# Should types of data (nominal/ordinal/interval/ratio) really be considered types of variables?

So for instance here are the definitions that I get from standard text books

Variable - characteristic of population or sample. ex. Price of a stock or grade on a test

Data - actual observed values

So for a two column report [Name | Income] the column names would be the variables and the actual observed values {dave | 100K} , {jim | 200K} would be the data

So if I say that the [Name] column is nominal data and that [income] is ratio data, wouldn't I be more accurate describing it as a type of variable instead of a type of data like most textbooks do? I understand that this might be semantics, and that's fine it that's all there is too it. But I fear that I might be missing something here.

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Doesn't strike me as a meaningful difference; I'd consider either phrasing acceptable, personally. The definition of "variable" seems a little off though. – Nick Stauner Jul 9 '14 at 22:12
@Nick I believe that if we translate the colloquial "characteristic" to the mathematical "real-valued function," we get part of the definition of a random variable. (The missing part, of course, is measurability with respect to a sigma field on the population.) Normally, though, we would translate "characteristic of a sample" into the technical term statistic: maybe that's what you are referring to as being a "little off." With these translations, variables do not have "types" at all in Stevens' sense (we can only distinguish discrete from continuous distributions)--but some data can. – whuber Jul 9 '14 at 22:30

Stevens' scale typology isn't necessarily some inherent characteristic of the variables, nor even data itself, but of how we treat the information - of what we're using it to mean.

In some circumstances, exactly the same value may be considered ratio, interval, ordinal or nominal, depending on what we're doing with it - it's a matter of what meaning we give the values, which can change from one analysis to the next. Stevens' typology has some value, but it doesn't do to be overly prescriptive about it.

This issue of the importance of scale as meaning dates back at least to Lord (1953), who offered an example where there were both nominal and interval interpretations of the same set of numbers.

This point was even more clearly made by Velleman and Wilkinson (1993), who offer an example of people receiving consecutive numbered tickets on entry to a reception with a prize being awarded to one of the tickets; depending on the use being made of the numbers on the tickets, they have interpretations on all four scales.

So, for example 'did I win?' is a question treating the number as nominal, while 'did I arrive too early to get the winning ticket?' is a question that treats it as ordinal; on the other hand (and I don't think this one is in the paper) using 5 random ticket numbers in order to estimate the number of people in the room would treat them as ratio (e.g. if there were 4 randomly drawn numbers that got consolation prizes, you'd have 5 random numbers altogether from which to estimate total attendance).

They argue that "good data analysis does not assume data types", "Stevens’s categories do not describe fixed attributes of data", "Stevens’s categories are insufficient to describe data scales" and "Statistics procedures cannot be classified according to Stevens’s criteria" (indeed each statement is also a section title).

Criticisms were also offered in several places by Tukey (e.g. in chapter 5 of Mosteller and Tukey's 1977 book Data analysis and regression); Mosteller and Tukey offered a typology - names, grades (ordered labels), ranks (starting from 1, which may represent either the largest or smallest), counted fractions (bounded by zero and one, these include percentages), counts (non-negative integers), amounts (non-negative real numbers), balances (unbounded, positive or negative values).

In my own work, I've seen situations where severe problems with analysis were caused by people failing to appreciate the great difference between variables relating to levels (sometimes called 'stock' variables) and flows - a simple example of these types is the difference in the kinds of analysis appropriate for the amounts of water actually in a storage tank in each of a sequence of periods, and the amount of water flowing into it. These would (in some of those cases) both be sub-categories of the Mosteller and Tukey 'amounts' type (and in those same cases, both ratio variables in Stevens' scheme), indicating that issues of typology may be quite subtle, but can still critically impact appropriate analyses.

P.F.Velleman and L.Wilkinson (1993),
"Nominal, Ordinal, Interval, and Ratio Typologies are Misleading,"
The American Statistician, vol.47 no.1 pp.65-72

(a working version seems to be available at the 2nd authors web page here)

Lord, F. (1953),
"On the statistical treatment of football numbers,"
American Psychologist, 8, pp.750-751

(The year of this paper is given wrongly in the references of the version of the Velleman and Wilkinson paper I linked to, but correctly referred to in the body of the paper)

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Thanks. Very thorough answer. I was thinking along those lines but when researching this stuff many times they make it seem as if it is concrete and consensus has been reached. That's why I ended up here. – BI Guy Jul 10 '14 at 19:59
Stevens' typology has been debated and disputed since it was first published. It's a sometimes-helpful framework, not a theorem. – Glen_b Jul 11 '14 at 5:41

The type of the data is related but not identical to the type of the variable. Most of the cases, they are the same but they don't have to be.

For example, if you collect N samples from a normal distribution. You would think it's a numerical (ratio or scale) data. But I can also say it's a categorical variable with N different categories, with frequency of 1 for each category. It looks stupid but it's also a valid variable.

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This seems a little at odds with Stevens (who is credited with formulating this typology), who wrote "the real issue is the meaning of measurement." Although you may always elect to treat such data as nominal, that does not make them nominal in Stevens' estimation. His paper is available at gaius.fpce.uc.pt/niips/novoplano/mip1/mip1_201314/scales/… . – whuber Jul 10 '14 at 13:18
The paper is not found!!!! – Student T Jul 10 '14 at 15:22
Thank you for pointing that out. There seems to be an "idiosyncrasy" in this site's parsing of urls: it mistakenly included the final period. By inserting a space before the period I found that the link now works. – whuber Jul 10 '14 at 15:26