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Nominal, ordinal, interval, & ratio data
Department of Psychology,
University of Glasgow.
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
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.
|Nominal||Just names, IDs|
|Ordinal||Have / represent rank order (e.g. fully agree, mostly
agree, somewhat agree)|
|Interval||Has a fixed size of interval between data points.
(E.g. degrees Centigrade)|
|Ratio||Has a true zero point (e.g. mass, length, degrees
Summary table: The four data types
|| Equal interval
|| Fully ordered, rank ordered
|| Unit size fixed
|| Zero or ref.pt fixed|
|| Count, Mode, chi-squared
|| + median, rank order correlation
|| + ANOVA, mean, SDev
|| + logs??|
|| Set of participants, makes of car
|| order of finishing a race
|| centigrade scale
|| Degrees Kelvin or absolute|
|Transformations/ rescaling allowed
|| Montonic (any curve that always increases)
|| Hash function
|| Add and multiply
|| 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)| > |(C-D)|
|Types of absolute
|| Identity of individual entities
|| order, seqeuence
|| intervals, differences
|| ratios, proportions|
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.)
|Grades ||ordered labels such as Freshman, Sophomore, Junior,
|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 |
P.F.Velleman & L.Wilkinson (1993)
"Nominal, Ordinal, Interval, and Ratio Typologies are
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)
Which are Lickert scales/data?
Fill in the table
Thurstone as a 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
No bimodal test.
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