Editing Correlation (section)

===Correlation and independence===
It is a corollary of the [[Cauchy–Schwarz inequality]] that the [[absolute value]] of the Pearson correlation coefficient is not bigger than 1. Therefore, the value of a correlation coefficient ranges between −1 and +1. The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship ('''anti-correlation'''),<ref>Dowdy, S. and Wearden, S. (1983). "Statistics for Research", Wiley. {{ISBN|0-471-08602-9}} pp 230</ref> and some value in the [[open interval]] <math>(-1,1)</math> in all other cases, indicating the degree of [[linear dependence]] between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.

If the variables are [[statistical independence|independent]], Pearson's correlation coefficient is 0. However, because the correlation coefficient detects only linear dependencies between two variables, the converse is not necessarily true. A correlation coefficient of 0 does not imply that the variables are independent{{Cn|date=May 2024}}.

<math display=block>\begin{align}
X,Y \text{ independent} \quad & \Rightarrow \quad \rho_{X,Y} = 0 \quad (X,Y \text{ uncorrelated})\\
\rho_{X,Y} = 0 \quad (X,Y \text{ uncorrelated})\quad & \nRightarrow \quad X,Y \text{ independent}
\end{align}</math>

For example, suppose the random variable <math>X</math> is symmetrically distributed about zero, and <math>Y=X^2</math>. Then <math>Y</math> is completely determined by <math>X</math>, so that <math>X</math> and <math>Y</math> are perfectly dependent, but their correlation is zero; they are [[uncorrelated]]. However, in the special case when <math>X</math> and <math>Y</math> are [[Joint normality|jointly normal]], uncorrelatedness is equivalent to independence.

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