Variance and standard deviation

The variance and standard deviation of a discrete random variable, the two formulas for variance, and the variance of a linear function aX + b.

Measuring spread

Variance

$$\operatorname{Var}(X) = E\big((X - \mu)^2\big) = E(X^2) - \mu^2 \qquad \sigma = \sqrt{\operatorname{Var}(X)}$$

The variance is the expected squared distance from the mean. The standard deviation $\sigma$ is back in the units of $X$. The second form, $E(X^2) - \mu^2$, is usually quicker.

Example 1: two coins●●

Find the variance and standard deviation of the number of heads when two coins are tossed.

$\mu = 1$ and $E(X^2) = 0 \times \tfrac14 + 1 \times \tfrac12 + 4 \times \tfrac14 = 1.5$.

$\operatorname{Var}(X) = 1.5 - 1 = 0.5$ and $\sigma = \sqrt{0.5} \approx 0.71$.

Example 2●●

Find the variance for the distribution with $x = 1, 2, 3, 4$ and probabilities $0.1, 0.3, 0.4, 0.2$.

$\mu = 2.7$ and $E(X^2) = 0.1 + 1.2 + 3.6 + 3.2 = 8.1$.

$\operatorname{Var}(X) = 8.1 - 2.7^2 = 0.81$, so $\sigma = 0.9$.

Example 3: a die●●●

Find the variance of the score on a fair die.

$E(X^2) = \dfrac{1 + 4 + 9 + 16 + 25 + 36}{6} = \dfrac{91}{6}$.

$\operatorname{Var}(X) = \dfrac{91}{6} - 3.5^2 = \dfrac{35}{12} \approx 2.92$.

Linear changes

Variance of aX + b

$$\operatorname{Var}(aX + b) = a^2\operatorname{Var}(X) \qquad \text{sd}(aX + b) = |a|\,\sigma$$

Adding $b$ shifts every value but does not change the spread. Multiplying by $a$ scales the spread.

Flashcards

Quick formula for variance

$E(X^2) - \mu^2$

Standard deviation

$\sigma = \sqrt{\operatorname{Var}(X)}$

$operatorname{Var}(aX + b)$

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

Practice

Find the variance of the number of heads when two coins are tossed.
In Example 2, find $E(X^2)$.
In Example 2, find $\operatorname{Var}(X)$.
In Example 2, find the standard deviation.
Find the variance of the score on a fair die.
$\operatorname{Var}(X) = 8.1$. Find $\operatorname{Var}(2X + 3)$.
$\sigma = 0.9$. Find the standard deviation of $3X - 1$.
Adding $5$ to every value of $X$:
A larger standard deviation means the values are: