Is "categorical data" a synonym of "nominal data"?

Question

So far, it's always been my understanding that nominal data was a type of categorical data, not a synonym of it. For me, categorical data included ordinal data, not just nominal data.

As of November 2024, Wikipedia says (bold's mine):

Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known.

This seems in line with Alan Agresti's An Introduction to Categorical Data Analysis (second edition, 2007). Page 2:

Categorical variables have two main types of measurement scales. [...] Categorical variables having ordered scales are called ordinal variables.

Categorical variables having unordered scales are called nominal variables.

On the other hand, the categorical-data tag on CrossValidated says (bold's mine):

Categorical (also called nominal) data can take on a limited number of possible values called categories. Categorical values "label", they do not "measure". [...]

For analysis, categorical values are considered as abstract entities without any mathematical structure such as an order or a topology, regardless of how they are coded and stored.

The UCLA Statistical Methods and Data Analytics website seems to agree with the CrossValidated definition (bold's mine):

A categorical variable (sometimes called a nominal variable) is one that has two or more categories, but there is no intrinsic ordering to the categories. [...] A purely nominal variable is one that simply allows you to assign categories but you cannot clearly order the categories.

So it looks like there's some discrepancy here. While I'm in general a bit wary of Wikipedia, I have no reason to doubt either of the other resources I'm mentioning, in particular when their definitions don't seem ambiguous.

Is it reflective of some lack of consensus on the definition of "categorical"? In other words, is "categorical" a term allowing some flexible use? Or it it me misinterpreting something or missing some essential piece of information that would reconcile these different references?

I'd be also interested in (preferably academic) references, if any, discussing the definition of "categorical" and possibly the issue of varying definitions.

The reason I'm asking this question is certainly not to nitpick, but to avoid possible misunderstandings when reading or discussing this subject with other people.

Answer 1

This is a little puzzling, but it can all be summarized:

Treated narrowly, categorical data is a synonym for nominal data. Treated widely, categorical data includes ordinal data too.

So watch out: different authors and different sites may be using the terms differently, so pick and choose according to tribal tradition or personal taste. I wouldn't want to try to count or estimate the relative abundance of these senses, but I jump for the wide sense myself.

The broader context is a terminological jungle for different kinds of data, and within that the continuing use of S.S. Stevens' distinction between nominal, ordinal, interval and ratio scales, which has sometimes helped but often hindered both clear communication and good analyses.

Entire books and papers have been written around this topic. The most contentious theme is an idea that variable type or scale determines what kind of analysis is appropriate (and which not).

The papers

Velleman, P. F., & Wilkinson, L. 1993. Nominal, ordinal, interval, and ratio typologies are misleading. The American Statistician 47(1): 65–72. https://doi.org/10.1080/00031305.1993.10475938 https://www.jstor.org/stable/2684788

Hand, D. J. 1996. Statistics and the theory of measurement. Journal of the Royal Statistical Society. Series A (Statistics in Society) 159(3): 445–492. https://doi.org/10.2307/2983326 https://www.jstor.org/stable/2983326

are both notable for vigorous discussion.

This isn't completely crazy. The ratio of two masses makes sense. The ratio of two Celsius temperatures really doesn't, although believe it or not, published work can be found implying that it does.

But experienced analysts don't buy nominal-ordinal-interval-ratio as both relevant and compelling as a total framework for analysis.

For one, modern categorical data analysis is centred on the idea that we may start out with cats and dogs as nominal categories, but modelling will first focus on counting those cats or dogs, modelling the counts using logarithmic link functions, and so forth. The original data may be "cat", "dog", or whatever, but the data analysed are counts.

Want to take averages of ordinal data? Feel free, but be careful.

Answer 2

As others have noted, this categorization comes from Stevens' scales. But he intended those as guidance, not a straitjacket (but see comments), and his scheme is not complete. I wrote a blog post about this (and other, much more illustrious statisticians have written about it, as well, but I have access to mine).

Answer 3

Measurement has been going on since math was a thing, but the more modern definitions of variables that you typically see in textbooks these days come from Stevens's 1946 article on the subject where here you see the following table provided:

And where he defines in his article as, where he does not consider them to have any numeric value and are strictly for procedures like counting:

The nominal scale represents the most unrestricted assignment of numerals. The numerals are used only as labels or type numbers, and words or letters would serve as well.

While this list has had its controversies (Michell, 1986), many tend to agree that nominal data and categorical data are essentially the same thing (myself included) while others may still disagree on that point for pedantic reasons. The most important distinction, which I have gathered from your comments, is whether or not we are talking about ordered categories. In this sense, as I'm sure you know, we often consider this data ordinal, and anything unordered as categorical/nominal (Bandalos, 2018). Whatever distinctions between "categorical" and "nominal" that exist are otherwise likely not real or useful distinctions, as I rarely hear people label nominal or categorical data in a way that they are actually conceptually different.

References

• Bandalos, D. L. (2018). Measurement theory and applications for the social sciences. The Guilford Press.

• Michell, J. (1986). Measurement scales and statistics: A clash of paradigms. Psychological Bulletin, 100(3), 398–407. https://doi.org/10.1037/0033-2909.100.3.398

• Stevens, S. S. (1946). On the theory of scales of measurement. Science, New Series, 103(2684), 677–680.