Prof. Candace Clark
Department of Sociology

 

Sociology 240,  Social Statistics

 
   
  

Levels of Measurement

Variables can be measured at different levels of precision.  Various statistics have been invented to deal with
each level of measurement.  In order to choose the proper statistics to examine data, we first have to figure out
at what level each variable is measured.

We decide at which level we think a variable is measured by thinking about its categories.  We try to think of
how the categories are related to each other and what patterns we can find.  Sometimes the categories are
numbers, and sometimes they are words.  Sometimes the categories have an inherent order to them, and
sometimes they do not.
 

                                                   the categories of the variable:
 
Level are names have an inherent order 
from more to less 
or higher to lower		 are numbers with
equal intervals
between them 		 are numbers that
have a theoretical
zero point
Nominal 
level		 X      
Ordinal 
level		 X X    
Interval 
level		 X X X  
Ratio 
level		 X X X X

 

The Nominal Level

The least precise level of measurement is the nominal level.  (The word "nominal" means "in name.")  Examples
of nominal-level variables are Sex (with the categories of male and female), ethnicity (categories could include
African American, Latino, and white), Political Party Identification (Democrat, Republican, Independent, etc.)
and Religion (Catholic, Protestant, Jewish, Hindu, Buddhist, etc.).  The categories of these variables are justnames for the pigeonholes we can create when we classify people by sex, ethnicity, or religion. The categories
don't have any particular order from more to less or higher to lower.  That is, someone in the category Latino
does not have more or less "ethnicity" than an African American or a white, just a different ethnicity.  A
Republican does not have more or less "party identification" than a Democrat.  A Catholic does not have more
or less "religion" than a Protestant or a Jew.  And a woman does not have more or less "sex" than a man.  Since
there is no order inherent in the categories, we treat these variables as nominal-level.  (Later we will learn about
special statistics to examine nominal-level variables.)
 

The Ordinal Level

If we had a variable whose categories did have an order, we might have an ordinal-level variable (assuming the
categories meet none of the additional criteria).  (The word "ordinal" means "in order.")  An example would be
the variable "fear of crime" with categories such as very afraid, somewhat afraid, and not afraid.  These
categories have names (as does a nominal-level variable), but they also have something more.  The categories
have an inherent order from more to less fear.  Another example is "social class," with categories such as
lower class, working class, middle class, and upper class.

Almost any method of measuring attitudes results in ordinal-level variables, even if the variables include only
two categories.  For example, we could categorize the variable "attitudes toward capital punishment" into those
who favor and those who oppose; and those who favor capital punishment hold more favorable attitudes, while
those who oppose hold less favorable attitudes.

Statistics that allow us to analyze ordinal-level variables are different from statistics for nominal-level variables
and for higher-level variables, as we will see later on.  In order to use these more powerful statistics, we might
try to reconceptualize (and rename) a variable so we can consider it to be ordinal-level.

For example, if our job were to categorize the reasons for calls made to 911 emergency dispatchers on a
particular day, we might come up the following categories:  noisy neighbors, fender-bender, heart attack.  One
way to think of these categories is as just names of problems, and the name of the variable could be "Type of
Problem."  In this case, we would be conceiving of our variable as nominal-level.  But another way to think of
the categories is to order them from least severe (noisy neighbors) to most severe (heart attack).  The name of
the variable, as it is being conceived this time, would be "Severity of Problem," and it would be an ordinal-level
variable.
 

The Interval Level

When the categories of a variable are legitimate numbers (not just code numbers attached to named categories,
such as 1 for male and 2 for female), the measurement is more precise than with nominal- and ordinal-level
variables.  If the categories are numbers, they probably have equal intervals between them, as in the figure
below.

___/___/___/___/___/___/___/___/___/___/
0   1    2    3     4    5    6     7    8    9    10

Take "temperature in degrees," the best example of an interval-level variable.  Temperature is measured in
degrees, and the degrees are not words (cold, super-cold, warm, etc.), but numbers corresponding to levels of
mercury in a thermometer.  The distance, or interval, between 1 degree and 2 degrees is exactly equal to the
distance between 2 degrees and 3 degrees, and indeed, between 78 and 79 degrees or 99 and 100 degrees.  The
intervals between any two adjacent categories are equal (exactly 1 degree).  In addition, what makes
temperature in degrees an interval-level measure is that it does not have what it takes to be a ratio-level
measure.  It does not have a theoretical zero point.  Actually, a thermometer does have a zero, but the zero
does not indicate a lack or absence of the variable, temperature.  Zero indicates "cold."  And one method of
measuring temperature (e.g., Fahrenheit) has a different spot for zero than others (e.g., Celsius).  These are
arbitrary zero points that are not intended to indicate a total lack of temperature.  It's really impossible to
imagine a lack of temperature.  With no true zero point, temperature in degrees must be considered only an
interval-level variable.

Almost no variables used in social science are interval-level variables, with the exception of time measured in
calendar years.  The interval between the categories 1902 and 1903 is one year, the same as the interval
between 1766 and 1767 or between 2002 and 2003.  So this variable has equal intervals.  But what about a zero
point?  When did time start?  Can we imagine an absence of time?  Philosophers or astronomers may have
answers for these questions, but in practical terms, there is no zero point.  Hence, time in years would be an
interval-level variable.  But for practical purposes, we will ignore interval-level variables and concentrate on
nominal-, ordinal-, and ratio-level measures.
 

The Ratio Level

The highest level of measurement is the ratio level.  Variables measured at the ratio level have all the
characteristics of nominal-, ordinal-, and interval-level measures (categories that have names, order, and equal
intervals), and the categories include a true zero point.  Even though the sample you are examining may
not include any cases in the category zero, zero is possible at least in theory.  An example is "income in
dollars."  The categories have names (1 dollar; 2 dollars; 500,923 dollars; etc.); the categories follow an order
from less income to more income; the intervals between the categories are equal (1 dollar); and it is possible to
have zero dollars.  Age is another example.  The categories have names (1 year old, 22 years old, etc.); the
categories have an inherent order from youngest to oldest; the intervals between the categories are equal (1
year); and it is possible to be 0 years old.  Other examples are numerous:  number of children, number of pets,
years of education, number of arrests, years on the job, number of marriages, and so on.

After the first three criteria are met, we then determine if the variable has a zero point.  If so, we consider the
variable to be ratio-level.  The zero point makes a ratio-level variable more precise than an interval-level variable.
The zero point means we can sensibly multiply and divide the categories of a ratio-level variable.  For instance,
we can say that someone who has $100 has twice as much income as someone who has $50 and half as much
income as another person who has $200.  With age, a person who is ten years old is twice as old as a five-year-
old and one third as old as someone who is 30.  A person who has one child  has half as many children as those
with two children.  These statements make sense.

But if we tried to do the same thing with a nominal-level variable, we would end up with gibberish.  It would
not make sense to say that a Protestant had twice as much religious affiliation as a Jew, or that a Latino had
three times as much ethnicity as an African-American.  When the categories are merely names, we can attach
code numbers to them; but those code numbers cannot be manipulated mathematically in the same way as the
categories of a ratio-level variable.

The same problem occurs with ordinal-level variables.   It would not make sense to say that an upper-class
person has twice as much social class as a working-class person.  Again, we can attach code numbers to the
categories, but we cannot sensibly multiply and divide the codes.

Even with interval-level variables, we cannot legitimately create ratios or make precise comparisons.  The key
difference between an interval- and a ratio-level variable lies in the zero point.  For this reason, we cannot say
that 100 degrees is twice as warm as 50 degrees or that 20 degrees is half as warm as 40 degrees.  Even though
the categories have equal distances between them, there is no zero point.  Also consider the variable Time,
measured in years.  Various calendars have arbitrarily designated one year or another to be the year 0.  But a
true zero point would indicate the absence of time.  Now, this is a concept even Einstein would have trouble
with!

Note that a having a zero point is not the only criterion that makes a variable ratio-level.  With the variable
"Fear of Crime," there could be people who have no fear.  So, this variable could have a zero point.  But that fact
would not make "Fear of Crime" a ratio-level variable, because the intervals between the categories are not
precise enough to be equal.  Remember that the categories of "Fear of Crime" were:  very afraid, somewhat
afraid, and not afraid.  What are the intervals between these categories?  We cannot say that somewhat afraid is
one "fear unit" above not afraid.  We don't know precisely what the interval or distance between these
categories is.  We only know that one category is higher or lower than the others.  Even if we assign code
numbers to these categories (e.g., 1 = not afraid, 2 = somewhat afraid, and 3 = very afraid), we cannot make
the variable any more precise.  It wouldn't make sense to say, "John's fear of crime is 3" the way we might say,
"John's number of children is 3."  The code numbers we assign to ordinal-level (or nominal-level) variables are
useful for having the computer deal with our data, but we should not make the mistake of assuming that the
intervals between such code numbers are equal.  In sum, to be considered to be at a particular level of
measurement, a variable's categories must meet all the criteria for the lower levels too.

Some statistics require us to make ratios with, multiply, and divide a variable's categories.  These statistics can
only be used with ratio-level variables.  If a variable is interval-level or lower, we need different statistics to
summarize the variable.  The statistics reserved for ratio-level variables are more powerful and yield more
information than statistics for nominal- or ordinal-level variables.  Thus researchers try to measure variables at
the ratio level whenever they can.  For instance, one could measure Education in terms of the categories:  less
than high school, high school only, some college, college degree, and advanced degree.  But the categories of
this variable do not have equal intervals between them.  Thus, it is an ordinal-level measure of education.  If,
however, we asked how many years of school the respondents had completed, we would have categories such
as 0, 1, . . . 11, 12, . . . 16, and so forth.  These categories have an inherent order from less to more education,
the intervals between the categories are equal, and it is possible to have 0 years of school.  We would have a
ratio-level measure of education, which would be amenable to analysis with more powerful statistics.

You can see that just because a variable could be measured at the ratio level does not mean that it has been.
Take, for example, "Family Income."  Researchers could interview a sample of people and ask them to indicate
their annual family income in dollars.  Or, at least, they could try.  It is very unlikely that people really know
exactly what their annual family income is, down to the dollar.  Another problem is that many Americans do
not like to tell people their incomes.  Obtaining a ratio-level measure of family income would be quite difficult.
Therefore, most researchers ask respondents to indicate where their income falls in specified ranges of dollars,
as in this hypothetical example:

               A.   $0 to $19,999
               B.   $20,000 to 39,999
               C.   $40,000 to 59,999
               D.   $60,000 to 79,999
               E.   $80,000 or higher

Given that the researcher has used this set of categories, at what level did s/he measure family income?  Look
carefully at the categories.  They have names (e.g., "$0 to $19,999"), and the categories follow an order from
least income to most income.  But we cannot say the intervals between the categories are equal.  Although
most of them are equal, the last category could include a family earning $80,001, and it could also include the
Microsoft magnate Bill Gates, whose income is in the millions every year.  So the highest level of measurement
at which we can think of this variable, as it is measured here, is ordinal-level.  The categories are ordered from
lowest income to highest, but the intervals between the categories are unequal.

A common practice among statistical analysts is to convert nominal-level variables to what are called dummy
variables so they can be used as if they were ratio-level.  A dummy variable has two categories, one of
which is coded 0 and the other is coded 1.  (Dummy variables are used only as independent variables, not
as dependent variables.)   For instance, with the variable "Sex," instead of coding males as 1 and females as 2 (or
the other way around), we could code males as 0 and females as 1.  Now we have to rethink our conception of
the variable.  It is no longer "sex," but "femaleness."  Males have 0 femaleness.


 

Homework Problems


1.   At what level are the following variables measured?

     F.   Income
               less than $10,000
               $10,000 to 29,999
               $30,000 to 59,999
               $60,000 or more

     G.   Sex  Male, Female

     H.   Attitude toward gun laws
               Very favorable
               Somewhat favorable
               Somewhat unfavorable
               Very unfavorable
               No answer

     I.   Ideal number of children

     J.   Family Income in dollars

     K.   Candidate voted for in 2000 election
               Gore
               Bush
               Other
               Not applicable, didn't vote
               No answer

2.   For each of the following variables, can you think of ways to measure it at the ratio level?
          If not, why not, and what is the highest level at which it can be measured?
          If so, how?

     A.   Fear of becoming a victim of crime in the area around one's home

     B.   Right or left handedness

     C.   Knowledge of statistics

     D.   Attitude toward abortion

     E.   Division of labor in the household

     F.   Defendants' risk of flight from prosecution