 StatLect

# Variance formula

The variance of a random variable can be computed using the definition of variance: where denotes the expected value operator. ## Formula for discrete variables

When the random variable is discrete the above formula becomes where is the set of all possible realizations of and is the probability mass function of . In other words, we need to compute a weighted average of the squared deviations of from its mean.

To see how to apply this formula, read some Solved exercises.

## Formula for continuous variables

When is continuous, the formula is where is the probability density function of .

To see how to apply this formula, read some Solved exercises.

## A simple variance formula

Instead of computing variance using these formulae, it is often easier to use the following equivalent variance formula: For example, when we know the moment generating function of , we can use it to compute the two moments and and then plug their values in this formula.

## More details

More details about this formula - as well as a proof of it and some solved exercises - can be found in the lecture entitled Variance.

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