# 1.7 Variance and Standard Deviation

Another important quantity related to a given random variable is its variance. The **variance** is a numerical description of the spread, or the *dispersion*, of the random variable. That is, the variance of a random variable *X* is a measure of how spread out the values of *X* are, given how likely each value is to be observed.

Variance and Standard Deviation of a Discrete Random Variable |
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The variance, Var(X), of a discrete random variable X is
where The |

Observe that the variance of a random variable is always nonnegative (since probabilities are nonnegative, and the square of a number is also nonnegative).

Observe also that much like the expectation of a random variable *X*, the variance (or standard deviation) is a weighted average of an expression of observable and calculable values. More precisely, notice that

## Example: Test Scores

Using the test scores example of the previous sections, calculate the variance and standard deviation of the random variable *X* associated to randomly selecting a single exam.

## Solution

The variance of the random variable *X* is given by

The standard deviation of *X* is then

## Interpretation of the Standard Deviation

For most "nice" random variables, i.e. ones that are not too wildly distributed, the standard deviation has a convenient informal interpretation. Consider the intervals

for some positive integer *m*. As we increase the value of *m*, these intervals will contain more of the possible values of the random variable *X*.

A good rule of thumb is that for "nicely distributed" random variables, all of the most likely possible values of the random variable will be contained in the interval *S*_{3}. Another way to say this is that, for discrete random variables, most of the PMF will live on the interval *S*_{3}. We will see in the next chapter that a similar interpretation holds for continuous random variables.