scale. To a person sitting in a room at 62 8 F, a jump to 64 8 F would be distinctly noticeable
(and welcome). The same cannot be said about the difference between room temperatures
of 92 8 F and 94 8 F. This points up the important fact that it is the underlying variable that
we are measuring (e.g., comfort), not the numbers themselves, that is important in defining
the scale. As a scale of comfort, degrees Fahrenheit do not form an interval scale—they
don’t even form an ordinal scale because comfort would increase with temperature to a
point and would then start to decrease.
There usually is no unanimous agreement concerning the measurement scale em-
ployed, so the individual user of statistical procedures must decide which scale best fits the
data. All that can be asked of the user is that he or she think about the problem carefully
before coming to a decision, rather than simply assuming that the standard answer is nec-
essarily the best answer.The Role of Measurement Scales
I stated earlier that writers disagree about the importance assigned to measurement scales.
Some authors have ignored the problem totally, whereas others have organized whole text-
books around the different scales. A reasonable view (in other words, my view) is that the
central issue is the absolute necessity of separating in our minds the numbers we collect
from the objects or events to which they refer. Such an argument was made for the exam-
ple of room temperature, where the scale (interval or ordinal) depended on whether we
were interested in measuring some physical attribute of temperature or its effect on people
(i.e., comfort). A difference of 2 8 F is the same, physically, anywhere on the scale, but a
difference of 2 8 F when a room is already warm may not feelas large as does a difference
of 2 8 F when a room is relatively cool. In other words, we have an interval scale of the
physical units but no more than an ordinal scale of comfort (again, up to a point).
Because statistical tests use numbers without considering the objects or events to which
those numbers refer, we may carry out any of the standard mathematical operations (addition,
multiplication, etc.) regardless of the nature of the underlying scale. An excellent, entertaining,
and highly recommended paper on this point is one by Lord (1953), entitled “On the Statisti-
cal Treatment of Football Numbers,” in which he argues that these numbers can be treated in
any way you like because, “The numbers do not remember where they came from” (p. 751).
The problem arises when it is time to interpret the results of some form of statistical
manipulation. At that point, we must ask whether the statistical results are related in any
meaningful way to the objects or events in question. Here we are no longer dealing with a
statistical issue, but with a methodological one. No statisticalprocedure can tell us whether
the fact that one group received higher scores than another on an anxiety questionnaire re-
veals anything about group differences in underlying anxiety levels. Moreover, to be satis-
fied because the questionnaire provides a ratio scale of anxiety scores(a score of 50 is
twice as large as a score of 25) is to lose sight of the fact that we set out to measure anxi-
ety, which may not increase in an orderly way with increases in scores. Our statistical tests
can apply only to the numbers that we obtain, and the validity of statements about the ob-
jects or events that we think we are measuring hinges primarily on our knowledge of those
objects or events, not on the measurement scale. We do our best to ensure that our meas-
ures relate as closely as possible to what we want to measure, but our results are ultimately
only the numbers we obtain and our faith in the relationship between those numbers and
the underlying objects or events.^48 Chapter 1 Basic Concepts
(^4) As Cohen (1965) has pointed out, “Thurstone once said that in psychology we measure men by their shadows.
Indeed, in clinical psychology we often measure men by their shadows while they are dancing in a ballroom
illuminated by the reflections of an old-fashioned revolving polyhedral mirror” (p. 102).