Would you say your pain level reading this blog post is:

- One to three – A minor annoyance, at worst
- Four to six – A real pain to get through
- Seven to ten – A public nuisance on the order of roaches or rats?

The above question and those like it, repeated *ad infinitum* in doctor’s offices, Gallup polls, and employee exit interviews, exemplify a data type commonly referred to as “ordinal” data. Why “ordinal,” and not “ranked” or simply “ordered,” stuck is presumably a byproduct of academics’ penchants for anachronistic language that borders on a holy mystery.

## Ordinal Data Versus Nominal Data: What’s the Difference?

- Ordinal data is data that can be ranked or ordered. Examples include data taken from a poll or survey.
- Nominal data is data that can be made to fit various categories. Examples include whether an animal is a mammal, fish, reptile, amphibian, or bird.

## What Is Ordinal Data?

Nevertheless, ordinal data is simply what the name imples: data that is ordered or ranked. When it comes to ordinal data, however, the tricky part is a lack of consistent spacing between ranks. To flesh this out a bit more, we’ll turn to Stanley Smith Stevens and the Level of Measurement scale he pioneered.

## Ordinal Versus Nominal Data

In an article in *Science* magazine, Stevens laid out a hierarchy of the types of data researchers collect. Stevens based his evaluation on measurement precision. At one extreme lay nominal data. To collect this information, researchers simply had to classify something into a pre-defined category. Categories showed no order apart from membership based on some criteria. So, whether someone is a Democrat, Republican, or independent voter, or whether an animal is a fish, bird, mammal, reptile, amphibian, or insect are both examples of nominal data.

On the other extreme of Stevens’ scale lay ratio data, which has a true zero, equal spacing between units of observation and a scale applicable to every observation that extends from small to large. Temperature, income, years of life, and speed are examples of ratio-scale data.

Ordinal data falls between these two extremes. Unlike its nominal cousin, ordinal categories have a rank structure. On the other hand, unlike its ratio-scale cousin, distances between ordinal ranks do not need to be equal or even standardized between rank levels.

## What Is an Example of Ordinal Data?

To further illustrate the difference, consider ways of assigning ranks to students in a classroom. One teacher (presumably a tall one) wants to rank students by their heights. The scale of measurement, whether it be in inches, feet, meters, or some other, is the same for each student and equally calibrated so that measurements are directly comparable.

Another teacher (presumably a cheery one) wants to rank students by happiness. Some students surely are happier than others. Or, rather, some students are much more discontent than their peers. But there is no standardized scale by which to measure student happiness. In lieu of a standardized, calibrated measurement, teachers must either rely on student self-reporting via a survey instrument or they must rely on external observers to judge student happiness based on some common criteria, i.e. a rubric or form. The challenge inherent in this is that the distance between one student’s being “happy” and “very happy” can be radically different from another student’s. This problem persists despite how the data are collected.

Ordinal scales beget different types of statistical inferences than do more precisely measured data. Ordinal data also make predictive inference more challenging. Challenging, that is, but not impossible. If enough ranked data can be collected in the same way, then researchers can construct empirical distributions of the frequencies of individuals’ rankings. This allows researchers to examine summary statistics like medians and interquartile ranges in ordinal-scale data. All of this flows from the natural rank structure of the ordinal scale, even though distances between ranks can differ between individuals! The simple fact that units are orderable, even though the orderings will be different from person to person, provides enough structure in the data to allow researchers to employ a range of empirical methodologies for getting at statistical regularities.

## How Is Ordinal Data Used?

Even further, because ordered data are categorized along a continuum, they can be assigned numerical likelihoods. If researchers can justifiably make distributional assumptions about these likelihoods, then they can compute probabilistic estimates of ordinal outcomes. This is especially powerful because it allows researchers to use information contained in precisely measured predictors to estimate how likely ordinal outcomes will be. To see why, let’s go back to the classroom-ranks conundrum.

If there is reason to suspect that taller people tend to be happier, student height measurements can provide a basis for probabilistic inference about how happy students are. For instance, if a student is 1.5 times taller than the class’s average height, the frequency of happiness levels reported allows teachers to claim that student is twice as likely to be “very happy” relative to other students, four times as likely to be “happy” or “very happy” relative to other students, and so on.

By exploiting the linkage between the rank-ordering of student happiness levels and student height, teachers can use movements on the height scale, which is more measurable, to decipher movements along the not-directly-measurable (or, switching back to academese, “latent”) student-happiness scale. As long as correlation exists between predictors and an ordinal outcome and researchers are willing to make assumptions (and justify them) about what distribution the ordinal outcomes likely exhibit, they can probabilistically predict ordinal outcomes.

## How Is Ordinal Data Collected?

The biggest pitfall to estimating ordinal outcomes is measurement error arising from data collection. Ordinal data collection suffers more than nominal data from measurement error because nominal data are simpler to collect. Categorization of items into classes is more straightforward than both classifying and ranking items. For instance, identifying car makes as Porsche, Ford, BMW, or Subaru is easier than attempting to rank-order them based on consumer preference. Ordinal data, unlike ratio data, also suffers from imprecision due to a lack of a standardized, equally spaced continuum of measurement. Thus, direct comparison is not on offer with respect to ordinal outcomes.

For these reasons, researchers must avoid measurement error when collecting ranked outcome data. Methods for doing so include standardizing both questions on surveys and how people can respond to them. Changing categories, vague cut points, or murky wording can make for erratic, noise-filled responses.

Measurement-error likelihood also increases with second- or third-party data collection. Surveyors and respondents can both be sources of error, as can disparate observers who record totally different outcomes for the same result. Also, predictive inference and statistical estimation are complicated by the lack of a direct scale applicable across all measurements. Assumptions play a greater role in statistically examining ordinal outcomes, and unwarranted, unstated, or unrealized assumptions researchers make about data can be treacherous.

## What Is Ordinal Data Used For?

Challenges notwithstanding, ordinal data, when well and consistently collected, allows researchers to bring statistical insight to bear on a variety of topics not broachable by more precise forms of scientific measurement. Important questions concerning overall contentment in life, subjective interpretations of job or family satisfaction, individual feelings of pain or depression, and many others can all be addressed on a large scale using ordinal outcomes. Understanding ranked information and subsequently using it to build accurate statistical models, to the extent researchers can, opens the door to a deeper understanding of subjective but common human well-being.