III. Levels of Measurement
III. LEVELS OF MEASUREMENT
Subtopics
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Introduction
Nominal Data
Ordinal Data
Interval Data
Ratio Data
Dichotomous
Variables
Why It Matters
Other Ways to
Classify Data
Key Concepts
Exercises
For Further Study
SPSS Tools
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New with this topic
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Getting Started
Introduction
When data are prepared for analysis by computer, values of variables are usually entered as
numbers. Sometimes such coding is natural — for example, the population of a country or the
number of votes received by a candidate. Sometimes, artificial numerical codes are created for
convenience of processing. In a file containing data on members of the U.S. Congress, for
example, Democrats might be coded numerically as 1, Republicans as 2, and independents as 3.
Numerical, however, is not the same thing as quantitative. In fact, whether data are coded
numerically or not, there are different levels of measurement.
Nominal Data
The values of a nominal variable do not indicate the amount of the thing being measured, nor are
they in any particular order. If coded numerically, the numbers chosen are arbitrary. For
example, if we list the regions of the United States as Northeast, South, Midwest, and West, we
are not indicating the amount of "regionness" each possesses, nor listing them in order of
"regionness." We may code the regions as "1," "2," "3," and "4" respectively, but this is done
merely for convenience and in no way quantifies what we are doing. Each value, numerical or
otherwise, is merely a label or name (hence the term "nominal").
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III. Levels of Measurement
Ordinal Data
Sometimes the values of a variable are listed in order. (Alternatively, we say that the values are
"ranked" or "rank ordered.") For example, the army orders (ranks) military personnel from
general to private. At a college or university, class standing of undergraduates (freshman to
senior) is another example of an ordinal variable. In both of these examples, the values of the
variable in question (military rank or class standing) are ranked from highest to lowest or vice
versa. There are other kinds of ordering. For example, respondents in a survey may be asked to
identify their political philosophy as "very liberal," "liberal," "moderate," "conservative," or
"very conservative," creating a scale rank ordered from most liberal to most conservative.
Interval Data
Sometimes, in addition to being ordered, the differences (or intervals) between any two adjacent
values on a measurement scale are the same. For example, the difference in temperature
between 80 degrees Fahrenheit and 81 degrees is the same as that between 90 degrees and 91
degrees. When each interval represents the same increment of the thing being measured, the
measure is called an interval variable.
Ratio Data
Finally, in addition to having equal intervals, some measures also have an absolute zero point.
That is, zero represents the absence of the thing being measured. Height and weight are obvious
examples. Physicists sometimes use the Kelvin temperature scale, in which zero means the
complete absence of energy. The same is not true of the Fahrenheit or Celsius (Centigrade)
scales. Zero degrees Celsius, for example, represents the freezing point of water at sea level, but
this does not mean that there is no temperature at this point. The choice to put zero degrees at
this point on the scale is arbitrary. There is no particular reason why scientists could not have
chosen instead the freezing point of beer in Golden, Colorado (other than that water is a more
common substance, at least for most successful scientists). With an absolute zero point, you can
calculate ratios (hence the name). For example, $20 is twice as much as $10, but 60 degrees
Fahrenheit is not really twice as hot as 30 degrees. Ratio data is fully quantitative: it tells us the
amount of the variable being measured. The percentage of votes received by a candidate, Gross
Domestic Product per Capita, and felonies per 100,000 population are all ratio variables.
Dichotomous Variables
Dichotomous variables (those with only two values) are a special case, and may sometimes be
treated as nominal, ordinal, or interval. Take, for example, political party affiliation in a two21
III. Levels of Measurement
party legislature. Party is, on its face, a pure example of a nominal variable, with the values of
the variable being simply the names of the parties (or arbitrary numbers used, for convenience, in
place of the names). On the other hand, we could treat party (and other dichotomous variables)
as ordinal, since there are only two possible ways for the values to be ordered, and it makes no
difference which way is chosen. There is, therefore, no way that they can be listed out of order.
For certain purposes, we can even treat dichotomous variables as interval, since there is only one
interval (the difference between Party A and Party B), which is obviously equal to itself.[1]
Why It Matters
Level of measurement is important because the higher the level of measurement of a variable
(note that "level of measurement" is itself an ordinal measure) the more powerful are the
statistical techniques that can be used to analyze it. With nominal data, you can count the
frequency with which each value of a variable occurs. A voter's choice in the 2008 presidential
race, for example, is a nominal variable (with the values of the variable being "McCain,
"Obama," “Nader," "Barr," "McKinney," etc.), and so you can count the number of votes
received by each candidate. You can also calculate the percentage of votes each candidate
received. You can calculate joint frequencies and percentages (how many and what percent of
votes were cast by Midwestern Obama supporters, for example). You can also use certain
measures that tell you how strong the relationship is between region and vote, and the likelihood
that the relationship occurred by chance.
On the other hand, there are other operations you cannot legitimately perform with nominal data.
Even if you use numbers to label candidates (e.g., 1 = Obama, 2 = McCain, 3 = Nader, etc.), you
cannot very well say that Obama plus McCain equals Nader, or that Nader divided by McCain is
half way between Obama and McCain. Unfortunately, there are many statistical techniques that
require higher levels of measurement.
With ordinal data, you can employ techniques that take into account the fact that the values of a
variable are listed in a meaningful order. With interval data, you can go even further and use
powerful techniques that assume a measurement scale of equal intervals. As it happens, there are
very few techniques in the social sciences that require ratio data, and so some textbooks ignore
the distinction between interval and ratio scales.
If you use a technique that assumes a higher level of measurement than is appropriate for your
data, you risk getting a meaningless answer. On the other hand, if you use a technique that fails
to take advantage of a higher level of measurement, you may overlook important things about
your data. (Note: in addition to level of measurement, many statistical techniques also require
other assumptions about your data. For example, even if a variable is interval, some otherwise
appropriate techniques may yield misleading results if the variable includes some values that are
extremely high or low relative to the rest of the distribution.)
The distinctions between levels of measurement are not always hard and fast.
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III. Levels of Measurement
Sometimes it depends on the underlying concept being measured. This applies, for example, to
the question of whether to treat a dichotomous variable as nominal or ordinal. Do our values
indicate two distinct categories (e.g., male and female), or do we think of them as two points
along a spectrum (e.g., for or against capital punishment, since some people may favor or oppose
capital punishment more strongly than others)?
In survey research, independents are often thought of as being somewhere in between Democrats
and Republicans, and so measures of party identification are usually treated as ordinal. On the
other hand, if you were studying the U.S. Senate, you would find that the only independents
currently serving (as of the 112th Congress, 2011-2012) are Bernie Sanders of Vermont and
Joseph Lieberman of Connecticut. While Lieberman might in some senses be considered
"between" the Republican and Democratic parties, the same could hardly be said of Sanders, one
of the most liberal members of the chamber. (Before coming to congress, he had run as a
Socialist in winning election as mayor of Bennington, Vermont.)
Sometimes the question of level of measurement hinges on the precise nature of the measure
itself. For example, the American National Election Study has for many years been using
"feeling thermometers." Respondents are asked to locate a person (e.g., a presidential candidate)
or a category of people (Democrats, Republicans, feminists, evangelical Christians, Latinos, etc.)
on a scale ranging from 0 to 100, with higher numbers representing warmer feelings toward the
person or category of people in question. Most researchers using these variables have treated
them as interval. Some, however, have raised doubts about this practice. For example, does the
difference between a rating of, say, 60 and 70 really mean the same thing as the difference
between 90 and 100?
In designing research, there can be tradeoffs between having data that are at a higher level of
measurement and other considerations. Aggregate data (data about groups of people) are
generally interval or ratio, but usually provide only indirect measures of how people think and
act. Individual data get at these things more directly, but are usually only nominal or ordinal.
Official election returns, for example, can provide us with ratio level data about the distribution
of votes in each precinct. These data, however, tell us little about why individual people vote the
way they do. Survey research (public opinion polling), which provides data that are for the most
part only nominal or ordinal, allows us to explore such questions much more extensively and
directly.
Other Ways to Classify Data
Sometimes you will find other terms used to describe the level of measurement of variables.
SPSS, for example, distinguishes among nominal, ordinal, and scale (that is, interval or ratio)
variables. Some texts distinguish between nonparametric (nominal or ordinal) and parametric
(interval or ratio) variables. In describing different statistical procedures, we will sometimes
distinguish between categorical and continuous variables. Categorical variables generally
consist of a small number of values, or categories, and are usually nominal or ordinal. The
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III. Levels of Measurement
values of continuous variables represent a large or even infinite number of possible points along
a scale, and are interval or ratio.
Key Concepts
categorical variable
continuous variable
dichotomous variable
interval variable
levels of measurement
nominal variable
nonparametric variable
ordinal variable
parametric variable
ratio variable
scale variable
Exercises
1,
Start SPSS, and open anes08s.sav. In Variable View, notice that SPSS uses three
categories of measurement: nominal, ordinal, and scale (equivalent to interval and ratio). Notice
also that almost all of the variables are either nominal or ordinal. This is typical with data on
individuals, such as survey data. Now open countries.sav and do the same. Notice that almost
all of the variables are scale, as is usually the case with aggregate data.
With each of the datasets included in this project, care has been taken to correctly categorize the
level of measurement of variables. Remember, however, that the level of measurement of some
variables may depend on how the variable is used. Also, when using datasets other than those
provided with POWERMUTT, do not assume without checking that the author has bothered to
verify each variable's level of measurement.
2.
Open the codebooks for the other datasets included in this project. Classify each variable
as nominal, ordinal, interval, or ratio. Check your answers by opening the dataset in SPSS and
examining the Variable View (noting again that SPSS uses the term “scale” for both interval and
ratio data). (In some cases, there may be more than one correct answer, depending on what
assumptions are made.)
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III. Levels of Measurement
For Further Study
Lane, David, et al. “Levels of Measurement,” Online Statistics: A Multimedia Course of Study.
http://onlinestatbook.com/chapter1/levels_of_measurement.html.
University of Cambridge. "Levels of Measurement," Universities' Collaboration in eLearning.
http://www.ucel.ac.uk/showroom/levels_of_measurement/Default.html.
[1] See the section on "dummy" variables under the regression analysis Topic.
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