Is there a relationship between "levels of measurement" and groups/rings/fields?

Question

Stevens's classification of scale provides "levels of measurement" which imply how values in a dataset may be understood/manipulated:

level	mode	median	mean	⊕	⊗	zero
nominal/categorical	×
ordinal	×	×
interval/cardinal	×	×	×	×
ratio	×	×	×	×	×	×

where ⊗ is the addition operation and ⊗ is the multiplication operation.

Is there a formal relationship between these levels of measurement and groups/rings/fields?

There doesn't seem to be a level for partial orderings (assuming "ordinal" refers to a total ordering) or non-commutative multiplication.

One of the reasons I'm interested in this is that some data I'm dealing with comes as real-valued scalar, vector, or tensor values; but sometimes the context implies that, for instance, they are nominal, ordinal, or interval measurements rather than ratio measurements. Since I cannot assume the level of measurement from the structure implied by the underlying group from which measurements are drawn, what information is best conveyed with the data to constrain how it can be analyzed/visualized?

Answer 1

Nominal / categorical means your measurements just form a set $X$ with no additional structure. Ordinal means $X$ has a total order as you say.

I don't quite understand what "interval / cardinal" means, but a minimal algebraic structure that lets you define means is the structure of a convex space, which lets you take convex linear combinations. This is a weakening of the concept of an affine space where you can take affine linear combinations. Notably this does not let you define the sum, which is not a convex or affine linear combination; for example the set $[0, 1]$ of probabilities is a convex space and it is not probabilistically meaningful to add two probabilities which sum to greater than $1$. So I don't agree with the third row that the ability to take means is necessarily paired with the ability to add.

I really don't understand what "ratio" means but it sounds like it implies that $X$ is a ring, probably even a commutative ordered ring since you are supposed to still have the ability to take medians. In mathematics it's clear that the concept of having an order and having addition or multiplication operations are logically independent; $X$ could have an order but no addition or multiplication, or it could have an addition or multiplication but no order. And the order could be partial or total and the multiplication could be non-commutative, at least in principle.

One of the reasons I'm interested in this is that some data I'm dealing with comes as real-valued scalar, vector, or tensor values; but sometimes the context implies that, for instance, they are nominal, ordinal, or interval measurements rather than ratio measurements.

I don't understand what this means; can you be more specific?