---
title: "Level of measurement"
chunk: 3/4
source: "https://en.wikipedia.org/wiki/Level_of_measurement"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:44:10.167435+00:00"
instance: "kb-cron"
---

== Debate over Stevens's typology ==
While Stevens's typology is widely adopted, it is still being challenged by other theoreticians, particularly in the cases of the nominal and ordinal types (Michell, 1986). Duncan (1986), for example, objected to the use of the word measurement in relation to the nominal type and Luce (1997) disagreed with Stevens's definition of measurement.
On the other hand, Stevens (1975) said of his own definition of measurement that "the assignment can be any consistent rule. The only rule not allowed would be random assignment, for randomness amounts in effect to a nonrule". Hand says, "Basic psychology texts often begin with Stevens's framework and the ideas are ubiquitous. Indeed, the essential soundness of his hierarchy has been established for representational measurement by mathematicians, determining the invariance properties of mappings from empirical systems to real number continua. Certainly the ideas have been revised, extended, and elaborated, but the remarkable thing is his insight given the relatively limited formal apparatus available to him and how many decades have passed since he coined them."
The use of the mean as a measure of the central tendency for the ordinal type is still debatable among those who accept Stevens's typology. Many behavioural scientists use the mean for ordinal data anyway. This is often justified on the basis that the ordinal type in behavioural science is in fact somewhere between the true ordinal and interval types; although the interval difference between two ordinal ranks is not constant, it is often of the same order of magnitude.
For example, applications of measurement models in educational contexts often indicate that total scores have a fairly linear relationship with measurements across the range of an assessment. Thus, some argue that so long as the unknown interval difference between ordinal scale ranks is not too variable, interval scale statistics such as means can meaningfully be used on ordinal scale variables. Statistical analysis software such as SPSS requires the user to select the appropriate measurement class for each variable. This ensures that subsequent user errors cannot inadvertently perform meaningless analyses (for example correlation analysis with a variable on a nominal level).
L. L. Thurstone made progress toward developing a justification for obtaining the interval type, based on the law of comparative judgment. A common application of the law is the analytic hierarchy process. Further progress was made by Georg Rasch (1960), who developed the probabilistic Rasch model that provides a theoretical basis and justification for obtaining interval-level measurements from counts of observations such as total scores on assessments.

=== Other proposed typologies ===
Typologies aside from Stevens's typology have been proposed. For instance, Mosteller and Tukey (1977) and Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. See also Chrisman (1998), van den Berg (1991).

==== Mosteller and Tukey's typology (1977) ====
Mosteller and Tukey noted that the four levels are not exhaustive and proposed seven instead:

Names
Grades (ordered labels like beginner, intermediate, advanced)
Ranks (orders with 1 being the smallest or largest, 2 the next smallest or largest, and so on)
Counted fractions (bound by 0 and 1)
Counts (non-negative integers)
Amounts (non-negative real numbers)
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.

==== Chrisman's typology (1998) ====
Nicholas R. Chrisman 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 Stevens's original work, leading to the introduction of six new levels of measurement, for a total of ten:

Nominal
Gradation of membership
Ordinal
Interval
Log-interval
Extensive ratio
Cyclical ratio
Derived ratio
Counts
Absolute
While some claim that the extended levels of measurement are rarely used outside of academic geography, graded membership is central to fuzzy set theory, while absolute measurements include probabilities and the plausibility and ignorance in Dempster–Shafer theory. Cyclical ratio measurements include angles and times. Counts appear to be ratio measurements, but the scale is not arbitrary and fractional counts are commonly meaningless. Log-interval measurements are commonly displayed in stock market graphics. All these types of measurements are commonly used outside academic geography, and do not fit well to Stevens's original work.

=== Scale types and Stevens's "operational theory of measurement" ===
The theory of scale types is the intellectual handmaiden to Stevens's "operational theory of measurement", which was to become definitive within psychology and the behavioral sciences, despite Michell's characterization as its being quite at odds with measurement in the natural sciences (Michell, 1999). Essentially, the operational theory of measurement was a reaction to the conclusions of a committee established in 1932 by the British Association for the Advancement of Science to investigate the possibility of genuine scientific measurement in the psychological and behavioral sciences. This committee, which became known as the Ferguson committee, published a Final Report (Ferguson, et al., 1940, p. 245) in which Stevens's sone scale (Stevens & Davis, 1938) was an object of criticism: