2typesofvariables.pdf Michael Hallstone, Ph.D.

hallston@hawaii.edu

Lecture 2: Types of Variables

Recap what we talked about last time

Recall how we study social world using populations and samples.

Recall that when we study the social world using statistics we need to define characteristics that we
wish to study so that they can be expressed using numbers. And when we do this we “code”
variables.

Types of variables
There are two ways to classify variables that will be important to us in this course. One is to decide
whether a variable is continuous or discrete and the other is to decide whether a variable is nominal,
ordinal, interval, or ratio.

continuous vs. discrete

Some of the statistical tests assume the variable is continuous or discrete. For example, the theory
behind a test called Chi-Square requires discrete variables. Therefore if the test you use assumes a
discrete variable you must use one, or you are violating a key assumption of the test. This is why we
need to learn to distinguish between continuous or discrete.

discrete variables
Consider the definition of a discrete variable first. A variable that is discrete is coded with “either/or”
categories where there is NOTHING “in between” the categories. There is nothing in-between these
categories. You are in one or the other but not both.

For example, imagine the variable T-shirt size: 1= small 2= medium 3=large.

There is no way to be in between categories in this variable, so it is discrete. You either wear a small
or a medium or a large and there is nothing in between.

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Any variable with a yes or no answer is discrete. Have you ever been convicted of a crime? 1= yes
0=no. The answer is either yes or no and there is nothing in between. Did you go to the doctor in the
past 30 days? 1= yes 0=no. The answer is either yes or no and there is nothing in between.

Any variable where the categories are “either/or” is discrete. Who is your health insurance provider?
1=HMSA 2= Kaiser 3=other 4= uninsured. You choose one of the four and there is nothing in
between.

To use a public administration example, pretend you had a variable the measured under which
branch of government a person’s job in the state government was organized. (Generally speaking,
there are three main branches of state or federal government under which all departments are
organized: executive, judiciary, and legislature). So pretend there was a variable branch: 1= executive
2= judiciary 3=legislative. Now pretend you had a respondent who worked in the prison system for the
State of Hawaii (in Hawaii the prisons are run by the Department of Public Safety that falls under the
executive branch of government). So that person would be coded 1= executive. The important thing is
these categories are discrete. There is nothing in-between these categories. You are in one or the
other but not both.

continuous variables
Continuous variables exist on a continuum, thus the term “continuous.” There are NOT discrete
categories in a continuous variable, but are an infinite number of responses along a number line.

So pretend you had a variable that measured length of longest fish ever caught ______(fill in the
blank). If you think about it, if you had a precise enough measuring instrument there are an infinite
number of possible responses between 68 and 69 inches. It is possible to have caught a fish that was
68.25 inches or 68.251 inches or 68.2512 inches, etc. etc. on and on and on.

Many variables that “code naturally” can be considered continuous variables: age _____, height____,
weight_____, years worked in your current department ________, etc.

Some times we “pretend” it is continuous

Now to complicate matters a bit. Take the variable age for example. Even if you measured a person’s
age to the second, the distance between seconds is, theoretically discrete. Or if you measured a
person’s age in whole years the distance between years is discrete. However, in practice it okay to

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use a variable age in whole years when using some tests that are designed for continuous variables.
(This is because in theory the variable is continuous, even though in this case the data collected was
collected in a discrete manner).

On a test I will make it very clear – with this one “trick” question

Bubble test questions will mostly look like this

Kelly measures the exact length of a person's surfboard. He uses a really sophisticated measuring device
that allows very precise measurement. He will not use either/or categories, but will measure them exactly.
Classify the variable. Here I am giving you the hint that the correct answer is continuous because he
does NOT use either/or categories.
Kelly measures many years a person has been a surfer. He collects the data in whole years. Classify the
variable. Here I am giving you the hint that the correct answer is discrete because he collects in
whole years.
Blu will measure how long prisoners have been incarcerated. Choose the best statement below regarding
this variable. There will be answers that say “this variable must be continuous” and this variable must be
“discrete” but the correct answer is Depending upon how he collects the information, it could be either
continuous or discrete.

Test yourself to see if you REALLY understand the concept continuous vs.
discrete

This is a bit theoretical, but it will create a self-test and increase your critical thinking skills. Classify
these variables as continuous or discrete:

number of deaths in US prisons last year _________ (insert #)

number of people treated in the Emergency Room in Queen’s Hospital last year ________ (insert #)

number of felony convictions in your life _________ (insert #)

At first glance all of these might appear to be continuous as they are number variables that “naturally
code.” And many times natural coding number-type variables are continuous in theory, but THESE
ARE NOT. Each of these are discrete as there is no such thing as “half a death” or “half a person
treated” or “half a felony conviction.” You should know this concept for the test, because I want to see
if you “really” understand the concept.

That being said, each of these variables could be used on most tests in this course that are designed
for continuous variables.

The really important way to classify variables for this course is next.

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Is the variable nominal, ordinal, interval, or ratio?
All of the statistical tests you will learn in this course will require you to classify variable into one of 4
categories nominal, ordinal, interval, or ratio. So for example, some statistical tests you learn are
designed for nominal level variables, for example something called chi-square. So when you do a
chi-square test you MUST use a nominal variable or you are violating a key assumption of this test
and your results will be, essentially, meaningless.

Thus, it is very important that you be able to decide whether a variable is either nominal, ordinal,
interval, or ratio.

Most important of all is whether or not the variable is fit into one of 3 categories: nominal, ordinal,
interval/ratio. (As you will see later in the course many of the statistical tests are designed for interval
or ratio level variables, for example, a t-test or a hypothesis test of means requires a variable that is at
least interval/ratio.) By the way, the statistical software program SPSS uses only three categories:
nominal, ordinal, and “scale” where scale includes interval and ratio level variables.

By the way, some books will simplify these four categories into something like “categorical” or
“attribute” and “numerical,” but let’s learn the “real way” instead. If you go on to use statistics later,
learning this way will help. (Essentially, it appears that nominal and ordinal are attribute and interval
and ratio are numerical.)

NOIR spells “black” in French!

The word NOIR is black in French and that allows you to keep these in their proper order. As we go
from Nominal to Ratio, the variables are getting more complex. You must keep the categories
nominal, ordinal, interval, or ratio in that exact order to understand how to classify these variables.

N - nominal
O - ordinal
I - interval
R -ratio

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N - nominal
A nominal level variable has categories only. There is no dimension or numbering to the categories.
The numbers assigned to each category are MEANINGLESS and are just a grouping procedure.

Consider the following variables, they are all nominal.

Gender: 1=male 2=female

Ethnicity: 1=Haole 2=Polynesian 3= Asian 4= other
Favorite plate lunch: 1=chicken katsu 2=mixed plate 3=kalbi ribs 4=other
Have you ever surfed: 1=yes 0=no

We assign numbers to each category, but the numbers do not mean anything. The numbers are
meaningless. You could swap the numbers any way you want and it would not matter. 1 is not “less”
ethnicity than 2 and vise versa. 1 is not “less gender” than 2. Any variable with a “yes/no” answer is
nominal.

O - ordinal
The numbers have an order! The order is meaningful. The numbers mean something. We have the
beginnings of a dimension but no common units of measurement -- no way exactly measure distance
between categories. We have an order but they are not real numbers (because there are not
common units to the numbers you cannot “do math” on the numbers”).

Consider the variable: t-shirt size: 1=small 2=medium 3=large 4=extra large.

A large t-shirt (3) is NOT ACTUALLY three times larger than a small (1) t-shirt is it? But the number 3
is actually “3 times the size of the number 1.” An extra large (4) is NOT ACTUALLY twice as large as
a medium (2), but the number 4 is actually twice a big as the number 2.

The professor explains things in an easy-to-understand manner.

1=strongly disagree
2=disagree
3=neutral
4=agree
5=strongly agree

As you “go up” in numbers you do have “more agreement” although you cannot say that strongly
agree (5)is five times as much agreement as strongly disagree (1).

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I - interval
An interval variable has categories that are “real numbers” with common units of measurement, but
there is NO TRUE ZERO. The zero does not indicate “a complete absence” of something. There are
not lot of truly interval variables. Two I know of are temperature in degrees (it does not matter
whether or not it is measured in Fahrenheit or Celsius scale) and calendar year (in relation to the birth
of Christ).

Temperature: _____ (insert # of degrees).

Year you were born:__________

For both variables, you have numbers and they are real numbers. For example 40 degrees is half the
temperature of 80 degrees. 81 degrees is one more degree than 80 degrees. The year 2000 is twice
as many years as 1000. You could do math on these numbers. You have real numbers however
there is no “true zero.” Zero degrees (F or C) does NOT indicate a “lack” or an “absence” of
temperature. Temperature will be on the test! As I say in class, “When you see temperature you say
interval.”

Years are also measured in exact intervals, meaning each year is exactly the same size. But the year
zero is not an absence of years. It is merely marking the year of Christ’s birth.

You could also take a ratio variable and recode it into categories that were exactly the same size and
you could turn a ratio variable into an interval variable. So you could recode age into five year
intervals and you’d have an interval variable. But age is not an inherently interval variable.

The fact that there are very few actual interval level variables is why, in practice, for many statistical
tests we simply need an interval or ratio level variable.

R -ratio
A ratio level variable is the same as an interval variable, but they have a “true zero.” You have real
numbers with real intervals and a zero means “a lack of” or an “absence of” what ever you are
measuring.

# of cars you own ________

Amount of money in your wallet right now_____
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# of felony convictions in your life
# of children you have ____
# of times you have been surfing ____
# of times you have jumped out of an airplane ____

In each of these cases if you answer 0, that means a complete absence of that thing. If you own zero
cars, that’s an absence of cars, if you have no money, that is an absence of money, if you have no
children that is an absence of children, and so on.

Both interval and ratio are “real numbers”

So you can see that both interval and ratio level variables are composed of “real numbers.” You can
“do math” on real numbers. This is why earlier I wrote, “it is most important to know whether or not the
variable is fit into one of 3 categories: nominal, ordinal, interval/ratio. (As you will see later in the
course many of the statistical tests are designed for interval or ratio level variables, for example, a t-
test or a hypothesis test of means requires a variable that is at least interval/ratio.) By the way, the
statistical software program SPSS uses only three categories: nominal, ordinal, and “scale” where
scale includes interval and ratio level variables.”

“Dummy” variables
“Dummy” nominal variables to pretend they are “ratio”
Sometimes we can code a nominal (or ordinal) variable into 1’s and 0’s for statistical techniques like
regression and “pretend” it is a sort of ratio variable.

When we cover multiple regression we recode gender into a dummy variable. And as I explain
elsewhere, I’m hip to the idea that gender can exist on a continuum, but to keep this simple we shall
use an example that can be easily understood. I’ll use a cis-gender manifestation of a socially
constructed concept as institutional sexism exists in many, but of course not all, parts of society.
(Facts exist whether we like it or not.) I’m not trying to offend sensibilities here, just teach statistics in
an easy-to-understand manner, so please let’s not politicize this. That being said, instead of male
and female as described below we could pretend we used “conforming and non-conforming” (and
then you’d have to assume that construct affected income).

Most students code gender into 1 and 2 in some fashion and for regression we need it coded into 1
and 0 to dummy it. This is true for any nominal variable by the way: we can recode it to 1 and 0,
where 1 is considered 100% of the characteristic and 0 as 0% of the characteristic. So if you have

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been paying attention, in American society women are often paid less than men even when
controlling for education and experience etc. (When you control like this it is evidence that
creates evidence that contributes to creating a “fact.”) There is no simplistic sound bite explanation I
can do here but the reasons are complex. But the fact remains, that as a group this is basically true
for women in many fields of employment.

So for a test problem for lecture 22 we will examine income and one of the variables we will use will
be gender. We use women as the “characteristic of interest” as they are the group who often sees
their salary affected, so we code female=1 and men=0. This way we can analyze what is called a
coefficient and say something like “When all other independent variables are held constant, when the
person is female (as compared to male), monthly income decreases by $1266.40.”

By the way there are examples when men are the characteristic of interest. Men are more likely to
have more criminal convictions than women and if we wanted to examine this variable in the context
of criminal convictions, we would code male=1 and female=0. Then we could say something like say
something like “When all other independent variables are held constant, when the person is male (as
compared to female), the number or criminal convictions increases by 1.2.”

This is true for any nominal variable by the way: we can recode it to 1 and 0 , where 1 is considered
100% of the characteristic and 0 as 0% of the characteristic. The nominal variable can have more
than 2 categories.

So pretend we had a simplistic coding of race in the US prison system. (I am also aware that race is
a social construction and not biological when you examine the DNA of the human genome. I won’t
bore you with the details but am happy to explain what this means verbally.) But systematic racism
exists in the US prison system in the sense that Black and Hispanics are over-represented in our
prisons compared to Whites. (These are the terms used by the Department of Justice for which I can
show data to prove this.). Again, there is no simplistic sound bite explanation I can do here but the
reasons are complex.

So pretend the Dept. of Justice coded race like this and we wanted to look at Blacks compared to
Whites and Hispanics: 1=White 2=Hispanic 3=Black. We would code Black =1 and Hispanics and
Whites would be coded 0. Then we could say something like “When all other independent variables
are held constant, when the person is Black (as compared to White or Hispanic), the incarceration
rate increases by...”
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You can dummy an ordinal variable in this same manner as noted below.

Dummy” ordinal variables to pretend they are “ratio”

Pretend you had an agreement variable like this and you wanted to compare people who agree to
some statement as those who are neutral or disagree.

1=disagree
2=neutral
3=agree
You recode so agree =1 and neutral and disagree =0. Then you can make a statement like “When all
other independent variables are held constant, when the person is agrees (as compared to those who
are neutral or who disagree) to the statement that ‘Professor Hallstone is really confusing me right
now’…”J

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