Outcome Variable Covariance

Why

We want a measure of the spread of a random variable.

Definition

The covariance (or variance) of a random variable $x: \Omega \to \R $ is $\E ((x - \E (x))^2)$, the expectation of the square of the random variable’s distance from its mean. The covariance measures the mean square difference from the mean.

Interpretation

The covariance of $x$ summarizes how “wide” the induced distribution of $x$ is. If the covariance is small, then the induced distribution is concentrated around its mean.¹

Notation

We denote the covariance of $x$ by $\cov(x)$. Another common notation is $\var(x)$.

Standard deviation

If $x$ has units meters, then $\cov(x)$ has units square meters. It can be useful to work instead with the standard deviation of $x$, defined as $\sqrt{\cov(x)}$, which has the same units as $x$. We denote the standard deviation of $x$ by $\std(x)$.

1. Future editions will give example. ↩︎