Probability and Statistics

Covariance

Measure of strength of the correlation between two or more sets of random variables i.e. it describes the uncertainty.

Covariance between population $X$ and population $Y$ with size $N$:

$$COV(X,Y)=\frac{1}{N}\sum _{i=1}^N(x_i-{\mu }_x)(y_i-{\mu }_y)$$

Covariance of a sample size $N$ is normalized by $N-1$ known as Bessel’s Correction:

$$COV(X,Y)=\frac{1}{N-1}\sum _{i=1}^N(x_i-{\mu }_x)(y_i-{\mu }_y)$$

Note

If $x$ and $y$ have no correlation then $COV(X,Y)=0$.

Blue and red subplots demonstrate uncorrelated measurements whereas green and cyan display positive and negative correlation respectively.