# Correlation and dependence

For other uses, see correlation (disambiguation). Correlation and dependence_sentence_1

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Correlation and dependence_sentence_2

In the broadest sense correlation is any statistical association, though it commonly refers to the degree to which a pair of variables are linearly related. Correlation and dependence_sentence_3

Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve. Correlation and dependence_sentence_4

Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. Correlation and dependence_sentence_5

For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. Correlation and dependence_sentence_6

In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. Correlation and dependence_sentence_7

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). Correlation and dependence_sentence_8

## Pearson's product-moment coefficient Correlation and dependence_section_0

Main article: Pearson product-moment correlation coefficient Correlation and dependence_sentence_9

### Definition Correlation and dependence_section_1

The most familiar measure of dependence between two quantities is the Pearson product-moment correlation coefficient (PPMCC), or "Pearson's correlation coefficient", commonly called simply "the correlation coefficient". Correlation and dependence_sentence_10

Mathematically, it is defined as the quality of least squares fitting to the original data. Correlation and dependence_sentence_11

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. Correlation and dependence_sentence_12

Mathematically, one simply divides the covariance of the two variables by the product of their standard deviations. Correlation and dependence_sentence_13

Karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton. Correlation and dependence_sentence_14

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. Correlation and dependence_sentence_15

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 dataset. Correlation and dependence_sentence_16

### Correlation and independence Correlation and dependence_section_3

If the variables are independent, Pearson's correlation coefficient is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. Correlation and dependence_sentence_17

Even though uncorrelated data does not necessarily imply independence, one can check if random variables are independent if their mutual information is 0. Correlation and dependence_sentence_18

## Example Correlation and dependence_section_5

For this joint distribution, the marginal distributions are: Correlation and dependence_sentence_19

This yields the following expectations and variances: Correlation and dependence_sentence_20

Therefore: Correlation and dependence_sentence_21

## Rank correlation coefficients Correlation and dependence_section_6

Main articles: Spearman's rank correlation coefficient and Kendall tau rank correlation coefficient Correlation and dependence_sentence_22

Rank correlation coefficients, such as Spearman's rank correlation coefficient and Kendall'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. Correlation and dependence_sentence_23

If, as the one variable increases, the other decreases, the rank correlation coefficients will be negative. Correlation and dependence_sentence_24

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. Correlation and dependence_sentence_25

However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the Pearson 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. Correlation and dependence_sentence_26

Correlation and dependence_description_list_0

• (0, 1), (10, 100), (101, 500), (102, 2000).Correlation and dependence_item_0_0

## Other measures of dependence among random variables Correlation and dependence_section_7

The information given by a correlation coefficient is not enough to define the dependence structure between random variables. Correlation and dependence_sentence_28

The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a multivariate normal distribution. Correlation and dependence_sentence_29

(See diagram above.) Correlation and dependence_sentence_30

In the case of elliptical distributions it characterizes the (hyper-)ellipses of equal density; however, it does not completely characterize the dependence structure (for example, a multivariate t-distribution's degrees of freedom determine the level of tail dependence). Correlation and dependence_sentence_31

Distance correlation was introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables; zero distance correlation implies independence. Correlation and dependence_sentence_32

The Randomized Dependence Coefficient is a computationally efficient, copula-based measure of dependence between multivariate random variables. Correlation and dependence_sentence_33

RDC is invariant with respect to non-linear scalings of random variables, is capable of discovering a wide range of functional association patterns and takes value zero at independence. Correlation and dependence_sentence_34

The correlation ratio, entropy-based mutual information, total correlation, dual total correlation and polychoric correlation are all also capable of detecting more general dependencies, as is consideration of the copula between them, while the coefficient of determination generalizes the correlation coefficient to multiple regression. Correlation and dependence_sentence_35

## Sensitivity to the data distribution Correlation and dependence_section_8

Further information: Pearson product-moment correlation coefficient § Sensitivity to the data distribution Correlation and dependence_sentence_36

Various correlation measures in use may be undefined for certain joint distributions of X and Y. Correlation and dependence_sentence_37

For example, the Pearson correlation coefficient is defined in terms of moments, and hence will be undefined if the moments are undefined. Correlation and dependence_sentence_38

Measures of dependence based on quantiles are always defined. Correlation and dependence_sentence_39

Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being unbiased, or asymptotically consistent, based on the spatial structure of the population from which the data were sampled. Correlation and dependence_sentence_40

Sensitivity to the data distribution can be used to an advantage. Correlation and dependence_sentence_41

For example, scaled correlation is designed to use the sensitivity to the range in order to pick out correlations between fast components of time series. Correlation and dependence_sentence_42

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 and dependence_sentence_43

## Correlation matrices Correlation and dependence_section_9

A correlation matrix appears, for example, in one formula for the coefficient of multiple determination, a measure of goodness of fit in multiple regression. Correlation and dependence_sentence_45

In statistical 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. Correlation and dependence_sentence_46

For example, in an exchangeable correlation matrix, all pairs of variables are modeled as having the same correlation, so all non-diagonal elements of the matrix are equal to each other. Correlation and dependence_sentence_47

On the other hand, an autoregressive matrix is often used when variables represent a time series, since correlations are likely to be greater when measurements are closer in time. Correlation and dependence_sentence_48

Other examples include independent, unstructured, M-dependent, and Toeplitz. Correlation and dependence_sentence_49

## Common misconceptions Correlation and dependence_section_11

### Correlation and causality Correlation and dependence_section_12

Main article: Correlation does not imply causation Correlation and dependence_sentence_50

See also: Normally distributed and uncorrelated does not imply independent Correlation and dependence_sentence_51

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. Correlation and dependence_sentence_52

This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. Correlation and dependence_sentence_53

However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations (tautologies), where no causal process exists. Correlation and dependence_sentence_54

Consequently, a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction). Correlation and dependence_sentence_55

A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Correlation and dependence_sentence_56

Does improved mood lead to improved health, or does good health lead to good mood, or both? Correlation and dependence_sentence_57

Or does some other factor underlie both? Correlation and dependence_sentence_58

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. Correlation and dependence_sentence_59

### Simple linear correlations Correlation and dependence_section_13

These examples indicate that the correlation coefficient, as a summary statistic, cannot replace visual examination of the data. Correlation and dependence_sentence_60

The examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a normal distribution, but this is not correct. Correlation and dependence_sentence_61