# Connexions

You are here: Home » Content » Variance Sum Law I

### Recently Viewed

This feature requires Javascript to be enabled.

# Variance Sum Law I

Module by: David Lane. E-mail the author

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." For example, suppose the two populations are the populations of 8-year old males and 8-year females in Houston Texas and that the variable of interest is memory span. You repeat the following steps thousands of times: (1) sample one male and one female, (2) measure the memory span of each, and (3) sum the two memory spans. After you have done this thousands of times, you compute the variance of the sum. It turns out that the variance of this sum can be computed according to the following formula: σ sum 2= σ M 2+ σ F 2 σ sum 2 σ M 2 σ F 2 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. Therefore, if the variances on the memory span test for the males and females respectively were 0.90.9 and 0.80.8, then the variance of the sum would be 1.701.70.

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. σ difference 2= σ M 2+ σ F 2 σ difference 2 σ M 2 σ F 2 More generally, the variance sum law can be written as follows: σ X±Y 2= σ M 2+ σ F 2 σ X±Y 2 σ M 2 σ F 2 which is read "The variance of XX plus or minus YY is equal the variance of XX plus the variance of YY."

## Note:

The formulas for the sum and difference of variables given above only apply when the variables are independent.
In this example, we have thousands of randomly-paired scores. Since the scores are paired randomly, there is no relationship between memory span of one member of the pair and the memory span of the other. Therefore the two scores are independent. Contrast this situation with one in which thousands of people are sampled and two measures (such as verbal and quantitative SAT) are taken from each. In this case, there would be a relationship between the two variables since higher scores on the verbal SAT are associated with higher scores on the quantitative SAT (although there are many examples of people who score high on one test and low on the other). Thus the two variables are not independent and the variance of the total SAT score would not be the sum of the variance of the verbal SAT and the quantitative SAT. The general form of the variance sum law is presented in a section in the chapter on correlation.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks