Deviation Score

The deviation score is the difference between a score in a distribution and the mean score of that distribution. The formula for calculating the deviation score is as follows:

where

X(called “X bar”) is the mean value of the group of scores, or the mean; and the X is each individual score in the group of scores.

Deviation scores are computed most often for the entire distribution. For example, for the following data set (see Table 1), there are columns representing scores on the variables X and Y for 10 observations. The deviation scores for X and Y have also been calculated. Notice that the means of the deviation score distributions are zero.

Thus, the deviation scores are simply a linear transformation of a variable. This can be demonstrated by calculating the Pearson correlations between X and Y and then between the deviation-X and deviation-Y scores. In both instances, the correlations are 0.866.

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Table 1 Raw and Deviation Scores on Two Variables, X and Y
Observation	X	Y	X − 4.8	Y − 4.2
1	2	1	−28	−3.2
2	3	4-1	8	−0.2
3	4	3	−0.8	−1.2
4	7	5	2.2	0.8
5	8	6	3.2	1.8
6	9	8	4.2	3.8
7	2	3	−2.8	−1.2
8	3	3	−1.8	−1.2
9	4	2	−0.8	−2.2
10	6	7	1.2	2.8
X¯;	X¯;= 4.8 X	X¯;= 4.2 X	X¯;= 0.0 X	X¯;= 0.0

Table 2 Raw Scores and Interaction Terms for Nondeviation and Deviation Scores
Observation	X	Y	(X)(Y)	X − 4.8	Y − 4.2	(X − 4.8)(Y-4.2)
1	2	1	2	−2.8	−3.2	8.96
2	3	4	12	−1.8	−0.2	0.36
3	4	3	12	−0.8	−1.2	0.96
4	7	5	35	2.2	0.8	1.76
5	8	6	48	3.2	1.8	5.76
6	9	8	72	4.2	3.8	15.96
7	2	3	6	−2.8	−1.2	3.36
8	3	3	9	−1.8	−1.2	2.16
9	4	2	8	−0.8	−2.2	1.76
10	6	7	42	1.2	2.8	3.36

The next question one might want to ask is, Why would one want to calculate such scores? The most frequent use of deviation scores is in conducting simultaneous solution regression analyses when there is an interest in the effects of interaction terms.

For example, assume one wants to predict a criterion (Z) with two main effects, X and Y, as well as their interaction. The interaction term is generated by multiplying X and Y, but this interaction term exhibits multicollinearity with each of the main effects, X and Y. However, if the interaction term is created from the deviation scores of X and Y, the multicollinearity no longer is a problem.

To demonstrate this, the data set shown earlier is used (see Table 2). The interaction terms have been generated for each score. The correlations between the nondeviation interaction, (X)(Y), and the main effects are .955 with X and .945 with Y. The correlations between the deviation interaction (X – 4.8)(Y – 4.2) and the main effects are .479 with X and .428 with Y. This feature of deviation scores is of immense utility when conducting simultaneous linear regression-based analyses (such as multiple regression, discriminant function analysis, logistic regression, and structural equation modeling).

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