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The Corruption of Measurement
The corruption stems from the claim that "ordinal measurement" and "nominal measurement" are kinds of measurement. They are not. The first is an oxymoron and a stolen concept (see my definition below). It is ranking, but not measurement. The second is an anti-concept: "an unnecessary and rationally unusable term designed to replace and obliterate some legitimate concept" (Ayn Rand). A key part of the corrupters' strategy was defining measurement as "the assignment of numerals to objects or events according to a rule." What rule? Any rule, which Stevens often appended to this definition. Such a definition and criterion leave the door wide open to subjectivity.
The Wikipedia article gives several logical, arithmetical, and statistical differences among Stevens' four categories. Also, numbers are found in interval and ratio measurement, whereas numbers in labeling or ranking are usually assigned. This is a very important distinction which his categories do not recognize. Indeed, all measurements are assigned per his definition.
My definition: "Measurement is a quantity in terms of real numbers and a standard of magnitude that serves as a unit, with multiples of the unit being additive and subtractive." This implies all units used to measure must be equal. A real number and such a standard unit are the distinguishing characteristics, or conceptual common denominator, of measurement. If P claims that X is measurement, then P should be able to name the unit. If P cannot do so, then X is not measurement.
Stevens' definition describes ranking, labeling, and counting. It trashes the unit standard essential for true measurement—ratio or interval—which is part of the conceptual common denominator. Thus it is a stolen concept. Indeed, numbers are not even needed for ranking, which the corrupters call "ordinal measurement." It can be done in other ways (examples below). Obviously numbers are not needed for labeling, which the corrupters even admit. They call labeling "nominal measurement." Real (not ordinal) numbers and an explicit, standard unit of magnitude are necessary for ratio and interval measurement. They are not at all required for "ordinal measurement" and "nominal measurement." Yet the corrupters want you to believe that labeling and ranking are "measuring."
Ranking may be done with ordinal numbers—1st, 2nd, 3rd, etc.—but there are other methods. Commonly a numerical ranking or rating scheme is prescribed in advance. For example, a number between 1-6 (decimals allowed) is assigned in figure skating competition, and 1-10 (decimals allowed) is assigned in gymnastics competition. The president's approval ratings use percents. A grading scale used for students—A, B, C, D, and F, maybe with + and - added—does not even use numbers. Ratings in martial arts are shown with different color belts. Military ranks use different insignia.
A numerical ranking may be computed. Suppose a figure skating judge assigns numbers for several factors like difficulty, skill in execution, and originality, and then uses an algebraic formula to compute a number that falls in the required range. Is the result a measurement? Not per my definition, for reasons already given. The same holds for utility functions in general. Doing some computation does not undo the fact that assigned numbers are used.
In some cases of numerical ranking, the method is quite similar to measurement. An example is the correlation coefficient of statistics—which is computed—when the two sets of numbers have no subjective elements. It quantifies the extent (and direction) of a linear relationship between two sets of numbers. However, there is no unit of magnitude like there is in true measurement, and a number by itself does not make a measurement. In my view the correlation coefficient is a quantifier but not a measurement.
To anticipate some questions, objections or confusion, true measurements may be the basis for ranking or categorizing. For example, one can rank 2 inches, 5 inches and 11 inches, or running times in a race. Such ranking is a separate, distinct act after the measuring; it is not measurement. As an example of the latter, a long distance race could categorize the contestants in age groups and award prizes in each age group. Counting may also provide the basis for ranking or categorizing. Stevens even confounds counting with measurement. However, ranking is very often subjective. Imagine people rating movies, songs, or the panelists on The Gong Show. Trying to equate ranking, labeling, or categorizing with true measurement is naive, sloppy, or corrupting.
Two important kinds of quantifying that Stevens' categories do not address are probabilities and statistics. Probabilities may be assigned, e.g., odds in a horse race; or found, as in frequency ratios or the probability of drawing three kings from a deck of playing cards. Frequency ratios follow the rules of arithmetic like ratio measures, but I would not call them a kind of measurement. There is no unit involved akin to inch, gram, miles per hour, etc., and the attribute of the numerator and denominator do not match (except when the probability equals 1). Probability is another kind of quantification, the latter being the wider category that also includes measurement and ranking. Statistics are also not measurements due to lack of a unit standard of magnitude.
In summary the differences between true measurement and ranking or labeling are huge, and the similarities are moderate at best and usually superficial. Using "measuring" to include them is as corrupt as a welfare-statist's use of "rights." Measurement is a hallmark of science, reason and objectivity. It should be defended. It is too important to allow it to be corrupted by subjectivism. How's that for KASS? :-)
For anyone interested in more about this, I recommend Measurement in Psychology by Joel Michell, a psychology professor and defender of true measurement.
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