## Nominal Ordinal Interval Ratio & Cardinal: Examples

Contents:

## 1. Nominal

Nominal: nominal is from the Latin

nomalis, which means “pertaining to names”. It’s another name for acategory.

**Examples**:

**Gender**: Male, Female, Other.**Hair Color**: Brown, Black, Blonde, Red, Other.**Type of living accommodation**: House, Apartment, Trailer, Other.**Genotype**: Bb, bb, BB, bB.**Religious preference**: Buddhist, Mormon, Muslim, Jewish, Christian, Other.

A **nominal variable** is another name for a categorical variable. Nominal variables have two or more categories without having any kind of natural order. they are variables with no numeric value, such as occupation or political party affiliation. Another way of thinking about nominal variables is that they are **named **(nominal is from Latin *nominalis*, meaning *pertaining to names*).

**Nominal variables:**

- Cannot be quantified. In other words, you can’t perform arithmetic operations on them, like addition or subtraction, or logical operations like “equal to” or “greater than” on them.
- Cannot be assigned any order.

## Examples of Nominal Variables

- Gender (Male, Female, Transgender).
- Eye color (Blue, Green, Brown, Hazel).
- Type of house (Bungalow, Duplex, Ranch).
- Type of pet (Dog, Cat, Rodent, Fish, Bird).
- Genotype ( AA, Aa, or aa).

Nominal variables are related to the **nominal scale**, where data is categorized without any order.

## The Nominal Scale

The nominal scale, sometimes called the* qualitative type*, places non-numerical data into **categories or classifications**. For example:

- Placing cats into breed type. Example: a Persian is a breed of cat.
- Putting cities into states. Example: Jacksonville is a city in Florida.
- Surveying people to find out if men or women have higher self-esteem.
- Finding out if introverts or extroverts are more likely to be philanthropic.

These pieces of information aren’t numerical. They are assigned a **category** (breeds of cat, cities in Florida, men and women, introvert and extrovert). Qualitative variables are measured on the nominal scale.

## Mean Mode and Median for the Nominal Scale

The nominal scale uses categories, so **finding the median makes no sense**. You *could* put the items in alphabetical order but even then, the middle item would have no meaning as a median. However, a mode (the most frequent item in the set) is possible. For example, if you were to survey a group of random people and ask them what the most romantic city in the World is, Venice or Paris might be the most common response (the mode).

The nominal scale is one of **four scales of measurement**. The other three are:

- The
**Ordinal Scale**: Rank order (1st, 2nd 3rd), dichotomous data that has two choices like true/false or guilty/innocent and non-dichotomous data with choices like “completely agree” “somewhat agree” “neutral” and “disagree.” - The
**Interval Scale**, sometimes called*Scaled Variable*: data with degrees of difference like time B.C. or Celsius. Interval scales have arbitrary zeros (for example, when B.C. began and ended has no real mathematical basis). - The
**Ratio Scale**: encompasses most measurements in physics and engineering like mass and energy. Ratio scales have meaningful zeros (zero energy means that energy does not exist).

The four scales were suggested by Stanley Smith Stevens in a 1946 Science article titled “On the Theory of Scales of Measurement.”

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## 2. Ordinal

Ordinal: meansin order. Includes “First,” “second” and “ninety ninth.”

**Examples**:

**High school class ranking**: 1st, 9th, 87th…**Socioeconomic status**: poor, middle class, rich.- The
**Likert Scale**: strongly disagree, disagree, neutral, agree, strongly agree. **Level of Agreement**: yes, maybe, no.**Time of Day:**dawn, morning, noon, afternoon, evening, night.**Political Orientation:**left, center, right.

Ordinal data is made up of ordinal variables. In other words, if you have a list that can be placed in “first, second, third…” order, you have ordinal data. It

*sounds*simple, but there are a couple of elements that can be confusing:

- You don’t have to have the exact words “first, second, third….” Instead, you can have different rating scales, like “Hot, hotter, hottest” or “Agree, strongly agree, disagree.”
- You don’t know if the intervals between the values are equal. We know that a list of cardinal numbers like 1, 5, 10 have a set value between them (in this case, 5) but with ordinal data you just don’t know. For example, in a marathon you might have first, second and third place. But if you don’t know the exact finishing times, you don’t know what the interval between first and second, or second and third is.

## Ordinal Scales.

Ordinal scales are made up of ordinal data. Some examples of ordinal scales:

- High school class rankings: 1st, 2nd, 3rd etc.
- Social economic class: working, middle, upper.
- The Likert Scale: agree, strongly agree, disagree etc.

The **Likert Scale** gives another example of how you can’t be sure about intervals with ordinal data. What is the interval between “strongly agrees” and “agrees”? It’s practically impossible to put any kind of number to that interval. Even if you could put a number to the interval, the gap between “strongly agree” and “agree” is likely to be much smaller than the gap between “agree” and “no opinion.” Think of someone being asked to rate a question like “Chocolate is irresistible.” Someone who likes chocolate a lot might have their pencil hover between answering “strongly agree” or “agree”, but their pencil never hovers over “no opinion.”

## Ordinal Scale Examples

The ordinal scale is a type of measurement scale that deals with **ordered **variables.

Let’s say you were asked to order five movies from your most favorite to your least favorite: Jaws, The Matrix, All Good Things, Children of Men and The Sound of Music. Creating the order of preference results in the movies being ordered on an ordinal scale:

- The Matrix.
- Jaws.
- Children of Men.
- The Sound of Music.
- All Good Things.

A second example of the ordinal scale: you might conduct a survey and ask people to rate their level of satisfaction with the choice of the following responses:

- Extremely satisfied.
- Satisfied.
- Neither satisfied nor dissatisfied.
- Dissatisfied.
- Extremely dissatisfied.

The choices from “extremely satisfied” to “extremely dissatisfied” follow a natural order and are therefore ordinal variables.

The ordinal scale is one of **four measurement scales**commonly used. The other three are:

- The Nominal Scale: Data that can be put into categories.
- The Interval Scale: Data with degrees of difference like time B.C. or degrees Celsius.
- The Ratio Scale: Encompasses most measurements in physics and engineering like mass and energy. Ratio scales have meaningful zeros (zero energy means that energy does not exist).

The ordinal scale and interval scales are very similar to each other and are often confused. If you assume that the differences between the variables are equal, or if the distances are measured precisely (for example, using the logarithmic scale) the scale is an interval scale.

## Disadvantage of the Ordinal Scale

A **major disadvantage** with using the ordinal scale over other scales is that the distance between measurements is not always equal. If you have a list of numbers like 1,2 and 3, you know that the distance between the numbers in this case is exactly 1. But if you had “very satisfied”, “satisfied” and “neutral”, there’s nothing to say if the different between the three ordinal variables is equal. In the list of five movies listed above, there’s a small difference in my preference for Jaws or Children of Men, but a huge difference between Children of Men (which I enjoyed…twice!) and The Sound of Music (which I do not like at all). This inability to tell how much is in between each variable is one reason why other scales of measurement are usually preferred in statistics.

## Ordinal Numbers in Set Theory.

Although “ordinal number” usually refer to values on a rating scale, it’s worth mentioning that they can have other meanings outside of arithmetic and statistics. For example, an ordinal number in formal set theory is defined as “the order type of a well ordered set” (Dauben 1990, p. 199; Moore 1982, p. 52; Suppes 1972, p. 129). In set theory, ordinal numbers are represented with Arabic numerals or lower case Greek letters.

## 3. Interval

Interval: has values of equal intervals that mean something. For example, a thermometer might have intervals of ten degrees.

**Examples**:

- Celsius Temperature.
- Fahrenheit Temperature.
- IQ (intelligence scale).
- SAT scores.
- Time on a clock with hands.

## 4. Ratio

Ratio: exactly the same as the interval scale except that the zero on the scale means:does not exist. For example, a weight of zero doesn’t exist; an age of zero doesn’t exist. On the other hand, temperature (with the exception of Kelvin) is not a ratio scale, because zero exists (i.e. zero on the Celsius scale is just the freezing point; it doesn’t mean that water ceases to exist).

**Examples**:

- Age.*
- Weight.
- Height.
- Sales Figures.
- Ruler measurements.
- Income earned in a week.
- Years of education.
- Number of children.

*It could be argued that age isn’t on the ratio scale, as age 0 is culturally determined. For example, Chinese people also have a nominal age, which is tricky to calculate.

## 5. Cardinal Numbers

A

cardinal number, sometimes called a “counting number,”is used for counting, like when you count 1, 2, 3. You use these numbers to answer the question “how many?”

Many times, sets of cardinal numbers create statistics. When this happens, the cardinal numbers disappear. For example, according to the 2010 U.S. Census, the average number of people per household in the U.S. is 2.58. This number was arrived at by taking the cardinal number of people in each household and then finding the mean. Once you’ve taken that set of cardinals and found its mean (2.58), the statistic is no longer cardinal.

**Cardinals are always positive (or zero)**, as they are used to count. For example, you can have 5 loaves of bread, but having minus five loaves makes no sense (at least, in the real world).

**The cardinals used in everyday language and those used in set theory are defined in different ways.**For example, in set theory, cardinals can represent negative numbers. The cardinal number of this set {-5, -99, -100} is three. Infinity is also a cardinal: the cardinal number of this set {1,2,3,…} is infinity.

**Trivia:** In the English language, cardinals come before the noun. For example, you say “three brothers.” In American sign language (ASL), they come either before or after the noun. For example, you can say “I have brother 3” in ASL.

## Set Theory, The Largest Cardinal Number and Cantor’s Theorem

Set theory describes how many elements are in a set and tells us how many cardinal numbers exist. Cardinality in set theory forms a generalization of the natural numbers, which extends into transfinite numbers. Transfinite numbers are close to infinity, but are not *exactly *infinite. Infinity is itself a difficult concept to grasp on its own, because most things we can see, feel, or hear are finite. But just when you think you can wrap your head around the concept of infinity, it actually gets a **lot more complicated** than that; Cantor demonstrated that there are an different sizes of infinity and in fact there are an infinite amount of infinities. Cantor’s theorem sheds a little light on the idea.

Cantor’s theorem tells us that **there is no largest cardinal number**. The theorem also tells is that there are infinite amounts of infinite cardinal numbers. The theorem basically says a set exists that contains all cardinal numbers. This set also has a *power set,* which is a collection of subsets.

As a very simple example, let’s start with a small set of cardinal numbers {1, 2, 3}.

The power set of {1, 2, 3} includes the empty set { } and all of the possible combinations of sets (this is very similar to the idea of combinations in statistics):

P(S) = { {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }.

Now imagine a set containing all possible cardinals {1,2,3…∞} Even for infinite-sized sets, the power set is always larger. Mathematicians call this a **paradox **(a contradiction). However, it also gives rise to the idea of multiple types of infinities.

## A More Precise Definition

Cardinal numbers get their name from the literal definition, which means “chief” or “primary.” The word cardinal comes from the Latin *cardin*, which means “stem” or “hinge”, so these are the numbers that all others stem from, or hinge (depend) upon (Svarney & Svarney, 2012). These are the fundamental **counting units**, without which mathematics wouldn’t be possible. In this basic sense, cardinals are easy to use and understand. However, the precise mathematical definition is a *lot *more complicated and involves some pretty hefty mathematics, including set theory. Hamilton (1982) calls the concept of cardinality a “difficult notion” to grasp and many other authors agree.

Part of the reason for the difficulty in forming an easy definition is that if we say it’s what you get when you count objects (1, 2, 3 etc.), then the process of counting is itself ordinal. “1” is the first number, “2” is the second, and so on. In addition, the sum *a *+* b* could refer to two ordinal numbers, or two cardinal numbers, and they do not end up with the same result. Another example of cross-contamination between systems: May 10 in the Hindu-Arabic numbering system (the one in common use in the U.S.) could be read as either a cardinal (May 10) or ordinal (May 10*th*). The same is true for Roman Numerals, where II could be read as a cardinal two, or an ordinal second (as in Charles II, Charles the second).

If you’re confused by this, you aren’t alone. Historically, the exact definition was (an perhaps, still is), quite the convoluted topic.

## Early Definitions

Gottlob Frege (in 1884) and Bertrand Russell (1903) defined the cardinal numbers as *the set of all sets equipollent to A* (Moore, 1982, p.153). In English, that’s saying that the cardinal number of a particular set is the aggregate of all sets you can match with it. Or. to put it another way, it’s that unique aspect of a set you can match to another set. “Matching” implies a *one-to-one correspondence*. For example, let’s say you had fifty people at a Bingo game (so the set of all people equals 50). And let’s further suppose that those fifty people purchased 50 Bingo cards. As the number of people (50) exactly matches to the number of Bingo cards (50), we say there is one-to-one correspondencee and the cardinality of the set is therefore 50.

Outside of mathematical philosophy, Frege and Russell’s definition didn’t stand the test of time. This may be because although the two men agreed on the wording of the definition, they did not agree on the philosophical *meaning *of the definition. Frege called numbers “self-subsistent objects” while Russell took it as “…enabling him to dispense with numbers as distinct from classes of equinumerous classes as unnecessary physical lumber” (Beaney, 2010). However, it was important as it set the stage for the idea that **cardinals are members of a universal set made up of smaller sets of members.**

## Cantor-Von Nuemann Definition

Another early definition was the Cantor-Von Nuemann definition, which is significantly more technical than the Frege-Russell definition. In brief, the theory states:

|A| is defined as the least (von Neumann) ordinal α such that A can be well-ordered with type α (Dasgupta, 2013).

To define arbitrary sets like the set of all reals (**R**), it requires the use of the Axiom of Choice, which has many forms. This (one of the simplest) is from Vanderbilt University’s Axiom of Choice:

Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.

This takes us down the rabbit hole of set theory, which is beyond this rudimentary discussion of cardinal numbers. If you’re interested, I recommend Abhijat Dasgupta’s excellent book *Set Theory: With an Introduction to Real Point Sets.*

## References

Beaney, M. (2010). The Analytic Turn: Analysis in Early Analytic Philosophy and Phenomenology. Routledge.

Dasgupta, A. (2013). Set Theory: With an Introduction to Real Point Sets. Springer Science & Business Media.

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.

Hamilton, A. (1982). Numbers, Sets and Axioms: The Apparatus of Mathematics. Cambridge University Press.

Hosch, W. (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group.

Levine, D. (2014). Even You Can Learn Statistics and Analytics: An Easy to Understand Guide to Statistics and Analytics 3rd Edition. Pearson FT Press

Moore, G. (1982). Zermelo’s Axoim of Choice. Springer.

Russel, B. (1903). Principles of Mathematics.

Svarney and Svarney (2012). The Handy Math Answer Book. Visible Ink Press.

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