Variance and Standard Deviation

Abstract

In Chapter 12, we show that the mean value µX = E[X] of a random variable locates the center of mass of the induced probability distribution. As the center of mass, the mean value provides important information about the distribution, hence about the values likely to be taken on by the random variable. However, this location of the center of mass is not a sufficient characterization of the distribution for most purposes. Quite different probability distributions may share the same mean value. For example, the random variable X uniformly distributed on the interval (-0.1,0.1) and the random variable Y uniformly distributed on the interval (-100,100) both have zero mean value. But there is far less uncertainty about where to look for an observed value of X than there there is for Y. The difference is the spread of probability mass (hence possible values) about the mean. It is highly desirable to have an appropriate measure of this spread or variation.