Covariance

The covariance between two variables $x$ and $y$ is a factor that tells us how (but not how much) these variables go together. It tells us the direction of the trend. Covariance is very similar to correlation, which is just a normalized covariance. Furthermore, correlation is a function of the covariance.

If we have a two dimensional dataset (each point has two coordinates), we can compute the variance on the $x$ and $y$ axis.

Having the two independent variances doesn’t tell us much about the trend of the dataset, that’s why we need the covariance between $x$ and $y$. We can compute it like this:

$$\text{Cov}(X,Y)=\mathbb{E}[(X-{\mu }_x)(Y-{\mu }_y)]\in (-\infty ,+\infty )$$

We are basically taking the expected value of the product between the differences of X and its mean, and Y and its mean.

A positive covariance mean that we have an increasing trend, while a negative value means that we have a decreasing trend (as X decreases, Y decreases).

Note

Covariance quantifies the direction of the trend, but doesn’t say how strong the trend is.

In order to know how strong the trend is, we need to use a Correlation coefficient, which is not subject to the scale of the variable, since it’s normalized.

Note

Covariance has the commutative property, so $\text{Cov}(x,y)=\text{Cov}(y,x)$

Related:

• Covariance Matrix

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statistics resources:

• Covariance Clearly Explained! - YouTube