## 3.8 Variance

SW 2.2

The next most important feature of the distribution of a random variable is its variance. The variance of a random variable $$X$$ is a measure of its “spread”, and we will denote it $$\mathrm{var}(X)$$ [You might also sometimes see the notation $$\sigma^2$$ or $$\sigma_X^2$$ for the variance.] The variance is defined as

$\mathrm{var}(X) := \mathbb{E}\left[ (X - \mathbb{E}[X])^2 \right]$ Before we move forward, let’s think about why this is a measure of the spread of a random variable.

• $$(X-\mathbb{E}[X])^2$$ is a common way to measure the “distance” between $$X$$ and $$\mathbb{E}[X]$$. It is always positive (whether $$(X - \mathbb{E}[X])$$ is positive or negative) which is a good feature for a measure of distance to have. It is also increasing in $$|X-\mathbb{E}[X]|$$ which also seems a requirement for a reasonable measure of distance.

• Then, the outer expectation averages the above distance across the distribution of $$X$$.

An alternative expression for $$\mathrm{var}(X)$$ that is often useful in calculations is

$\mathrm{var}(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2$

Sometimes, we will also consider the standard deviation of a random variable. The standard deviation is defined as

$\textrm{sd}(X) := \sqrt{\mathrm{var}(X)}$ You might also see the notation $$\sigma$$ or $$\sigma_X$$ for the standard deviation.

The standard deviation is often easier to interpret than the variance because it has the same “units” as $$X$$. Variance “units” are squared units of $$X$$.

That said, variances more often show up in formulas/derivations this semester.