Ordinal Scale
The ordinal scale is the second level in Stanley Smith Stevens's 1946 typology of measurement: values can be put in a meaningful order — first, second, third; mild, moderate, severe; strongly disagree through strongly agree — but the distances between adjacent ranks are not defined or not equal. A race result tells who beat whom without saying by how much, and that missing magnitude is precisely what separates ordinal data from the interval and ratio scales above it.
What ordinal data adds to nominal data is order, so any transformation that preserves rank is permissible: a monotonic increasing change such as relabeling 1, 2, 3 as 10, 20, 500 leaves all the information intact. Stevens called this the isotonic group. The corollary is that quantities depending on spacing are suspect: the "gap" between a first-place and second-place finish may be a hundredth of a second while the gap between second and third is a minute, and a mean of ranks assumes the gaps are equal. Two classic examples are the Mohs hardness scale from 1812, where diamond is 10 and corundum is 9 though diamond is far harder than the one-step difference suggests, and the stages of cancer or severity grades in medicine, where stage III is worse than stage II in an unquantified way.
Statistics that use only order — the median, percentiles, the interquartile range, and rank-based methods — are the natural tools for ordinal data. The median of a Likert item is meaningful because it depends only on which response sits in the middle. Charles Spearman's 1904 rank correlation, Kendall's tau (Maurice Kendall, 1938), the Mann-Whitney U test (1947), the Wilcoxon tests (Frank Wilcoxon, 1945), and the Kruskal-Wallis test (1952) all operate on ranks, which is why they are called nonparametric and why they are robust. For ordered categories regression variants such as ordinal logistic regression (the proportional-odds model of Peter McCullagh, 1980) respect the order without assuming equal spacing.
Ordinal data is ubiquitous in surveys, ratings, medicine, and ranking systems, and in machine learning it motivates ordinal-aware modeling. Rensis Likert's 1932 attitude scales produced the five-point format that fills questionnaires, product reviews (one to five stars), and satisfaction surveys. Search engines and recommenders are evaluated on ranked lists with metrics like NDCG that care about order rather than spacing, and ordinal regression, or cumulative-link models, predict ordered targets such as credit rating tiers or disease severity without pretending the steps are equal. When a target has ordered classes, treating them as nominal wastes the ordering and treating them as a number imposes uniform spacing, so ordinal-specific losses sit between the two.
The central controversy is whether to compute means of ordinal data, and the honest answer is that it depends on how well the equal-spacing assumption holds. Stevens's rules were criticized almost immediately, including by Frederic Lord in 1953 ("On the statistical treatment of football numbers") and later by Velleman and Wilkinson in 1993, who argued that the data's measurement level does not by itself dictate the right analysis. Still, a mean of a single five-point item can mislead: two groups with the same average of 3 may be one cluster of neutrals and one split between extremes. Another pitfall is the ceiling effect, when most responses crowd the top category and the scale loses discrimination, and a third is assuming rankings reveal magnitude, as with a university league table where rank 5 and rank 6 may be nearly tied.
Read the ordinal scale as measurement that knows who is ahead but not by how much: order is the one structure it guarantees, so rank-based summaries are always safe, while any statistic that leans on equal spacing is an assumption someone must be prepared to defend.