or disagree with each of several items. To prevent the subjects from simply checking the same
point on the scale all the way down the page without thinking, we phrase half of our questions
in the positive direction and half in the negative direction. Thus, given a 5-point scale where 5
represents “strongly agree” and 1 represents “strongly disagree,” a 4 on “I hate movies” would
be comparable to a 2 on “I love plays.” If we want the scores to be comparable, we need to
rescore the negative items (for example), converting a 5 to a 1, a 4 to a 2, and so on. This pro-
cedure is called reflectionand is quite simply accomplished by a linear transformation. We
merely write The constant (6) is just the largest value on the scale plus 1. It
should be evident that when we reflect a scale, we also reflect its mean but have no effect on
its variance or standard deviation. This is true by Rule 3 in the preceding list.Standardization
One common linear transformation often employed to rescale data involves subtracting the
mean from each observation. Such transformed observations are called deviation scores,
and the transformation itself is often referred to as centeringbecause we are centering the
mean at 0. Centering is most often used in regression, which is discussed later in the book.
An even more common transformation involves creating deviation scores and then dividing
the deviation scores by the standard deviation. Such scores are called standard scores,and
the process is referred to as standardization.Basically, standardized scores are simply
transformed observations that are measured in standard deviation units. Thus, for example,
a standardized score of 0.75 is a score that is 0.75 standard deviation above the mean; a stan-
dardized score of 2 0.43 is a score that is 0.43 standard deviation below the mean. I will
have much more to say about standardized scores when we consider the normal distribution
in Chapter 3. I mention them here specifically to show that we can compute standardized
scores regardless of whether or not we have a normal distribution (defined in Chapter 3).
People often think of standardized scores as being normally distributed, but there is ab-
solutely no requirement that they be. Standardization is a simple linear transformation of the
raw data, and, as such, does not alter the shape of the distribution.Nonlinear Transformations
Whereas linear transformations are usually used to convert the data to a more meaningful
format—such as expressing them on a scale from 0 to 100, putting them in standardized
form, and so on, nonlinear transformationsare usually invoked to change the shape of a
distribution. As we saw, linear transformations do not change the underlying shape of a dis-
tribution. Nonlinear transformations, on the other hand, can make a skewed distribution
look more symmetric, or vice versa, and can reduce the effects of outliers.
Some nonlinear transformations are so common that we don’t normally think of them
as transformations. Everitt (in Hand, 1994) reported pre- and post-treatment weights for
29 girls receiving cognitive-behavior therapy for anorexia. One logical measure would be
the person’s weight after the intervention (Y). Another would be the gain in weight from
pre- to post-intervention, as measured by (Y– X). A third alternative would be to record the
weight gain as a function of the original score. This would be (Y– X))/Y. We might use this
measure because we assume that how much a person’s score increases is related to how un-
derweight she was to begin with. Figure 2.16 portrays the histograms for these three meas-
ures based on the same data.
From Figure 2.16 you can see that the three alternative measures, the second two of
which are nonlinear transformations of Xand Y, appear to have quite different distributions.
In this case the use of gain scores as a percentage of pretest weight seem to be more nearly
normally distributed than the others. (We will come back to this issue when we come toXnew= 62 Xold.54 Chapter 2 Describing and Exploring Data
deviation scores
centering
standard scores
standardization
reflection
nonlinear
transformations