The variance of a given event is the "average distance" between each individual event and the average.

Sample

Given as the number of events the variance of a given sample is given by the following equation:

Random Variable

The variance for a given random variable can be given by the following equation:

Alternatively it can be written as

When the random variable is squared and then the expected value is applied it ends up just being , where the probability density function stays unchanged.

Properties

Addition by a Constant

Multiplication by a Constant

Addition of Two Random Variables