Stat Trek

Teach yourself statistics

Stat Trek

Teach yourself statistics


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
2
3
   (    [ 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   =   
3 5 1
9 1 4

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 ]   
3 5 1
9 1 4
   (    [ 1 1 ]   
1
1
  )-1
m'  =  
3+9   5+1   1+4
   (    [ 1 1 ]   
1
1
  )-1
m'   =   
12   6   5
  *  0.5
m'   =   
6   3   2.5

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.