Information Technology Reference
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In any of these cases, you must reverse the scale before averaging these percentages with
other percentages where higher numbers are better. For example, with the rating scale
just shown, you would subtract each value from 6 (the maximum) to reverse the scale.
So 0 becomes 6 and 6 becomes 0.
8.1.3 Combining Metrics Based on Z Scores
Another technique for transforming scores on different scales so that they can be
combined is using z scores. (See, for example, Martin & Bateson, 1993, p. 124.)
These are based on the normal distribution and indicate how many units above
or below the mean of the distribution any given value is. When you transform a
set of scores to their corresponding z scores, the resulting distribution by defini-
tion has a mean of 0 and standard deviation of 1. This is the formula for trans-
forming any raw score to its corresponding z score:
z
=−
(
µσ
)
/
,
where x is the score to be transformed, μ is the mean of the distribution of those
scores, and σ is the standard deviation of the distribution of those scores.
This transformation can also be done using the “=STANDARDIZE” function
in Excel. Data in Table 8.2 could also be transformed using z scores, as shown
in Tab le 8.7 .
Table 8.7 Sample data from Table 8.2 transformed using z scores a .
Time
Per Task
(sec)
Tasks
Completed
(of 15)
z
Time*
( 1)
Rating
(0-4)
z
Tasks
z
Rating
Participant #
z Time
Average
1
65
7
2.4
0.98
0.98
0.91
0.46
0.78
2
50
9
2.6
0.02
0.02
0.05
0.20
0.06
3
34
13
3.1
1.01
1.01
1.97
0.43
1.14
4
70
6
1.7
1.30
1.30
1.39
1.35
1.35
5
28
11
3.2
1.39
1.39
1.01
0.56
0.99
6
52
9
3.3
0.15
0.15
0.05
0.69
0.20
7
58
8
2.5
0.53
0.53
0.43
0.33
0.43
8
60
7
1.4
0.66
0.66
0.91
1.73
1.10
9
25
9
3.8
1.59
1.59
0.05
1.32
0.98
10
55
10
3.6
0.34
0.34
0.53
1.07
0.42
Mean
0.0
0.0
0.0
0.00
0.00
Standard
deviation 1.0 1.0 1.0 1.00 0.90
a For each original score, the z score was determined by subtracting the mean of the score's distribution from it and then dividing by the standard
deviation. This z score tells you how many standard deviations above or below the mean that score is. Since you need all the scales to have
higher numbers better, the scale of the z scores of times is reversed by multiplying by (-1).
Table 8.7 Sample data from Table 8.2 transformed using z scores a .
a For each original score, the z score was determined by subtracting the mean of the score's distribution from it and then dividing by the standard
deviation. This z score tells you how many standard deviations above or below the mean that score is. Since you need all the scales to have
higher numbers better, the scale of the z scores of times is reversed by multiplying by (-1).
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