Answer to Question #161819 in Statistics and Probability for Lemchi Divine

Question

What is correlation

Answer 1

Сorrelation in statistics is an instrument to measure dependence between two random variables.

If X and Y are two random variables then, by definition, the correlation between them is $\rho_{X,Y} ={\rm corr}(X, Y) = {E((X-\mu_X) (Y-\mu_Y))}/({\sigma_X\sigma_Y})=(E(XY)-E(X)E(Y))/({\sigma_X\sigma_Y})$

where $\mu_X, \mu_Y$ are expected values of X, Y and $\sigma_X, \sigma_Y$ are their standard deviations (expected values and standard deviations are assumed to be finite).

If the variables X and Y are independent then their correlation is zero.

If the correlation between X and Y is 0, they may be either dependent or independent.

For any variables X and Y $|{\rm corr}(X, Y)|\leq 1$ (that is a corollary of the Cauchy–Schwarz inequality) and ${\rm corr}(X, X)=1$.

The higher the absolute value of the correlation coefficient, the more significant the relationship between the random variables.

The correlation coefficient can be considered as the cosine of the angle between random variables X and Y in a suitable Euclidean vector space. If the correlation is positive, the angle is sharp and the vectors (i.e. variables) are close to each other, their directions are nearly co-directed. If the correlation is negative, the angle is obtuse and the vectors (i.e. variables) are nearly anti-directed. The positive value of the correlation coefficient is similar to monotonously increasing relationship, whereas negative value of the correlation coefficient is similar to monotonously decreasing relationship.