Topics: Probability

As a Moment


Let be a random variable.

The variance of , denoted , is defined as:

That is, the variance of is its second moment.


Notice that is also equivalent to:

As a Measure of Dispersion


Let be observations of a random variable . The variance of is:

…where is the mean of .


Variance quantifies how spread out the variable’s mass is with respect to its mean.

Compare, for instance, the mass of the following two populations with the same mean but different variances (blue has ; red has ):

(courtesy of Wikimedia Commons)


Let be a random variable whose expected value exists, and let . Then: