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Yes.

In the main type the measurement is a ratio of two quantities with the same units so the dimensions cancel out. Nevertheless, the ratio may have a unit. Mach, the ratio of the speed of an object in a medium compared to the speed of sound in that medium. Note that an object travelling at Mach2 in air is travelling much slower than an object at Mach2 in water.

Also, I am not sure how you would treat measurements on arbitrary or subjective scales where there are no explicit units. Examples: Beaufort scale for wind force, or Richter scale for earth tremors. Events are often referred to in the form "an earthquake measuring 6 on the Richter scale", or "winds gusting at 10 or the Beaufort scale".

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