Level of measurement

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Not to be confused with Level measurement.

Level of measurement or scale of measure is a classification that describes the nature of information within the numbers assigned to variables.[1] Psychologist Stanley Smith Stevens developed the best known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio.[1][2] Other classifications include those by Chrisman[3] and by Mosteller and Tukey.[4] This framework of distinguishing levels of measurement originated in psychology and is widely criticized by scholars in other disciplines.[5]

Stevens's typology[edit]



S. S. Stevens (1946, 1951, 1975) claimed that what counted was having an interval or ratio scale. Subsequent research has given meaning to this assertion, but given his attempts to invoke scale type ideas it is doubtful if he understood it himself ... no measurement theorist I know accepts Stevens's broad definition of measurement ... in our view, the only sensible meaning for 'rule' is empirically testable laws about the attribute.




Measure Property Mathematical






Nominal Classification, Membership =, != Grouping Mode
Ordinal Comparison, Level >, < Sorting Median
Interval Difference, Affinity +, - Yardstick Mean,


Ratio Magnitude, Amount *, / Ratio Geometric Mean,

Coeff. of Variation

Nominal level[edit]


Examples of these classifications include gender, nationality, ethnicity, language, genre, style, biological species, and form.[6][7] In a university one could also use hall of affiliation as an example. Other concrete examples are

Nominal scales were often called qualitative scales, and measurements made on qualitative scales were called qualitative data. However, the rise of qualitative research has made this usage confusing. The numbers in nominal measurement are assigned as labels and have no specific numerical value or meaning. No form of mathematical computation (+,- x etc.) may be performed on Nominal measures. Nominal level is the lowest measurement level used from a statistical point of view.

Mathematical operations[edit]

Equality and other operations that can be defined in terms of equality, such as inequality and set membership, are the only non-trivial operations that generically apply to objects of the nominal type.

Central tendency[edit]


Ordinal scale[edit]

Further information: Ordinal data


Central tendency


In 1946, Stevens observed that psychological measurement, such as measurement of opinions, usually operates on ordinal scales; thus means and standard deviations have no validity, but they can be used to get ideas for how to improve operationalization of variables used in questionnaires. Most psychological data collected by psychometric instruments and tests, measuring cognitive and other abilities, are ordinal, although some theoreticians have argued they can be treated as interval or ratio scales. However, there is little prima facie evidence to suggest that such attributes are anything more than ordinal (Cliff, 1996; Cliff & Keats, 2003; Michell, 2008).[9] In particular,[10] IQ scores reflect an ordinal scale, in which all scores are meaningful for comparison only.[11][12][13] There is no absolute zero, and a 10-point difference may carry different meanings at different points of the scale.[14][15]

Interval scale[edit]

The interval type allows for the degree of difference between items, but not the ratio between them. Examples include temperature with the Celsius scale, which has two defined points (the freezing and boiling point of water at specific conditions) and then separated into 100 intervals, date when measured from an arbitrary epoch (such as AD), percentage such as a percentage return on a stock,[16] location in Cartesian coordinates, and direction measured in degrees from true or magnetic north. Ratios are not meaningful since 20 °C cannot be said to be "twice as hot" as 10 °C, nor can multiplication/division be carried out between any two dates directly. However, ratios of differences can be expressed; for example, one difference can be twice another. Interval type variables are sometimes also called "scaled variables", but the formal mathematical term is an affine space (in this case an affine line).

Central tendency and statistical dispersion[edit]


Ratio scale[edit]


Central tendency and statistical dispersion


Debate on Stevens's typology[edit]






Scale types and Stevens's "operational theory of measurement"[edit]






The Canadian measurement theorist William Rozeboom (1966) was an early and trenchant critic of Stevens's theory of scale types.

Same variable may be different scale type depending on context[edit]

Another issue is that the same variable may be a different scale type depending on how it is measured and on the goals of the analysis. For example, hair color is usually thought of as a nominal variable, since it has no apparent ordering.[19] However, it is possible to order colors (including hair colors) in various ways, including by hue; this is known as colorimetry. Hue is an interval level variable.

Chrisman's typology[edit]

Nicholas R. Chrisman wrote an article, "Rethinking Levels of Measurement for Cartography",[3] in which he introduced an expanded list of levels of measurement to account for various measurements that do not necessarily fit with the traditional notions of levels of measurement. Measurements bound to a range and repeating (like degrees in a circle, clock time, etc.), graded membership categories, and other types of measurement do not fit to Steven's original work, leading to the introduction of six new levels of measurement, for a total of ten:

  1. Nominal
  2. Graded membership
  3. Ordinal
  4. Interval
  5. Log-Interval
  6. Extensive Ratio
  7. Cyclical Ratio
  8. Derived Ratio
  9. Counts
  10. Absolute

The extended levels of measurement are rarely used outside of academic geography.[20]

Mosteller and Tukey's typology[edit]

Mosteller and Tukey[4] noted that the four levels are not exhaustive and proposed:

  1. Names
  2. Grades (e.g. freshmen, sophomores etc.)
  3. Counted fractions bound by 0 and 1
  4. Counts (non-negative integers)
  5. Amounts (non-negative real numbers)
  6. Balances (any real number)

For example, percentages (a variation on fractions in the Mosteller-Tukey framework) do not fit well into Stevens’s framework: No transformation is fully admissible.[17]

See also[edit]


  1. ^ a b Kirch, Wilhelm, ed. (2008). "Level of measurement". Encyclopedia of Public Health. 2. Springer. pp. 851–852. doi:10.1007/978-1-4020-5614-7_1971. ISBN 978-1-4020-5613-0. Retrieved 14 August 2015. 
  2. ^ a b Stevens, S. S. (7 June 1946). "On the Theory of Scales of Measurement". Science. 103 (2684): 677–680. doi:10.1126/science.103.2684.677. PMID 17750512. Retrieved 7 February 2016. 
  3. ^ a b Chrisman, Nicholas R. (1998). "Rethinking Levels of Measurement for Cartography". Cartography and Geographic Information Science. 25 (4): 231–242. doi:10.1559/152304098782383043. ISSN 1523-0406. Retrieved 14 August 2015.  – via Taylor & Francis (subscription required)
  4. ^ a b Mosteller, Frederick (1977). Data analysis and regression : a second course in statistics. Reading, Mass: Addison-Wesley Pub. Co. ISBN 978-0201048544. 
  8. ^ Manikandan, S (2011). "Measures of central tendency: Median and mode". Journal of Pharmacology and Pharmacotherapeutics. 2 (3): 214, 215. doi:10.4103/0976-500X.83300. PMC 3157145free to read. 
  9. ^ *Lord, Frederic M.; Novick, Melvin R.; Birnbaum, Allan (1968). Statistical Theories of Mental Test Scores. Reading (MA): Addison-Wesley. p. 21. LCCN 68011394. Lay summary (24 June 2013). Although, formally speaking, interval measurement can always be obtained by specification, such specification is theoretically meaningful only if it is implied by the theory and model relevant to the measurement procedure. 
  16. ^ = "Ken Black 2009" Black, Ken (2010). "Introduction". Business Statistics: Contemporary Decision Making (6th ed.). Hoboken, NJ: Wiley. pp. 2–135. ISBN 0-470-40901-0. 
  17. ^ a b Velleman, Paul F.; Wilkinson, Leland (1993). "Nominal, ordinal, interval, and ratio typologies are misleading". The American Statistician. American Statistical Association. 47 (1): 65–72. doi:10.2307/2684788. JSTOR 2684788. 
  18. ^ Lord, Frederic M. (December 1953). "On the Statistical Treatment of Football Numbers". American Psychologist. 8 (12): 750–751. doi:10.1037/h0063675. 
  19. ^ "What is the difference between categorical, ordinal and interval variables?". Institute for Digital Research and Education. University of California, Los Angeles. Retrieved 7 February 2016. 
  20. ^ Wolman, Abel G (2006). "Measurement and meaningfulness in conservation science". Conservation biology. 

Further reading[edit]

See also reprints in:

External links[edit]