How to Compute Deviation Scores
This lesson explains how to use matrix
methods to transform raw scores to deviation scores.
We show the transformation to deviation scores for
vectors and for
matrices.
Deviation Scores: Vectors
A deviation score is the difference between a raw score
and the mean.
d_{i} = x_{i}  x
where
d_{i} is the deviation score for the ith observation in
a set of observations
x_{i} is the raw score for the ith observation in
a set of observations
x is the mean of all the observations in
a set of observations
Often, it is easier to work with deviation scores than with raw scores.
Use the following formula to transform a
vector of
n raw scores into a vector of n deviation scores.
d =
x 
1'x1
( 1'1 )^{1} =
x 
1'x1
( 1/n )
where
1 is an n x 1 column
vector
of ones
d is an n x 1 column vector
of deviation scores: d_{1}, d_{2}, . . . ,
d_{n}
x is an n x 1 column vector
of raw scores: x_{1}, x_{2}, . . . ,
x_{n}
To show how this works, let's transform the raw scores in vector
x to deviation scores in vector d.
For this example, let x' = [ 1 2 3 ].
d = 
x 
 
1' 

x 

1 

( 

1' 

1 
)^{1} 
d = 

 
[ 1 1 1 ] 





( 

[ 1 1 1 ] 


)^{1} 
Note that the mean deviation score is zero.
Deviation Scores: Matrices
Let X
be an r x c
matrix
holding raw scores; and let
x be the corresponding r x c matrix
holding deviation scores.
When transforming raw scores from X into deviation
scores for x, we often want to compute deviation
scores separately within columns, consistent with the
equation below.
x_{r}_{c} =
X_{r}_{c}  X_{c}
where
x_{r}_{c} is the deviation score from row r
and column c of matrix x
X_{r}_{c} is the raw score from row r
and column c of matrix X
X_{c}
is the mean score, based on all r scores from
column c of matrix X
To transform the raw scores from matrix X into
deviation scores for matrix x, we use this matrix equation.
x = X 
11'X
( 1'1 )^{1} =
X 
11'X
( 1 / r )
where
1 is an r x 1 column
vector
of ones
x is an r x c matrix
of deviation scores: x_{1}_{1},
x_{1}_{2}, . . . ,
x_{r}_{c}
X is an r x c matrix
of raw scores: X_{1}_{1},
X_{1}_{2}, . . . ,
X_{r}_{c}
Note: Deviation score matrices are often denoted by a lowercase, boldface
letter, such as x. This can cause confusion, since
vectors are also denoted by lowercase, boldface letters; but usually the
meaning is clear from the context.
Test Your Understanding
Problem 1
Consider matrix X.
Using matrix methods, create a 2 x 3 vector D, such that
the elements of D are deviation scores based on elements from
X. That is,
D = 

3  X_{1} 
5  X_{2} 
1  X_{3} 

9  X_{1} 
1  X_{2} 
4  X_{3} 

where X_{c} is the mean of elements
from column c of matrix X.
Solution
To solve this problem, we use the following equation:
D = X 
11'X
( 1 / r ).
Each step in the computation is shown below.
D = 
X 
 
1 

1' 

X 

( 1/r ) 

D =


 


[ 1 1 ] 



( 1/r ) 
Thus, matrix D has the deviation scores, based on raw scores from
matrix X. Note that the mean and sum of each column in matrix
D is zero.