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 = Σx_{i} / n
where x is the mean of observations, Σx_{i} 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: x_{1}, x_{2}, . . . ,
x_{n}
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.
X_{c} = Σ X_{i}_{c} / r
where
X_{c}
is the mean of a set of r scores from column c
Σ X_{i}_{c}
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,
[ X_{1}
X_{2} ...
X_{c} ]
1 is an r x 1 column
vector
of ones
X is an r x c matrix of scores:
X_{1}_{1}, X_{1}_{2}, . . . ,
X_{r}_{c}
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' = [ X_{1} X_{2} X_{3} ]
where X_{i} 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.