Define and distinguish among nominal, ordinal, interval, and ratio
scales

Identify a scale type

Discuss the type of scale used in psychological measurement

Give examples of errors that can be made by failing to understand the
proper use of measurement scales

Types of Scales

Before we can conduct a statistical analysis,
we need to measure our dependent
variable. Exactly how the measurement is carried out depends
on the type of variable involved in the analysis. Different types
are measured differently. To measure the time taken to respond
to a stimulus, you might use a stop watch. Stop watches are of
no use, of course, when it comes to measuring someone's attitude
towards a political candidate. A rating scale is more appropriate
in this case (with labels like "very favorable," "somewhat
favorable," etc.). For a dependent variable such as "favorite
color," you can simply note the color-word (like "red")
that the subject offers.

Although procedures for measurement differ in many
ways, they can be classified using a few fundamental categories.
In a given category, all of the procedures share some properties
that are important for you to know about. The categories are
called "scale types," or just "scales," and
are described in this section.

Nominal scales

When measuring using a nominal scale, one simply
names or categorizes responses. Gender, handedness, favorite color,
and religion are examples of variables measured on a nominal scale.
The essential point about nominal scales is that they do not imply
any ordering among the responses. For example, when classifying
people according to their favorite color, there is no sense in
which green is placed "ahead of" blue. Responses are
merely categorized. Nominal scales embody the lowest level of
measurement.

Ordinal scales

A researcher wishing to measure consumers'
satisfaction with their microwave ovens might ask them to specify
their feelings as either "very dissatisfied," "somewhat
dissatisfied," "somewhat satisfied," or "very
satisfied." The items in this scale are ordered, ranging
from least to most satisfied. This is what distinguishes
ordinal from nominal scales. Unlike nominal scales, ordinal
scales allow comparisons of the degree to which two
subjects possess the dependent variable. For example, our
satisfaction ordering makes it meaningful to assert that one
person is more satisfied than another with their microwave
ovens. Such an assertion reflects the first person's use of
a verbal label that comes later in the list than the label
chosen by the second person.

On the other hand, ordinal scales fail to capture
important information that will be present in the other scales
we examine. In particular, the difference between two levels
of an ordinal scale cannot be assumed to be the same as the
difference between two other levels. In our satisfaction scale,
for example, the difference between the responses "very
dissatisfied"
and "somewhat dissatisfied" is probably not
equivalent to the difference between "somewhat dissatisfied" and "somewhat
satisfied." Nothing in our measurement procedure allows
us to determine whether the two differences reflect the same
difference in psychological satisfaction. Statisticians express
this point by saying that the differences between adjacent
scale values do not necessarily represent equal intervals on
the underlying scale giving rise to the measurements. (In our
case, the underlying scale is the true feeling of satisfaction,
which we are trying to measure.)

What if the researcher had measured satisfaction
by asking consumers to indicate their level of satisfaction by
choosing a number from one to four? Would the difference between
the responses of one and two necessarily reflect the same difference
in satisfaction as the difference between the responses two and
three? The answer is No. Changing the response format to numbers
does not change the meaning of the scale. We still are in no position
to assert that the mental step from 1 to 2 (for example) is the
same as the mental step from 3 to 4.

Interval scales

Interval scales are numerical scales in which
intervals have the same interpretation throughout. As an example,
consider the Fahrenheit scale of temperature. The difference between
30 degrees and 40 degrees represents the same temperature difference
as the difference between 80 degrees and 90 degrees. This is because
each 10-degree interval has the same physical meaning (in terms
of the kinetic energy of molecules).

Interval scales are not perfect, however. In particular,
they do not have a true zero point even if one of the scaled values
happens to carry the name "zero." The Fahrenheit scale
illustrates the issue. Zero degrees Fahrenheit does not represent
the complete absence of temperature (the absence of any molecular
kinetic energy). In reality, the label "zero" is applied
to its temperature for quite accidental reasons connected to the
history of temperature measurement. Since an interval scale has
no true zero point, it does not make sense to compute ratios of
temperatures. For example, there is no sense in which the ratio
of 40 to 20 degrees Fahrenheit is the same as the ratio of 100
to 50 degrees; no interesting physical property is preserved across
the two ratios. After all, if the "zero" label were
applied at the temperature that Fahrenheit happens to label as
10 degrees, the two ratios would instead be 30 to 10 and 90 to
40, no longer the same! For this reason, it does not make sense
to say that 80 degrees is "twice as hot" as 40 degrees.
Such a claim would depend on an arbitrary decision about where
to "start" the temperature scale, namely, what temperature
to call zero (whereas the claim is intended to make a more fundamental
assertion about the underlying physical reality).

Ratio scales

The ratio scale of measurement is the most informative
scale. It is an interval scale with the additional property that
its zero position indicates the absence of the quantity being
measured. You can think of a ratio scale as the three earlier
scales rolled up in one. Like a nominal scale, it provides a name
or category for each object (the numbers serve as labels). Like
an ordinal scale, the objects are ordered (in terms of the ordering
of the numbers). Like an interval scale, the same difference at
two places on the scale has the same meaning. And in addition,
the same ratio at two places on the scale also carries the same
meaning.

The Fahrenheit scale for temperature has an arbitrary zero point and is therefore not a ratio scale. However, zero on the Kelvin scale is absolute zero. This makes the Kelvin scale a ratio scale. For example, if one temperature is twice as high as another as measured on the Kelvin scale, then it has twice the kinetic energy of the other temperature.

Another example of a ratio scale is the amount of money
you have in your pocket right now (25 cents, 55 cents, etc.). Money
is measured on a ratio scale because, in addition to having the
properties of an interval scale, it has a true zero point: if
you have zero money, this implies the absence of money. Since
money has a true zero point, it makes sense to say that someone
with 50 cents has twice as much money as someone with 25 cents
(or that Bill Gates has a million times more money than you do).

What level of measurement is used for psychological
variables?

Rating scales are used frequently in psychological
research. For example, experimental subjects may be asked to rate
their level of pain, how much they like a consumer product, their
attitudes about capital punishment, their confidence in an answer
to a test question. Typically these ratings are made on a 5-point
or a 7-point scale. These scales are ordinal scales since there
is no assurance that a given difference represents the same thing
across the range of the scale. For example, there is no way to
be sure that a treatment that reduces pain from a rated pain level
of 3 to a rated pain level of 2 represents the same level of relief
as a treatment that reduces pain from a rated pain level of 7
to a rated pain level of 6.

In memory experiments, the dependent variable
is often the number of items correctly recalled. What scale of
measurement is this? You could reasonably argue that it is a ratio
scale. First, there is a true zero point: some subjects may get
no items correct at all. Moreover, a difference of one represents
a difference of one item recalled across the entire scale. It
is certainly valid to say that someone who recalled 12 items recalled
twice as many items as someone who recalled only 6 items.

But number-of-items recalled is a more complicated
case than it appears at first. Consider the following example
in which subjects are asked to remember as many items as possible
from a list of 10. Assume that (a) there are 5 easy items and
5 difficult items, (b) half of the subjects are able to recall
all the easy items and different numbers of difficult items, while
(c) the other half of the subjects are unable to recall any of
the difficult items but they do remember different numbers of easy items.
Some sample data are shown below.

Subject

Easy Items

Difficult Items

Score

A

0

0

1

1

0

0

0

0

0

0

2

B

1

0

1

1

0

0

0

0

0

0

3

C

1

1

1

1

1

1

1

0

0

0

7

D

1

1

1

1

1

0

1

1

0

1

8

Let's compare (1) the difference between Subject A's score of
2 and Subject B's score of 3 with (2) the difference between
Subject C's score of 7 and Subject D's score of 8. The former
difference is a difference of one easy item; the latter difference
is a difference of one difficult item. Do these two differences necessarily
signify the same difference in memory? We are
inclined to respond "No" to this question since only a little
more memory may be needed to retain the additional easy item
whereas a lot more memory may be needed to retain the additional
hard item. The general point is that it is often inappropriate
to consider psychological measurement scales as either interval
or ratio.

Consequences of level of measurement

Why are we so interested in the type of scale
that measures a dependent variable? The crux of the matter is
the relationship between the variable's level of measurement and
the statistics that can be meaningfully computed with that variable.
For example, consider a hypothetical study in which 5 children
are asked to choose their favorite color from blue, red, yellow,
green, and purple. The researcher codes the results as follows:

Color

Code

Blue
Red
Yellow
Green
Purple

1
2
3
4
5

This means that if a child said her favorite color was "Red,"
then the choice was coded as "2," if the child said
her favorite color was "Purple," then the response was
coded as 5, and so forth. Consider the following hypothetical
data:

Subject

Color

Code

1
2
3
4
5

Blue
Blue
Green
Green
Purple

1
1
4
4
5

Each code is a number, so nothing prevents us
from computing the average code assigned to the children. The
average happens to be 3, but you can see that it would be senseless
to conclude that the average favorite color is yellow (the color
with a code of 3). Such nonsense arises because favorite color
is a nominal scale, and taking the average of its numerical labels
is like counting the number of letters in the name of a snake
to see how long the beast is.

Does it make sense to compute the mean of numbers
measured on an ordinal scale? This is a difficult question, one
that statisticians have debated for decades. You will be able
to explore this issue yourself in a simulation shown in the next
section and reach your own conclusion. The prevailing (but by
no means unanimous) opinion of statisticians is that for almost
all practical situations, the mean of an ordinally-measured variable
is a meaningful statistic. However, as you will see in the simulation,
there are extreme situations in which computing the mean of an
ordinally-measured variable can be very misleading.