# Level of measurement

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The "**levels of measurement**", or **scales of measure** are expressions that typically refer to the **theory of scale types** developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 *Science* article titled "On the theory of scales of measurement".^{[1]} In that article, Stevens claimed that all measurement in science was conducted using four different types of scales that he called "nominal", "ordinal", "interval" and "ratio".

## The theory of scale types

Stevens (1946, 1951) proposed that measurements can be classified into four different types of scales. These are shown in the table below as: nominal, ordinal, interval, and ratio.

Scale Type | Permissible Statistics | Admissible Scale Transformation | Mathematical structure |
---|---|---|---|

nominal (also denoted as categorical) | mode, Chi-squared | One to One (equality (=)) | standard set structure (unordered) |

ordinal | median, percentile | Monotonic increasing (order (<)) | totally ordered set |

interval | mean, standard deviation, correlation, regression, analysis of variance | Positive linear (affine) | affine line |

ratio | All statistics permitted for interval scales plus the following: geometric mean, harmonic mean, coefficient of variation, logarithms | Positive similarities (multiplication) | one-dimensional vector space |

### Nominal scale

*Key Concept: Nominal scales are data that record categories.*

At the nominal scale, i.e., for a nominal category, one uses labels; for example, rocks can be generally categorized as igneous, sedimentary and metamorphic. For this scale, some valid operations are equivalence and set membership. Nominal measures offer names or labels for certain characteristics.

Variables assessed on a nominal scale are called **categorical variables**; see also categorical data. Categorically typed random variables that have only two possible outcomes (often termed "yes" vs. "no" or "success" vs. "failure") are known as **binary variables** (or **Bernoulli variables**), and characterized using a Bernoulli distribution. A categorical variable with 3 or more outcomes is sometimes termed **multi-way** (or **K-way** for some specific value of *K*), and characterized by a categorical distribution.

Stevens(1946, p. 679) must have known that claiming nominal scales to measure obviously non-quantitative things would have attracted criticism, so he invoked his theory of measurement to justify nominal scales as measurement:

“ | …the use of numerals as names for classes is an example of the assignment of numerals according to rule. The rule is: Do not assign the same numeral to different classes or different numerals to the same class. Beyond that, anything goes with the nominal scale. | ” |

The central tendency of a nominal attribute is given by its mode; neither the mean nor the median can be defined.

We can use a simple example of a nominal category: first names. Looking at nearby people, we might find one or more of them named Aamir. Aamir is their label; and the set of all first names is a nominal scale. We can only check whether two people have the same name (equivalence) or whether a given name is in on a certain list of names (set membership), but it is impossible to say which name is greater or less than another (comparison) or to measure the difference between two names. Given a set of people, we can describe the set by its most common name (the mode), but cannot provide an "average name" or even the "middle name" among all the names. However, if we decide to sort our names alphabetically (or to sort them by length; or by how many times they appear in the US Census), we will begin to turn this nominal scale into an ordinal scale.

### Ordinal scale

*Key Concept: Ordinal scales record information about the rank order of scores.*

Rank-ordering data simply puts the data on an ordinal scale. Ordinal measurements describe order, but not relative size or degree of difference between the items measured. In this scale type, the numbers assigned to objects or events represent the rank order (1st, 2nd, 3rd, etc.) of the entities assessed. An example of an ordinal scale is the result of a horse race, which says only which horses arrived first, second, or third but include no information about race times. Another example is military ranks; they have an order, but no well-defined numerical difference between ranks.

When using an ordinal scale, the central tendency of a group of items can be described by using the group's mode (or most common item) or its median (the middle-ranked item), but the mean (or average) cannot be defined.

In 1946, Stevens observed that psychological measurement usually operates on ordinal scales, and that ordinary statistics like means and standard deviations do not have valid interpretations. Nevertheless, such statistics can often be used to generate fruitful information, with the caveat that caution should be taken in drawing conclusions from such statistical data.

Psychometricians like to theorize that psychometric tests produce interval scale measures of cognitive abilities (e.g. Lord & Novick, 1968; von Eye, 2005) but there is little prima facie evidence to suggest that such attributes are anything more than ordinal for most psychological data (Cliff, 1996; Cliff & Keats, 2003; Michell, 2008). In particular,^{[2]} IQ scores reflect an ordinal scale, in which all scores are only meaningful for comparison, rather than an interval scale, in which a given number of IQ "points" corresponds to a unit of intelligence.^{[3]}^{[4]}^{[5]} Thus it is an error to write that an IQ of 160 is just as different from an IQ of 130 as an IQ of 100 is different from an IQ of 70.^{[6]}^{[7]}

In mathematical order theory, an ordinal scale defines a total preorder of objects (in essence, a way of sorting all the objects, in which some may be tied). The scale values themselves (such as labels like "great", "good", and "bad"; 1st, 2nd, and 3rd) have a total order, where they may be sorted into a single line with no ambiguities. If numbers are used to define the scale, they remain correct even if they are transformed by any monotonically increasing function. This property is known as the order isomorphism. A simple example follows:

Judge's scorex
| Score minus 8x-8
| Tripled score3x
| Cubed scorex
^{3} | |
---|---|---|---|---|

Alice's cooking ability | 10 | 2 | 30 | 1000 |

Bob's cooking ability | 9 | 1 | 27 | 729 |

Claire's cooking ability | 8.5 | 0.5 | 25.5 | 614.125 |

Dana's cooking ability | 8 | 0 | 24 | 512 |

Edgar's cooking ability | 5 | -3 | 15 | 125 |

Since x-8, 3x, and x^{3} are all monotonically increasing functions, replacing the ordinal judge's score by any of these alternate scores does not affect the relative ranking of the five people's cooking abilities. Each column of numbers is an equally legitimate ordinal scale for describing their abilities. However, the numerical (additive) difference between the various ordinal scores has no particular meaning.

See also Strict weak ordering.

### Interval scale

*Key Concept: Interval scales tell us about the order of data points, and the size of the intervals in between data points.*

Quantitative attributes are all measurable on interval scales, as any difference between the levels of an attribute can be multiplied by any real number to exceed or equal another difference. A highly familiar example of interval scale measurement is temperature with the Celsius scale. In this particular scale, the unit of measurement is 1/100 of the temperature difference between the freezing and boiling points of water under a pressure of 1 atmosphere. The "zero point" on an interval scale is arbitrary; and negative values can be used. The formal mathematical term is an affine space (in this case an affine line). Variables measured at the interval level are called "interval variables" or sometimes "scaled variables" as they have units of measurement.

Ratios between numbers on the scale are not meaningful, so operations such as multiplication and division cannot be carried out directly. But ratios of differences can be expressed; for example, one difference can be twice another.

The central tendency of a variable measured at the interval level can be represented by its mode, its median, or its arithmetic mean. Statistical dispersion can be measured in most of the usual ways, which just involved differences or averaging, such as range, interquartile range, and standard deviation. Since one cannot divide, one cannot define measures that require a ratio, such as studentized range or coefficient of variation. More subtly, while one can define moments about the origin, only central moments are useful, since the choice of origin is arbitrary and not meaningful. One can define standardized moments, since ratios of differences are meaningful, but one cannot define coefficient of variation, since the mean is a moment about the origin, unlike the standard deviation, which is (the square root of) a central moment.

### Ratio scale

*Key Concept: A ratio scale is an interval scale with a true zero point.*

Most measurement in the physical sciences and engineering is done on ratio scales. Mass, length, time, plane angle, energy and electric charge are examples of physical measures that are ratio scales. The scale type takes its name from the fact that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind (Michell, 1997, 1999). Informally, the distinguishing feature of a ratio scale is the possession of a zero value. For example, the Kelvin temperature scale has a non-arbitrary zero point of absolute zero, which is denoted 0K and is equal to -273.15 degrees Celsius. This zero point is accurately representing the particles that compose matter at this temperature having zero kinetic energy.

Examples of ratio scale measurement in the behavioral sciences are all but non-existent. Luce (2000) argues that an example of ratio scale measurement in psychology can be found in rank and sign dependent expected utility theory.

All statistical measures can be used for a variable measured at the ratio level, as all necessary mathematical operations are defined. The central tendency of a variable measured at the ratio level can be represented by, in addition to its mode, its median, or its arithmetic mean, also its geometric mean or harmonic mean. In addition to the measures of statistical dispersion defined for interval variables, such as range and standard deviation, for ratio variables one can also define measures that require a ratio, such as studentized range or coefficient of variation.

## Debate on classification scheme

There has been, and continues to be, debate about the merits of the classifications, particularly in the cases of the nominal and ordinal classifications (Michell, 1986). Thus, while Stevens' classification is widely adopted, it is by no means universally accepted.^{[8]}

Duncan (1986) observed that Stevens' classification *nominal measurement* is contrary to his own definition of *measurement*. Stevens (1975) said on 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". However, so-called nominal measurement involves arbitrary assignment, and the "permissible transformation" is any number for any other. This is one of the points made in Lord's (1953) satirical paper *On the Statistical Treatment of Football Numbers*.

Among those who accept the classification scheme, there is also some controversy in behavioural sciences over whether the mean is meaningful for ordinal measurement. In terms of measurement theory, it is not, because the arithmetic operations are not made on numbers that are measurements in units, and so the results of computations do not give numbers in units. However, many behavioural scientists use means for ordinal data anyway. This is often justified on the basis that ordinal scales in behavioural science are really somewhere between true ordinal and interval scales; 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 a 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 PSPP require 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 interval-level measurements based on the law of comparative judgment. For a common application of the law, see 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.

Another issue is derived from Nicholas R. Chrisman's article "Rethinking Levels of Measurement for Cartography",^{[9]} in which he introduces an expanded list of levels of measurement to account for various measurements that do not necessarily fit with the traditional notion of levels of measurement. Measurements bound to a range and repeat (like degrees in a circle, time, etc.), graded membership categories, and other types of measurement do not fit to Steven's original work, leading to the introduction of 6 new levels of measurement leading to: (1) Nominal, (2) Graded membership, (3) Ordinal, (4) Interval, (5) Log-Interval, (6) Extensive Ratio, (7) Cyclical Ratio, (8) Derived Ratio, (9) Counts and finally (10) Absolute. The extended levels of measurement are rarely used outside of academic geography.

## Scale types and Stevens' "operational theory of measurement"

The theory of scale types is the intellectual handmaiden to Stevens' "operational theory of measurement", which was to become definitive within psychology and the behavioral sciences,^{[citation needed]} 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' sone scale (Stevens & Davis, 1938) was an object of criticism:

“ | …any law purporting to express a quantitative relation between sensation intensity and stimulus intensity is not merely false but is in fact meaningless unless and until a meaning can be given to the concept of addition as applied to sensation. | ” |

That is, if Stevens' *sone* scale was genuinely measuring the intensity of auditory sensations, then evidence for such sensations as being quantitative attributes must be produced. The evidence needed was the presence of *additive structure* - a concept comprehensively treated by the German mathematician Otto Hölder (Hölder, 1901). Given the physicist and measurement theorist Norman Robert Campbell dominated the Ferguson committee's deliberations, the committee concluded that measurement in the social sciences was impossible due to the lack of concatenation operations. This conclusion was later rendered false by the discovery of the theory of conjoint measurement by Debreu (1960) and independently by Luce & Tukey (1964). However, Stevens' reaction was not to conduct experiments to test for the presence of additive structure in sensations, but instead to render the conclusions of the Ferguson committee null and void by proposing a new theory of measurement:

“ | Paraphrasing N.R. Campbell (Final Report, p.340), we may say that measurement, in the broadest sense, is defined as the assignment of numerals to objects and events according to rules (Stevens, 1946, p.677).
| ” |

Stevens was greatly influenced by the ideas of another Harvard academic, the Nobel laureate physicist Percy Bridgman (1927), whose doctrine of *operationism* Stevens used to define measurement. In Stevens' definition for example, it is the use of a tape measure that defines length (the object of measurement) as being measurable (and so by implication quantitative). Critics of operationism object that it confuses the relations between two objects or events for properties of one of those of objects or events (Hardcastle, 1995; Michell, 1999; Moyer, 1981a,b; Rogers, 1989).

The Canadian measurement theorist William Rozeboom (1966) was an early and trenchant critic of Stevens' theory of scale types. But it was not until much later with the work of mathematical psychologists Theodore Alper (1985, 1987), Louis Narens (1981a, b) and R. Duncan Luce (1986, 1987, 2001) did the concept of scale types receive the mathematical rigour that it lacked at its inception. As Luce (1997, p. 395) bluntly stated:

“ | 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' broad definition of measurement…in our view, the only sensible meaning for 'rule' is empirically testable laws about the attribute. | ” |

## See also

- Cohen's kappa
- Category theory
- Inter-rater reliability
- Measure (mathematics)
- Quantitative data
- Qualitative data
- Ramsey–Lewis method
- Scale
- Scaling (testing)
- List of analyses of categorical data

## References

- ↑
Stevens, S. S. (1946). On the Theory of Scales of Measurement.
*Science***103**(2684): 677–680. - ↑ Sheskin, David J. (2007).
*Handbook of Parametric and Nonparametric Statistical Procedures*, Fourth, Boca Raton (FL): Chapman & Hall/CRC. "Although in practice IQ and most other human characteristics measured by psychological tests (such as anxiety, introversion, self esteem, etc.) are treated as interval scales, many researchers would argue that they are more appropriately categorized as ordinal scales. Such arguments would be based on the fact that such measures do not really meet the requirements of an interval scale, because it cannot be demonstrated that equal numerical differences at different points on the scale are comparable." - ↑ Mussen, Paul Henry (1973).
*Psychology: An Introduction*, Lexington (MA): Heath. "The I.Q. is essentially a rank; there are no true "units" of intellectual ability." - ↑ Truch, Steve (1993).
*The WISC-III Companion: A Guide to Interpretation and Educational Intervention*, Austin (TX): Pro-Ed. "An IQ score is not an equal-interval score, as is evident in Table A.4 in the WISC-III manual." - ↑ Bartholomew, David J. (2004).
*Measuring Intelligence: Facts and Fallacies*, Cambridge: Cambridge University Press. "When we come to quantities like IQ or g, as we are presently able to measure them, we shall see later that we have an even lower level of measurement—an ordinal level. This means that the numbers we assign to individuals can only be used to rank them—the number tells us where the individual comes in the rank order and nothing else." - ↑ Eysenck, Hans (1998).
*Intelligence: A New Look*, 24–25, New Brunswick (NJ): Transaction Publishers. "Ideally, a scale of measurement should have a true zero-point and identical intervals. . . . Scales of hardness lack these advantages, and so does IQ. There is no absolute zero, and a 10-point difference may carry different meanings at different points of the scale." - ↑ Mackintosh, N. J. (1998).
*IQ and Human Intelligence*, 30–31, Oxford: Oxford University Press. "In the jargon of psychological measurement theory, IQ is an ordinal scale, where we are simply rank-ordering people. . . . It is not even appropriate to claim that the 10-point difference between IQ scores of 110 and 100 is the same as the 10-point difference between IQs of 160 and 150" - ↑ Velleman, Paul F., Wilkinson,Leland (1993). Nominal, Ordinal, Interval, and Ratio Typologies Are Misleading.
*The American Statistician***47**(1): 65–72. - ↑ Chrisman, Nicholas R. (1998). Rethinking Levels of Measurement for Cartography.
*Cartography and Geographic Information Science*, vol. 25 (4), pp. 231-242

## Further reading

- Alper, T. M. (1985). A note on real measurement structures of scale type (m, m + 1).
*Journal of Mathematical Psychology*,*29*, 73–81. - Alper, T.M. (1987). A classification of all order-preserving homeomorphism groups of the reals that satisfy finite uniqueness.
*Journal of Mathematical Psychology*,*31*, 135–154. - Briand, L. & El Emam, K. & Morasca, S. (1995). On the Application of Measurement Theory in Software Engineering.
*Empirical Software Engineering*,*1*, 61–88. [On line] http://www2.umassd.edu/swpi/ISERN/isern-95-04.pdf - Babbie, E. (2004).
*The Practice of Social Research*, 10th edition, Wadsworth, Thomson Learning Inc., ISBN 0-534-62029-9 - Cliff, N. (1996).
*Ordinal Methods for Behavioral Data Analysis*. Mahwah, NJ: Lawrence Erlbaum. ISBN 0-8058-1333-0 - Cliff, N. & Keats, J. A. (2003).
*Ordinal Measurement in the Behavioral Sciences*. Mahwah, NJ: Erlbaum. ISBN 0-8058-2093-0 - Lord, Frederic M (December 1953). On the Statistical Treatment of Football Numbers.
*American Psychologist***8**(12): 750–751.

- See also reprints in:

*Readings in Statistics*, Ch. 3, (Haber, A., Runyon, R.P., and Badia, P.) Reading, Mass: Addison-Wesley, 1970.

(2007) "Chapter 31" Maranell, Gary Michael*Scaling: A Sourcebook for Behavioral Scientists*, 402–405, Aldine Transaction. URL accessed 16 September 2010.

- Lord, F.M., & Novick, M.R. (1968).
*Statistical theories of mental test scores*. Reading, MA: Addison-Wesley. - Luce, R.D. (1986). Uniqueness and homogeneity of ordered relational structures.
*Journal of Mathematical Psychology*,*30*, 391–415. - Luce, R.D. (1987). Measurement structures with Archimedean ordered translation groups.
*Order*,*4*, 165–189. - Luce, R.D. (1997). Quantification and symmetry: commentary on Michell 'Quantitative science and the definition of measurement in psychology'.
*British Journal of Psychology*,*88*, 395–398. - Luce, R.D. (2000).
*Utility of uncertain gains and losses: measurement theoretic and experimental approaches*. Mahwah, N.J.: Lawrence Erlbaum. - Luce, R.D. (2001). Conditions equivalent to unit representations of ordered relational structures.
*Journal of Mathematical Psychology*,*45*, 81–98. - Luce, R.D. & Tukey, J.W. (1964). Simultaneous conjoint measurement: a new scale type of fundamental measurement.
*Journal of Mathematical Psychology*,*1*, 1–27. - Michell, J. (1986). Measurement scales and statistics: a clash of paradigms.
*Psychological Bulletin*,*3*, 398–407. - Michell, J. (1997). Quantitative science and the definition of measurement in psychology.
*British Journal of Psychology*,*88*, 355–383. - Michell, J. (1999).
*Measurement in Psychology – A critical history of a methodological concept*. Cambridge: Cambridge University Press. - Michell, J. (2008). Is psychometrics pathological science?
*Measurement – Interdisciplinary Research & Perspectives*,*6*, 7–24. - Narens, L. (1981a). A general theory of ratio scalability with remarks about the measurement-theoretic concept of meaningfulness.
*Theory and Decision*,*13*, 1–70. - Narens, L. (1981b). On the scales of measurement.
*Journal of Mathematical Psychology*,*24*, 249–275. - Rasch, G. (1960).
*Probabilistic models for some intelligence and attainment tests*. Copenhagen: Danish Institute for Educational Research. - Rozeboom, W.W. (1966). Scaling theory and the nature of measurement.
*Synthese*,*16*, 170–233. - Stevens, S.S (June 7, 1946). On the Theory of Scales of Measurement.
*Science***103**(2684): 677–680. - Stevens, S.S. (1951). Mathematics, measurement and psychophysics. In S.S. Stevens (Ed.),
*Handbook of experimental psychology*(pp. 1–49). New York: Wiley. - Stevens, S.S. (1975).
*Psychophysics*. New York: Wiley. - von Eye, A. (2005). Review of Cliff and Keats,
*Ordinal measurement in the behavioral sciences*.*Applied Psychological Measurement*,*29*, 401–403.

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