Nominal variables are not scales because no single continuum underlies them.

Evidence

This makes nominal “scales” fundamentally different from the other types. The key question is whether there is an underlying continuum; for all the other scale types there is an underlying conceptual continuum. For nominal variables there are implicitly two underlying continua, one for each concept. To call a nominal scale a scale—hence the scare quotes in the title of this subsection—is fundamentally misleading. — Gary Goertz, Social Science Concepts and Measurement, 777

Categorization

• QUESTION: If a nominal variable implies two separate continua, what does that do to standard statistical operations assumed to require one shared metric?