Nominal Scale
The nominal scale is the lowest level of measurement in the typology published by the psychologist Stanley Smith Stevens in his 1946 paper "On the Theory of Scales of Measurement" in the journal Science: numbers or labels are used purely as names to classify things into mutually exclusive categories, with no order, distance, or zero. Eye color, species, blood group, nationality, and the player numbers on sports jerseys are nominal; the only information they carry is "same" or "different."
The only structure a nominal scale preserves is identity, so the permissible transformation is any one-to-one relabeling: renaming red as blue and blue as red, or swapping the numeric codes 1, 2, 3 for 30, 10, 20, changes nothing about the information. Stevens called this the permutation group, and it is the reason arithmetic is forbidden here. If cities are coded 1 for Paris, 2 for Lima, and 3 for Oslo, the code for Oslo being "larger" tells nothing, and a mean code of 2 corresponds to no city at all. Any analysis whose answer changes when the labels are shuffled is not licensed by the scale, and that shuffle test is the practical way to spot nominal data wearing numerals.
The statistics that survive such relabeling are the ones built on counting: frequencies, proportions, the mode, and tests of association. The mode is the only measure of center defined, because it depends solely on which label is most common. Pearson's chi-square test of independence (1900), Cramér's V (Harald Cramér, 1946), and the contingency coefficient quantify association between nominal variables, and Cohen's kappa (1960) measures agreement between two raters assigning nominal labels. Entropy and the Gini impurity used in decision trees are also built from category proportions and are therefore legitimate on this scale. Median, mean, and standard deviation are not meaningful, and a correlation coefficient computed on arbitrary codes is an artifact of the arbitrary numbering.
Nominal variables appear throughout applied work, and their treatment in modeling is where the scale's limits become concrete. Classification targets in machine learning (spam or not, which of ten digits, which disease) are nominal, as are features like browser, merchant category, or genotype. Because there is no order, algorithms need the categories represented without implying one: one-hot or dummy encoding for linear and neural models, native categorical splits in tree libraries. Multiclass metrics such as the confusion matrix and macro-averaged F1 treat the classes symmetrically, which fits the scale's lack of ordering. The binary special case, such as treated versus control, is still nominal, though coding it 0 and 1 lets proportions be handled as means.
Common pitfalls involve numerals that pose as quantities and categories that are quietly not nominal. Zip codes, phone numbers, product codes, and database keys are numerals with only identity content, so computing the average zip code is meaningless even though software will happily do it. In the other direction, a coarse variable such as "education level" is often analyzed as nominal when it is ordinal, discarding rank information and with it statistical power. Category definitions also matter: they must be exhaustive and mutually exclusive, and ethnicity or occupation schemes are social constructions that vary by country and decade. Finally, merely having labels does not make a variable nominal forever, because the same data can be reclassified by purpose.
Read the nominal scale as numbers and words used strictly as names: the only questions it answers are whether two things are the same kind and how often each kind occurs, and any calculation that would give a different answer after the labels are shuffled has claimed more than the scale can support.