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 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.
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 |
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.