What measurement scale is the angle measurement assigned to (interval or ratio)?

I am a physiotherapist and am going to do clinical research based on RCT model. I am going to use a test based on degrees for ANGLES measurements. I am going to use a goniometer for such measurements. My question is in the title.

• It may help to think in terms of cosine similarity, well reviewed in this wiki posting... en.wikipedia.org/wiki/Cosine_similarity
– user234562
Commented Sep 26, 2019 at 14:11

You would — or should — regard an angle of 10 degrees as twice 5 degrees, which hangs with

• zero degrees is not an arbitrary zero

• ratios make sense.

Hence angle is ratio scale. But what hinges on the difference between that and interval scale?

Note: Although angles may be measured in degrees (or radians) sometimes that is only a measurement convention. Sometimes a trigonometric function of angle is closer to the real problem. I can think of fields in which sine, cosine, tangent are each better measures mathematically, statistically or physically.

• If you are contemplating angles larger than 360 degrees and will not accept negative angles, maybe you could justify calling them a kind of ratio scale. But such situations are rare: usually when we refer to angles we think of them modulo 360 degrees. That's neither a ratio nor an interval scale: it's something altogether new.
– whuber
Commented Sep 26, 2019 at 14:31
• Again, a goniometer as I understand it does not allow anything of the sort. Commented Sep 26, 2019 at 15:43
• I see what you mean: the goniometer is limited to values from 0 to 360 degrees. As such the angle is really a value limited to an interval. It's still not, strictly speaking, either an interval or ratio measurement in Stephens' terminology.
– whuber
Commented Sep 26, 2019 at 16:23
• You mean (Stanley Smith) Stevens.... I don't recall any statement about unboundedness as a criterion, but I have not read the entire span from Stevens' 1946 paper in Science to his posthumous 1975 book. He kept tweaking the scheme in small ways. Do you think (say) proportion of restaurants worth visiting is not ratio scale because it can't exceed 1 or fall below 0? (The best discussion of nominal-ordinal-interval-ratio that I can recall is Otis Dudley Duncan's book on social measurement, which gently points out all kinds of important distinctions that the scheme misses.) Commented Sep 26, 2019 at 16:33
• ... as in my question What hinges on the difference? Commented Sep 26, 2019 at 20:42

If the angles are all positive and not very large (all of them below $$180^{\circ}$$), then I agree that a ratio scale makes sense.

However, if you are adding angles, in particular to values above $$360^{\circ}$$, note that you should take care when computing quantities like the mean and the standard deviation. In that case, we should consider the measures to be a cyclical ratio, according to Chrisman's typology.

• I agree if the angles are orientations of vectors. The implication of a goniometer measurement I take to be otherwise. Commented Sep 26, 2019 at 12:34

Indeed, there might be some problems with a ratio scale, but I tend to agree with Nick. Ratios make sense and there is a meaningful 0.

Regarding negative angles, maybe this is similar to speed and velocity. Velocity is a vector, so strictly speaking it is not on a ratio scale. But practically direction and amount can be treated separately. Speed is obviously on a ratio scale.

Another perspective is considering the allowed transformations. For angles only $$a\cdot X$$ is allowed. For instance from degrees to radians: $$π/180 \cdot X$$

Stevens wrote (1946, p. 679):

Ratio scales are those most commonly encountered in physics and are possible only when there exist operations for determining all four relations: equality, rank-order, equality of intervals, and equality of ratios. Once such a scale is erected, its numerical values can be transformed (as from inches to feet) only by multiplying each value by a constant.

Seems fine to me with a goniometer. All four relations can be easily established.

There is also the perspective of measurement theory, which, for quantity would require: transitivity, antisymmetry, strong connexity, associativity, commutativity, monotonicity, solvability, positivity and the Archimedian condition. I guess, these indeed create problems when the interval is bounded. Consider monotonicity:

$$X\geq Y$$ if and only if $$X+Z \geq Y+Z$$

If X=360° and we add something to it, what is the result? If we say this is not allowed then we need to include another restriction, right? Similar problems occur for positivity and the Archimedian condition. So there is a point to whuber's "it's something altogether new". But the way angles are used still very much feels ratio-scale to me. I guess there are also natural bounds for some physical attributes, but they are also treated as ratio-scales.