlation and causality
The conventional dictum that "correlation does not imply causation" is treated in the article titled spurious relationship. See also correlation implies causation. However, correlations are not presumed to be acausal, though the causes may not be known.
[edit] Correlation and linearity

While Pearson correlation indicates the strength of a linear relationship between two variables, its value alone may not be sufficient to evaluate this relationship, especially in the case where the assumption of normality is incorrect.
The image on the right shows scatterplots of four different pairs of variables, first described by Francis Anscombe.[1] The four y variables have the same mean (7.5), standard deviation (4.12), correlation (0.81) and regression line (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 correspond 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, and Pearson correlation coefficient is not relevant. In the third case (bottom left), the linear relationship is perfect, except for one outlier which exerts enough influence to lower the correlation coefficient from 1 to 0.81. 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.
This example indicates that the correlation coefficient, as a summary statistic, can not replace an individual examination of the data.
[edit] Computing correlation accurately in a single pass
The following algorithm (in pseudocode) will estimate correlation with good numerical stability
sum_sq_x = 0 sum_sq_y = 0 sum_coproduct = 0 mean_x = x[1] mean_y = y[1] for i in 2 to N: sweep = (i - 1.0) / i delta_x = x[i] - mean_x delta_y = y[i] - mean_y sum_sq_x += delta_x * delta_x * sweep sum_sq_y += delta_y * delta_y * sweep sum_coproduct += delta_x * delta_y * sweep mean_x += delta_x / i mean_y += delta_y / i pop_sd_x = sqrt( sum_sq_x / N ) pop_sd_y = sqrt( sum_sq_y / N ) cov_x_y = sum_coproduct / N correlation = cov_x_y / (pop_sd_x * pop_sd_y)
For an enlightening experiment, check the correlation of {900,000,000 + i for i=1...100} with {900,000,000 - i for i=1...100}, perhaps with a few values modified. Poor algorithms will fail.
[edit] See also
[edit] Notes
- ^ Anscombe, Francis J. (1973) Graphs in statistical analysis. American Statistician, 27, 17–21.
[edit] References
- Abdi, H. "[1] ((2007). Coefficients of correlation, alienation and determination. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.".
- Abdi, H. "[2] ((2007). Part and partial correlation. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.".
- Abdi, H. "[3] ((2007). Multiple correlation coefficient. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.".
- Abdi, H. "[4] ((2007). The Kendal rank correlation coefficient. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.".
- Cohen, J., Cohen P., West, S.G., & Aiken, L.S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences. (2nd ed.) Hillsdale, NJ: Lawrence Erlbaum Associates.
[edit] External links
- Understanding Correlation - Introductory material by a U. of Hawaii Prof
- Statsoft Electronic Textbook
- Pearson's Correlation Coefficient - How to calculate it fast
- Learning by Simulations - The distribution of the correlation coefficient