Covariance

Covariance is a measure used to determine the extent to which two variables change in tandem. It’s a critical statistic for understanding the relationship between two variables. The covariance ($\mathrm{Cov}<!--\; FUNCTION\; APPLICATION\; -->(X,Y)$) between two variables $X$ and $Y$ is given by:

where ${\mu }_X$ and ${\mu }_Y$ are the means of $X$ and $Y$, respectively. A positive covariance indicates that the variables tend to move in the same direction, while a negative covariance indicates that they move in opposite directions.

If the (real) random variable pair $(X,Y)$ can take on the values $(x_i,y_i)$ for $i=1,\dots ,n$, with equal probabilities $p_i=\frac{1}{n}$, then the covariance can be equivalently written in terms of the means $E<!--\; FUNCTION\; APPLICATION\; -->[X]$ and $E<!--\; FUNCTION\; APPLICATION\; -->[Y]$ as