home

Variance, Covariance, and Composites

1. Basic Concepts

1.1 Basic Definitions

Name

Population

Sample

Binary

Computation

Mean

p

Variance

pq

Standard

Deviation

 

1.2 Algebra of summations

The deviate

Summation of a constant

 

Multiplication by a constant

 

Addition

 

Multiplication

Note that the sum of products DOES NOT EQUAL the product of the sums.

A numerical example of multiplication by a constant in which X is a variable and k (the constant) is 2, addition of distributions and multiplication of distributions.

 

X

kX

X+kX

X*kX

 

1

2

3

2

 

2

4

6

8

 

2.5

5

7.5

12.5

 

3

6

9

18

sum

8.5

17

25.5

40.5

2. Effects of linear transformations on the mean and variance of a distributions

2.1 Effect on mean of adding a constant to the distribution (X = X+a):

 

 

2.2 Effect on mean of multiplying the distribution by a constant (X = aX)

2.3 Effect on variance of adding a constant to the distribution (X = X+a):

 

2.4 Effect on variance of multiplying the distribution by a constant

Define variance multiplied by a

carry out multiplication; expand terms

collect a2 outside the parens by the distributive property

carry the a2 outside the summation sign by algebra of summations

substitute variance symbol for formula. The variance of the new distribution equals the variance of the old distribution times a2.

 

3. Composites are variables composed of the sum of other variables, such as test scores formed of multiple items; subtest scores, or scores on entire test batteries.

3.1 The mean of a composite

C = X1 + X2

Big C, Big X, raw scores

c = x1 + x2

little c, little x, deviation form

define the mean as the sum of the variables

distribute summation by algebra of summations

substitute mean symbols for computations. The mean of the composite is the sum of the means of the constituent elements.

 

3.2 The variance of a composite

 

This says that the variance of a composite is the sum of the deviations from the composite mean squared, which is also equal to the sum of the item or element deviations squared.

Let's consider what the variance of a composite of two elements would be:

 

The result is that the variance of the composite is the sum of the variances of the elements plus twice an additional term. You can see that the additional term is the average cross product. The average cross product is called a covariance.

It turns out that the variance of a composite is always equal to the sum of the elements in the variance-covariance matrix, that is, the sum of the variances plus twice the sum of the covariances. For example, suppose we had the following covariance matrix:

element

1

2

3

4

1

2

2

1

1

2

2

2

2

1

3

1

2

2

2

4

1

1

2

2

sum

6

7

7

6

If we formed a composite of the 4 elements in this matrix, the variance of the composite would be the sum of all the elements in the matrix, namely, 26. [We can add by columns, 6+7+7+6 = 26, or we can add double the lower part 2(2+1+1+2+1+2)=18 plus the diagonal (2+2+2+2) = 18+8 = 26.]