Overview
On this page, we discuss the variance of a discrete random variable, as well as the standard deviation and other socalled moments.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

Know the precise definition of the variance and the standard deviation of a discrete random variable and be able to calculate both by hand (at least for random variables which attain only very few values).

Understand that the standard deviation only in a certain sense quantifies the average distance from the mean, whereas the actual average of all distances from the mean is given by the socalled mean absolute deviation (which turns out to be less useful than the standard deviation). See the footnote “Why Square?” of this page for a more detailed explanation (in a uniformlydistributed situation).
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Perform calculations more efficiently using the alternative formula for the variance.

Understand the fundamental properties of variance.
To prepare for class

Watch the following video (by MIT OpenCourseWare) which defines the variance of a discrete random variable and shows some of its properties:

Watch the following video (by Khan Academy) which shows a detailed example calculation of a variance and standard deviation: