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Nominal, ordinal, interval, & ratio data

By Steve Draper,   Department of Psychology,   University of Glasgow.

Introduction

The point of this page is to have a note to hand of the traditional categories: to remind me about the widely used 4-way categorisation of data types for stats. purposes. (It was introduced by Stevens (1946).) They have been criticised (by Tukey, and more recently by Velleman et al.) but are still very widely referred to: widely adopted but not universally accepted. Basically: if you don't think about the category / type of your data then you are very likely to choose a chart blindly and sub-optimally, and quite likely to choose the wrong stats test. This 4-way typing is usually necessary but often not sufficient: you may have to go further into the type. The criticisms mostly amount to a) the need to recognise additional types (e.g. percentages must be between 0 and 100, and do not count as interval data which you can shift by adding a constant to); b) The type is in fact not a property of the data, but of the questions you ask of the data. For instance, you might label your participants with an ID numbers (1,2,3...) which is used as categorial data that distinguishes any participant from any other; but if you wonder whether the participants you recruited later are different in kind from earlier ones, then their ordinal nature is important e.g. test whether the data from the first half of the participants, as shown by their lower ID numbers, is significantly different from the second half.

Basic definitions

NominalJust names, IDs
OrdinalHave / represent rank order (e.g. fully agree, mostly agree, somewhat agree)
IntervalHas a fixed size of interval between data points. (E.g. degrees Centigrade)
RatioHas a true zero point (e.g. mass, length, degrees Kelvin)

Summary table: The four data types

Attribute Nominal Ordinal Interval Ratio
Name2 Categorical Sequence Equal interval Ratio
Name3 Set Fully ordered, rank ordered Unit size fixed Zero or ref.pt fixed
Statistics Count, Mode, chi-squared + median, rank order correlation + ANOVA, mean, SDev + logs??
Example1 Set of participants, makes of car order of finishing a race centigrade scale Degrees Kelvin or absolute
Transformations/ rescaling allowed Rename Montonic (any curve that always increases) Linear -
Transformations 2 Hash function Montonic Add and multiply multiply?
Transformation examples 1:1 mapping, Assign colours for lines on a chart Sorting. Log or exp Z-transform, renormalise IQ scores Scale (zoom in or out)
Types of relativity A≠B A>B |(A-B)|  >  |(C-D)| ?
Types of absolute Identity of individual entities order, seqeuence intervals, differences ratios, proportions

Transformations

  • Thurstone scaling takes in ordinal data and generates an interval scale.
  • Spreadsheet (re)sorting takes any kind of data and generates ordinal data as represented, say, by the row number after sorting.
  • Log (or log-log, or exp()) transformations create interval data out of ratio or other interval data. This corresponds to the fact that even when a measurement scale has a zero (a ratio scale), the measure of interest may not e.g. may be a difference.

    An alternative list of types

    (Attributed to Mosteller & Tukey.)
    Names
    Grades   ordered labels such as Freshman, Sophomore, Junior, Senior
    Ranks   starting from one, which may represent either the largest or smallest
    Counted fractions   bounded by zero and one. These include percentages, for example.
    Counts   non-negative integers
    Amounts   non-negative real numbers
    Balances   unbounded, positive or negative values.
    Circles   (partially ordered, but in a circle). e.g. the points of the compass

    Links

  • P.F.Velleman & L.Wilkinson (1993) "Nominal, Ordinal, Interval, and Ratio Typologies are Misleading" The American Statistician (1993), vol.47 no.1 pp.65-72
  • wikip entry on this
  • Mosteller, Frederick & Tukey, John W. (1977) Data analysis and regression. A second course in statistics ch.5 Addison-Wesley Series in Behavioral Science: Quantitative Methods, (Reading, Mass.: Addison-Wesley)
  • http://www.graphpad.com/support/faqid/1089/
  • Which are Lickert scales/data?

    To Do

    Fill in the table

Thurstone as a transf?
Z-transf
Curve shapes e.g. log-exp.  v.diff graphs, but ..

Has a zero / not.
Has a unit / not
Proportions preserved; intervals preserved; Diffs are absolute
Full ordering / partial ordering / UK house numbering

BUT:
No bimodal test.


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