Variance Sum Law I

Prerequisites
Variance

Learning Objectives

  1. Compute the variance of the sum of two uncorrelated variables
  2. Compute the variance of the difference between two uncorrelated variables

As you will see in later sections, there are many occasions on which it is important to know the variance of the sum of two variables. Consider the following situation: (a) you have two populations, (b) you sample one number from each population, (c) you add the two numbers together. The question is, "What is the variance of this sum."

It turns out that the variance of this sum can be computed according to the following formula:

where the first term is the variance of the sum, the second term is the variance of the males and the third term is the variance of the females.

The formula for the variance of the difference between the two variables (memory span in this example) is shown below. Notice that expression for the difference is the same as the formula for the sum.

More generally, the variance sum law can be written as follows:

which is read "The variance of X plus or minus Y is equal the variance of X plus the variance of Y.

The formulas for the sum and difference of variables given above only apply when the variables are independent.

The general form of the variance sum law is presented in a section in the chapter on correlation.