statistical vs set/category theoretic "equivalence"?

Equivalence is basic to mathematics, e.g. https://en.wikipedia.org/wiki/Equivalence_relation.

"Equivalence tests" are common in statistics, e.g. https://en.wikipedia.org/wiki/Equivalence_test.

Is the name "statistical equivalence" used intentionally to reflect a particular set-theoretic equivalence (and where is this best explained)?

Any views/references welcome. Thanks!

• It is a recognized pedagogical problem in statistics that many basic statistical terms like "Normal," "conditional," "independent," and so on differ so much from their colloquial meanings or their definitions in other fields (like mathematics) that it is erroneous to apply those "alien" concepts to help understand their statistical cognates. "Equivalence," although not so basic, suffers from the same problem.
– whuber
Commented Aug 10, 2022 at 14:51

Let’s say that equivalence is the means being within $$2$$ of each other.
$$\bar x_A =0.0\\ \bar x_B =1.5\\ \bar x_C =3.0$$
$$A$$ is equivalent to $$B$$, and $$B$$ is equivalent to $$C$$, but $$A$$ is not equivalent to $$C$$.