Measurement Scales: Nominal and Ordinal (Notes)

Nominal scale

• Definition and role

  • Qualitative type of measurement that uses labels to categorize data
  • Non-numeric; no mathematical operations are appropriate
  • Also described as a labeling or categorization scheme

• Characteristics

  • Is qualitative and non-numeric
  • Data are simply labels or categories without intrinsic order
  • No arithmetic meaning for differences between categories

• Examples

  • Survey responses: yes, no, and undecided
  • Colors of cars driven by college students: red, black, white, etc.
  • Classification of a sample of college instructors according to the subject taught
  • Classification of residents according to ZIP codes (labels)

• Note on the scope of the excerpt

  • Transcript states there are four common types of scales but the excerpt only details Nominal (and Ordinal) scales; other types are not described here.

Ordinal scale

• Definition and role

  • Qualitative form of measurement that imposes order or ranking on categories
  • Data are arranged in a meaningful order (ranked categories) but with no guaranteed equal intervals

• Characteristics

  • Categories have a rank or order
  • Precise differences between ranks do not necessarily exist or are not meaningful
  • Differences between ranks are not quantitatively comparable in the same way as interval/ratio scales
  • May still be treated as qualitative data, with emphasis on order rather than magnitude

• Examples

  • Bands in a state-wide music competition rated 1, 2, 3, or 4
  • College grades: A, B, C, D, or F
  • Magazine livability rankings: first, second, third, etc.
  • Guest speaker evaluations: superior, average, or poor
  • Additional note in transcript: English, history, psychology, mathematics, etc. (represented as courses or subjects in a ranking/context)

• Important caveat (interpretation of differences)

  • NOTE: Meaningful differences cannot be calculated with Ordinal data.
  • For example, we cannot determine differences between letter grades or placements.
  • We know that an "A" is higher than a "B," but we cannot subtract B from A to know the difference.
  • The same limitation applies to placements: yes, we can subtract first place from second place, but it is not an exact quantity that can be compared to other such differences.

• Mathematical representation (conceptual)

  • Rankings are represented by positions r_i ∈ {1, 2, …, k}
  • The difference between ranks, ri - rj, is not meaningful in the ordinal context
  • Practical implication: ordinal data supports order, not precise quantification

• Significance and implications

  • Ordinal data captures order information but not equal spacing
  • Useful for defining preferences, ratings, or hierarchical classifications
  • Limits on arithmetic operations and summary statistics (avoid computing means of ordinal categories)

Cross-cutting notes

• The transcript contrasts qualitative versus quantitative labeling in the context of these scales; nominal and ordinal are described as qualitative scales, with ordinal providing ordered categories but not precise numerical differences.
• Practical examples tie to real-world contexts: student grades, city livability rankings, and evaluations of speakers.
• The note emphasizes that while ordinal data can be ordered, the magnitude of differences between ranks is not necessarily consistent or meaningful across comparisons.
• Ethical/philosophical considerations: not explicitly discussed in the excerpt; the notes focus on measurement properties and limitations.

Quick reference summaries

• Nominal scale

  • Qualitative, labels only, non-numeric
  • Examples: yes/no/undecided; car colors; subject taught; ZIP-code labels
  • No mathematical operations on category differences

• Ordinal scale

  • Qualitative with an implicit order/rank
  • Examples: bands 1–4; A–F grades; livability rankings; speaker quality
  • Differences between ranks are not necessarily equal or meaningful
  • Some subtractions (e.g., first vs second) reflect order, not a true numeric distance

ri - rj is not meaningful for ordinal data; this distinction guides appropriate data analysis decisions.