# Is the level of measurement of variables always preserved?

Is it conceptually possible for multiple variables of lower level measurement (i.e. ordinal data) to be combined to i.e. interval data OR is the level of measurement always preserved?

If asked, I would argue that the former is true.

Example: The ranks within an organization is as follows: 1 = sales assistant, 2 = junior sales manager, 3 = senior sales manager, = 4 = sales department manager, 5 = CEO. Let's take a look at the organization and sum it's ranks:

> myOrganization <- c(1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5)
> sum(myOrganization)
[1] 25


Is the ordinal nature of the individual variables is preserved?

• I am really confused by this question. What do you mean by "is the ordinal nature of the individual variables is preserved?" Is there some other way you can ask this question? Or either be a little more verbose about the example you gave, or give another example of what you are getting at?
– jds
Aug 13, 2017 at 15:45
• I don't know how to make it any clearer than: Is the level of measurement of the compound variable myOrganization the same (in this case: ordinal) as of the individual variables and if so, is this always the case? Aug 13, 2017 at 22:28
• Ahhhh... by compound you mean some operation that turns multiple individual measurements into some overall measurement of the group?
– jds
Aug 13, 2017 at 23:08
• yes _________________ Aug 15, 2017 at 0:24
• Why is the number 25 (or the sum you give) "ordinal in nature"? Understanding this will help direct a better answer to your question.
– jds
Aug 15, 2017 at 13:16

The answer seems to me to be no. It seems as if you are associating integer with ordinal as you have stated that the number 25 is "ordinal". By counter example:

> myOrganization <- c(1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5)
> mean(myOrganization)
[1] 2.272727


That is, the mean rank is no longer an integer.

I agree my above response is a triviality based on an interpretation of your representation of a (logical/mathematical) system. If you want an answer that is representation invariant you may have to move from math/statistics to formal logic. Baring my suspicion that your question is a tautological truth, you may want to look at Godel's incompleteness theorems. While I am not an expert on formal logic, I believe these theorems would support the answer is no; within any representation you choose there will be operations that are inconsistent with your described logical/mathematical system.

Alternatively, It sounds like you may be asking a question about sets that are closed to group operations (in your case ordinal measurements which can be identified with the natural numbers and you are asking whether the set is closed to any "meaningful" operation on the set). In this case you may want to look at the concept of a semi-group which is the minimal algebraic structure that guarantees this behavior (groups, monoids, vector spaces, etc... are all on some level semi-groups). NOTE: Viewed in this light, my above "counter-example" associates all ordinal measurements with the set of natural numbers and then shows that the arithmetic mean is an operation that this set is not closed to (hence not a semi-group). That said, if your answer is that this is simply a non-meaningful operation, I would then suggest that your question is tautological because you are associating "meaningful" with some concept related to semi-groups.

Even another option: (I am just trying to understand your question and respond appropriately). If we say myOrganization1 <- c(1,4) and myOrganization1 <- c(2,3) you may be asking are the new variables myOrganization1 and myOrganization2 "ordinal". The answer is that it depends on the operation you choose to define what ordering is. "Ordinal" refers to data that is equipped with an "comparison" operation $$f: f(x,y) \rightarrow x?y$$ where $$?$$ can take on the following values $$<,\leq, =, \geq, >$$ (e.g., less than, less or equal, equal, greater or equal, greater than). So again, it is a tautological question that depends on your chosen comparison operation.

• How exactly is this a counterexample?
– whuber
Aug 15, 2017 at 4:35
• It doesn't follow that a variable is cardinal just because an integer is assigned to it. The question of level of measurement is conceptual and concerns what transformations makes sense. My example is chosen to be ordinal on purpose. A social hierarchy is clearly an ordinal scaled variable, since it doesn't make sense to compute it's mean, but it's median. Aug 15, 2017 at 7:20

It is certainly possible to perform an operation on ordinal level data that produces an interval level number. For example, you could apply a function that counts the number of occurrences of each rank. In your example, this gives [4, 3, 2, 1, 1] for ranks [1, 2, 3, 4, 5]. These counts are clearly interval level, since we can add them up while preserving their meaning (e.g. we can add the counts for ranks 1 & 2 to get a total of 7 employees with rank 2 or lower), and since equal differences between values correspond to equal differences in the quantity they represent. In fact these data are even ratio scale since ratios between values are also meaningful (8 employees really is twice as many as 4, which is twice as many as 2, etc.), and there is a meaningful zero-point (0 employees means no employees).

If you restrict yourself to affine operations (i.e. adding, subtracting, multiplying and dividing), though, then I don't think you can change the level of measurement. In fact, it is questionable whether you can ever meaningfully perform those operations on data that isn't at least interval level. In your example, the fact that these ranks are ordinal means the difference between ranks isn't necessarily equal, and so even though you use numbers to represent them, these numerical labels don't behave like numbers in a mathematical sense. So a statement like "the sum of rank 1 and rank 2" doesn't really make sense - the operation of adding is simply not well defined on ordinal data. You can still treat your data like real numbers of course and do arithmetic with them, but then the trouble starts when you have to interpret the outcome of your calculations (e.g. what does that number 25 you got out of your sum actually mean?).