Correlation coefficient

The correlation coefficient for two variables $X$ and $Y$ is computed as:

$$\text{Corr}(X,Y)=\frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)}\sqrt{\text{Var}(Y)}}\in [-1,1]$$

The product of the squared root of variances is a normalizing factor, in order for the correlation to not be affected by the variables scaling.

If the variable grow in the same direction (we have an increasing trend), then we will have a correlation close to 1, otherwise it will be close to -1. The closeness tells us the strength of the trend.

A value close to 0 means that there is no linear relationship between the variables.

Note

Remember that we can use also the covariance alone to compute the direction of the trend, but it doesn’t tells us anything about its strength

Important

Remember that CORRELATION DOES NOT MEAN CAUSATION, since there can be a third variable $z$ unknown to us that is actually causing the values of $x$ and $y$.

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statistics resources:

• Covariance Clearly Explained! - YouTube