Basic Empirical Operations
Determination of equality
Mathematical Group Structure
Permutation group
x' = f(x)
f(x) means any one-to-one substitution
Permission Statistics (invariantive)
Number of cases
Mode
Contingency correlation
The nominal scale represents the most unrestricted assignment of numerals. The numerals are used only as labels or type numbers, and words or letters would serve as well. Two types of nominal assignments are sometimes distinguished, as illustrated (a) by the 'numbering'of football players for the identification of the individuals, and (b) by the 'numbering' of types or classes, where each member of a class is assigned the same numeral. Actually, the first is a special case of the second, for when we label our football players we are dealing with unit classes of one member each.Since the purpose is just as well served when any two designating numerals are interchanged, this scale form remains invariant under the general substitution or permutation group (sometimes called the symmetric group of transformations). The only statistic relevantto nominal scales of Type A is the number of cases, e.g. the number of players assigned numerals.But once classes containing several individuals have been formed (Type B), we can determine the most numerous class (the mode), and under certain conditions we can test, by the contingency methods,hypotheses regarding the distribution of cases among the classes.
The nominal scale is a primitive form, and quite naturally there are many who will urge that it is absurd to attribute to this process of assigning numerals the dignity implied by the term measurement. Certainly there can be no quarrel with this objection, for the naming of things is an arbitrary business. However we christen it, the use of numerals as names for classes is an example of the "assignment of numerals according to rule." The rule is: Do not assign the same numeral to different classes or different numerals to the same class. Beyond that, anything goes with the nominal scale.
