correlation

Correlation

Correlation is the normalized covariance.

$$\text{Corr}(X,Y)=\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$$

Correlation stays the same under affine transformations, which does not hold for covariance.

$$\text{Corr}(aX+b,cY+d)=\text{Corr}(X,Y)$$

Why this does not hold for covariance:

$$\text{Cov}(aX+b,cY+d)=\mathbb{E}[(aX+\cancel{b}-\mathbb{E}[aX\cancel{+b}])(cY\cancel{+d}-\mathbb{E}[cY\cancel{+d}])]=ac\text{Cov}(X,Y)$$