# Dictionary Definition

correlated adj : mutually related [syn: correlative, correlate]

# User Contributed Dictionary

## English

### Verb

correlated- past of correlate

### Adjective

- mutually related in a correlation
- The variable "age15-24" was not included as this is highly correlated with the variable "students" and "students" were thought to be a more distinct group than the whole of the 15-24 age group.

# Extensive Definition

- This article is about the correlation coefficient between two 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,
(often measured as a 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. 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.

## Pearson's product-moment coefficient

### Mathematical properties

The correlation coefficient ρX, Y between two random variables X and Y with expected values μX and μY and standard deviations σX and σY is defined as:- \rho_= =,

- \rho_=\frac.

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, uncorrelatedness is equivalent to
independence.

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.

### 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". The Pearson correlation coefficient is then the best
estimate of the correlation of X and Y . The
Pearson correlation coefficient is written:

r_=\frac=\frac .

r_=\frac,

where \bar and \bar are the sample means of
X and Y , sx and sy are the
sample standard
deviations of X and Y and the sum is from i
= 1 to n. As with the population correlation, we may rewrite this
as

r_=\frac=\frac .

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

- r_^2=1-\frac,

where sy|x2 is the square of the error
of a linear
regression of xi on yi by the equation y = a + bx:

- s_^2=\frac\sum_^n (y_i-a-bx_i)^2,

and sy2 is just the variance of
y:

- s_y^2=\frac\sum_^n (y_i-\bar)^2.

Note that since the sample correlation
coefficient is symmetric in xi and yi , we will
get the same value for a fit of yi to xi :

- r_^2=1-\frac.

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

- r^2=1-\frac.

The distribution of the correlation coefficient
has been examined by R. A.
Fisher and A. K. Gayen.

### 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:

- \cos \theta = \frac = \frac = 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

- \cos \theta = \frac = \frac = 1 = \rho_,

as expected.

### Motivation for the form of the coefficient of correlation

Another motivation for correlation comes from
inspecting the method of simple linear
regression. As above, X is the vector of independent variables,
x_i, and Y of the dependent variables, y_i, and a simple linear
relationship between X and Y is sought, through a least-squares
method on the estimate of Y:

- \ Y = X\beta + \varepsilon.\,

Then, the equation of the least-squares line can
be derived to be of the form:

(Y - \bar) = \frac (X - \bar)

which can be rearranged in the form: (Y -
\bar)=\frac (X-\bar)

where r has the familiar form mentioned above :
\frac .

### 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, in the table on the right.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.

Along this vein, it is important to remember that
"large" and "small" should not be taken as synonyms for "good" and
"bad" in terms of determining that a correlation is of a certain
size. For example, a correlation of 1.0 or −1.0 indicates
that the two variables analyzed are equivalent modulo scaling.
Scientifically, this more frequently indicates a trivial result
than an earth-shattering one. For example, consider discovering a
correlation of 1.0 between how many feet tall a group of people are
and the number of inches from the bottom of their feet to the top
of their heads.

## Non-parametric correlation coefficients

Pearson's correlation coefficient is a parametric statistic and when distributions are not normal it may be less useful than non-parametric correlation methods, such as Chi-square, Point biserial correlation, Spearman's ρ and Kendall's τ. 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.## 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 the entropy-based mutual information/total correlation which is capable of detecting even more general dependencies. The latter are sometimes referred to as multi-moment correlation measures, in comparison to those that consider only 2nd moment (pairwise or quadratic) dependence.The polychoric
correlation is another correlation applied to ordinal data that
aims to estimate the correlation between theorised latent
variables.

## 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 a copula between them. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the cumulative distribution functions are the multivariate normal distributions. In the case of elliptic distributions it characterizes the (hyper-)ellipses of equal density, however, it does not completely characterize the dependence structure (for example, the a multivariate t-distribution's degrees of freedom determine the level of tail dependence).## 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
positive-semidefinite
matrix.

The correlation matrix is symmetric because the
correlation between X_i and X_j is the same as the correlation
between X_j and X_i.

## Removing correlation

It is always possible to remove the correlation
between zero-mean random variables with a linear transform, even if
the relationship between the variables is nonlinear. Suppose a
vector of n random variables is sampled m times. Let X be a matrix
where X_ is the jth variable of sample i. Let Z_ be an r by c
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, unit variance, and zero
correlation with all other variables. The transformed variables
will be uncorrelated, even though they may not be independent.

- D = X -\frac Z_ X

- T = D (D^T D)^

where an exponent of -1/2 represents the matrix
square root of the inverse of
a matrix. The covariance matrix of T will be the identity matrix.
If a new data sample 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:

- d = x - \frac Z_ X

- t = d (D^T D)^.

## Common misconceptions about correlation

### Correlation and causality

The conventional dictum that "correlation does not imply causation" means that correlation cannot be validly used to infer a causal relationship between the variables. This dictum should not be taken to mean that correlations cannot indicate causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown. Consequently, establishing a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction).Here is a simple example: hot weather may cause
both a reduction in purchases of warm clothing and an increase in
ice-cream purchases. Therefore warm clothing purchases are
correlated with ice-cream purchases. But a reduction in warm
clothing purchases does not cause ice-cream purchases and ice-cream
purchases do not cause a reduction in warm clothing
purchases.

A correlation between age and height in children
is fairly causally transparent, but a correlation between mood and
health in people is less so. Does improved mood lead to improved
health? Or does good health lead to good mood? Or does some other
factor underlie both? Or is it pure coincidence? 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 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 Anscombe's
quartet, a set of four different pairs of variables created by
Francis
Anscombe. 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 corresponds to what one
would expect when considering two variables correlated and
following the assumption of normality. The second one (top right)
is not distributed normally; while an obvious relationship between
the two variables can be observed, it is not linear, and the
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.

These examples indicate that the correlation
coefficient, as a summary statistic, cannot replace the individual
examination of the data.

## Computing correlation accurately in a single pass

The following algorithm (in pseudocode) will calculate Pearson 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)

## See also

- Autocorrelation
- Association (statistics)
- Cross-correlation
- Coefficient of determination
- Fraction of variance unexplained
- Kendall's tau
- Linear correlation (wikiversity)
- Pearson product-moment correlation coefficient
- Point-biserial correlation coefficient
- Partial correlation
- Spearman's rank correlation coefficient
- Statistical arbitrage
- Currency correlation

## Notes and references

## Further reading

- Cohen, J., Cohen P., West, S.G., & Aiken, L.S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences. (3rd ed.) Hillsdale, NJ: Lawrence Erlbaum Associates.

## External links

- Understanding Correlation - Introductory material by a U. of Hawaii Prof.
- Statsoft Electronic Textbook
- Pearson's Correlation Coefficient - How to calculate it quickly
- Learning by Simulations - The distribution of the correlation coefficient
- Correlation measures the strength of a linear relationship between two variables.
- MathWorld page on (cross-) correlation coefficient(s) of a sample.

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# Synonyms, Antonyms and Related Words

affiliate, affiliated, allied, associate, associated, bound, bracketed, collateral, conjugate, connected, corelated, corelational, corelative, correlational, correlative, coupled, implicated, interlinked, interlocked, interrelated, involved, joined, knotted, linked, of that ilk, of that
kind, parallel,
related, spliced, tied, twinned, wed, wedded, yoked