How to Compute Vector Means
This lesson explains how to use matrix methods to compute the means of vector elements and the means of matrix columns.
Mean Scores: Vectors
In ordinary algebra, the mean of a set of observations is computed by adding all of the observations and dividing by the number of observations.
x = Σxi / n
where x is the mean of observations, Σxi is the sum of all observations, and n is the number of observations.
In matrix algebra, the mean of a set of n scores can be computed as follows:
x = 1'x ( 1'1 )-1 = 1'x ( 1/n )
where
x is the mean of a set of n scores
1 is an n x 1 column
vector
of ones
x is an n x 1 column
vector
of scores: x1, x2, . . . ,
xn
To show how this works, let's find the mean of elements of vector x, where x' = [ 1 2 3 ].
| x = | 1' | x | ( | 1' | 1 | )-1 |
| x = | [ 1 1 1 ] |
|
( | [ 1 1 1 ] |
|
)-1 |
x = 6/3 = 2
Thus, the mean of the elements of x is 2.
Mean Scores: Matrices
You can think of an r x c matrix as a set of c column vectors, each having r elements. Often, with matrices, we want to compute mean scores separately within columns, consistent with the equation below.
Xc = Σ Xic / r
where
Xc
is the mean of a set of r scores from column c
Σ Xic
is the sum of elements from column c
In matrix algebra, a vector of mean scores from each column of matrix X can be computed as follows:
m' = 1'X ( 1'1 )-1 = 1'X ( 1/r )
where
m' is a row vector of column means,
[ X1
X2 ...
Xc ]
1 is an r x 1 column
vector
of ones
X is an r x c matrix of scores:
X11, X12, . . . ,
Xrc
The problem below shows how everything works.
Test Your Understanding
Problem 1
Consider matrix X.
| X = |
|
Using matrix methods, create a 1 x 3 vector m', such that the elements of m' are the mean of column elements from X. That is,
m' = [ X1 X2 X3 ]
where Xi is the mean of elements from column i of matrix X.
Solution
To solve this problem, we use the following equation: m' = 1'X ( 1'1 )-1. Each step in the computation is shown below.
| m' = | 1' | X | ( | 1' | 1 | )-1 | ||||||||||||||||
| m' = | [ 1 1 ] |
|
( | [ 1 1 ] |
|
)-1 |
| m' = |
|
( | [ 1 1 ] |
|
)-1 |
| m' = |
|
* 0.5 |
| m' = |
|
Thus, vector m has the mean column scores from matrix X. The mean score for column 1 is 6, the mean score for column 2 is 3, and the mean score for column 3 is 2.5.