Science:MATH105 Probability/Lesson 1 DRV/1.05 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.
Definition: Variance and Standard Deviation of a Discrete Random Variable 

The variance, Var(X), of a discrete random variable X is
The integer N is the number of possible values of X. The standard deviation, σ, is the positive square root of the variance:

Observe that the variance of a distribution is always nonnegative (p_{k} is 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
Students in MATH 105 are expected to memorize the formulas for variance and standard deviation.
Example: Grade Distributions
Using the grade distribution example of the previous page, calculate the variance and standard deviation of the random variable 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 most of the PDF will live on the interval S_{3}.
For our grade distribution example, notice that all possible values of X are contained in the interval S_{3}. In fact, all possible values of X are contained in S_{2} for this particular example.