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### Correlation

If one wants to know the degree of a relationship, the correlation between two variables can be examined. Correlations can be quantified by computing a correlation coefficient. This entry first describes a concept central to correlation, covariance, and then discusses calculation and interpretation of correlation coefficients.

Covariance indicates the tendency in the linear relationship for two random variables to covary (or vary together) that is represented in deviations measured in the unstandardized units in which X and Y are measured. Specifically, it is defined as the expected product of the deviations of each of two random variables from its expected values or means.

The population covariance between two variables, X and Y, can be written by:

$\begin{array}{c}cov\left(X,Y\right)=E\left[\left(X-E\left(X\right)\right)\left(Y-E\left(Y\right)\right)\right]\\ =E\left[XY-XE\left(Y\right)-E\left(X\right)Y-E\left(X\right)E\left(Y\right)\right]\\ =E\left(XY\right)-E\left(X\right)E\left(Y\right)-E\left(X\right)E\left(Y\right)-E\left(X\right)E\left(Y\right)\\ =E\left(XY\right)-E\left(X\right)E\left(Y\right)\end{array}$

where E is the expected value or population mean.

Similarly, the sample covariance ...