Expected value describes the center of a distribution, but two distributions with the same mean can have very different amounts of spread. Variance measures how far values typically lie from the mean by averaging squared deviations.

If $\mu=E[X]$, then

$$\operatorname{Var}(X)=E[(X-\mu)^2].$$

Why the deviations are squared

The ordinary deviations $X-\mu$ have mean zero, because positive and negative deviations cancel. Squaring makes all contributions nonnegative and gives larger deviations more weight.

An equivalent formula is

$$\operatorname{Var}(X)=E[X^2]-E[X]^2.$$

For calculations this form is often simpler.

Standard deviation

Variance is measured in squared units. The standard deviation returns to the original units of $X$:

$$\sigma_X=\sqrt{\operatorname{Var}(X)}.$$

A small standard deviation means values are concentrated near the mean; a large one means the distribution is more spread out.

Shifting and scaling

Adding a constant changes the center but not the spread:

$$\operatorname{Var}(X+b)=\operatorname{Var}(X).$$

Multiplying by $a$ scales deviations by $a$, so

$$\operatorname{Var}(aX)=a^2\operatorname{Var}(X),$$

and therefore

$$\operatorname{SD}(aX)=|a|\operatorname{SD}(X).$$

Variance and standard deviation describe spread around the expected value, not the full shape of the distribution.