A representation theorem tells us that a valid numerical assignment exists; the uniqueness theorem tells us how many valid assignments exist. If the measurement function φ is a valid representation, the uniqueness theorem characterizes all other valid representations φ′ — that is, the family of transformations f such that φ′ = f(φ) is also a valid representation. This family of admissible transformations defines the scale type.
Scale Types from Uniqueness
Interval: φ′ = α·φ + β (α > 0)
Ratio: φ′ = α·φ (α > 0)
Absolute: φ′ = φ (unique assignment)
For ordinal scales, any monotone transformation yields an equally valid representation — only rank-order information is meaningful. For interval scales, affine transformations are admissible: the zero point and unit are arbitrary (like Celsius vs. Fahrenheit), but ratios of differences are invariant. For ratio scales, only the unit is arbitrary (kilograms vs. pounds), and ratios of values are meaningful. Absolute scales (like probability or counting) have a unique representation.
Meaningful Statements
A numerical statement about measured quantities is meaningful if and only if its truth value is invariant under all admissible transformations. "Alice's IQ is twice Bob's" is not meaningful if IQ is an interval scale (the statement's truth depends on the arbitrary zero point). "Alice is 30 IQ points above Bob" is meaningful because differences are invariant under affine transformations.
This framework has practical consequences for statistics: computing a mean is meaningful for interval and ratio data but not for ordinal data; computing a geometric mean requires a ratio scale; computing a coefficient of variation requires a ratio scale with a meaningful zero.