“On the Theory of Scales of Measurement”

Nominal Scales

The nominal scale is at the lowest level of Stevens's hierarchy. Scales at this level assign numbers only as labels that can be used to distinguish whether attributes of different objects are the same. For example, the numbers “1” and “2” might be assigned to males and females as a shorthand means of differentiating the two. Here, the numbers used are not meaningful in any numerical sense, but are simply chosen for convenience. Other numbers, such as 100 and 200, would serve equally well. Permissible transformations of numbers on a nominal scale include any one-to-one substitution. As an example, assigning the numbers 3 and 4 instead of 1 and 2 would not alter our ability to distinguish between males and females, as long as we knew the “rule” governing this assignment. Because the numbers assigned have no numerical meaning, the only statistics considered to be permissible are those based on counts of the number in each category. Thus, we could determine how many males and females there were, or whether males or females were more numerous. However, we have no way of determining whether one gender is larger or greater than another because numbers in the nominal scale are arbitrary and do not support such a numerical meaning.

Ordinal Scales

In addition to the property of distinctiveness, ordinal scales must have the property of order, or of determining whether one object has more or less of an attribute than another. Thus, ordinal scales provide a means for ordering objects along a continuum of some sort. One familiar example is the outcome of a race. Contestants are ranked according to their order in crossing the finish line, with the first finisher regarded as faster than the second, and so on. Note that such ranks do not provide any information regarding the magnitude of difference between those with different ranks. In other words, based on an ordinal scaling, we have no way of knowing how much faster the first-place winner was than the second. Any transformation that preserves the original order is permissible for ordinal scales. For example, a constant such as two could be added to each rank, or different constants could be added to different ranks, as long as doing so did not change the original ordering. Statistical operations considered permissible for ordinal data include computation of the median, percentiles (although see Stevens, 1946, p. 679), and semi-interquartile range. Computations of statistics such as the mean or standard deviation are not considered permissible because they treat the intervals between scale points as equally spaced, which is not a property of ordinal scales.

...