Statistics Definitions > Ordinal Numbers, Variables and Ordinal Data

## Ordinal Numbers: Definition

An ordinal number in math tells you where a number is in order. For example, first, second, third, or fourth are all ordinal numbers. They are almost the same as the cardinal numbers (1, 2, 3…) we first learn in school.

However, in statistics, things get a little more confusing (statisticians like to confuse!). A set of ordinal numbers is called ordinal data, and that collection of numbers can be placed on an ordinal scale.

## Ordinal Data and variables.

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…etc.” 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** used in stats. 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.

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