Editing Level of measurement (section)

== Stevens's typology ==
=== Overview ===
Stevens proposed his typology in a 1946 ''[[Science (journal)|Science]]'' article titled "On the theory of scales of measurement".<ref name="Stevens 1946"/> 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", unifying both "[[Qualitative property|qualitative]]" (which are described by his "nominal" type) and "[[Quantitative property|quantitative]]" (to a different degree, all the rest of his scales). The concept of scale types later received the mathematical rigour that it lacked at its inception with the work of mathematical psychologists Theodore Alper (1985, 1987), Louis Narens (1981a, b), and [[R. Duncan Luce]] (1986, 1987, 2001). As Luce (1997, p.&nbsp;395) wrote:
{{blockquote|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&nbsp;... '''no measurement theorist I know accepts Stevens's broad definition of measurement&nbsp;...''' in our view, the only sensible meaning for 'rule' is empirically testable laws about the attribute.}}

==== Comparison ====
{| class="wikitable"
! Incremental<br/> progress
! Measure property
! Mathematical<br/> operators
! Advanced<br/> operations
! Central<br/> tendency
! Variability
|-
| Nominal
| Classification, membership
| =, ≠
| [[aggregate data|Grouping]]
| [[Mode (statistics)|Mode]]
| [[Qualitative variation]]
|-
| Ordinal
| Comparison, level
| >, <
| [[Sorting]]
| [[Median]]
| [[Range (statistics)|Range]],<br/> [[interquartile range]]
|-
| Interval
| Difference, affinity
| +, −
| [[Measurement#Methodology|Comparison to a standard]]
| [[Arithmetic mean]]
| [[Deviation (statistics)|Deviation]]
|-
| Ratio
| Magnitude, amount
| ×, /
| [[Ratio]]
| [[Geometric mean]],<br/> [[harmonic mean]]
| [[Coefficient of variation]],<br/> [[studentized range]]
|}

=== Nominal level === <!--Do not change this name. It is pointed to by other Wikipedia articles.-->
A nominal scale consists only of a number of distinct classes or categories, for example: [Cat, Dog, Rabbit]. Unlike the other scales, no kind of relationship between the classes can be relied upon. Thus measuring with the nominal scale is equivalent to [[classifying]]. 

Nominal measurement may differentiate between items or subjects based only on their names or (meta-)categories and other qualitative classifications they belong to. Thus it has been argued that even [[dichotomy|dichotomous]] data relies on a [[constructivist epistemology]]. In this case, discovery of an exception to a classification can be viewed as progress. 

Numbers may be used to represent the variables but the numbers do not have numerical value or relationship:  for example, a [[globally unique identifier]].

Examples of these classifications include gender, nationality, ethnicity, language, genre, style, biological species, and form.<ref>Nominal measures are based on sets and  depend on categories, a la Aristotle: {{cite web |url=http://www4.uwsp.edu/geo/faculty/gmartin/geog476/Lecture/BeySt.htm |title=Beyond Stevens: A revised approach to measurement for geographic information |first=Nicholas |last=Chrisman |date=March 1995 |access-date=2014-08-25}}</ref><ref>"Invariably one came up against fundamental physical limits to the accuracy of measurement. ... The art of physical measurement seemed to be a matter of compromise, of choosing between reciprocally related uncertainties. ...  Multiplying together the conjugate pairs of uncertainty limits mentioned, however, I found that they formed invariant products of not one but two distinct kinds. ... The first group of limits were calculable ''a priori'' from a specification of the instrument. The second group could be calculated only ''a posteriori'' from a specification of what was ''done'' with the instrument. ... In the first case each unit [of information] would add one additional ''dimension'' (conceptual category), whereas in the second each unit would add one additional ''atomic fact''.", – pp. 1–4: MacKay, Donald M. (1969), ''Information, Mechanism, and Meaning'', Cambridge, MA: MIT Press, {{ISBN|0-262-63-032-X}}</ref> In a university one could also use residence hall or department affiliation as examples. Other concrete examples are 
* in [[grammar]], the [[parts of speech]]: noun, verb, preposition, article, pronoun, etc.
* in politics, [[power projection]]: hard power, soft power, etc.
* in biology, the [[taxonomic rank]]s below domains: kingdom, phylum, class, etc.
* in [[software engineering]], type of [[Trap (computing)|fault]]: specification faults, design faults, and code faults

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. If numbers are assigned as labels in nominal measurement, they have no specific numerical value or meaning. No form of arithmetic computation (+,&nbsp;−,&nbsp;×,&nbsp;etc.) may be performed on nominal measures. The nominal level is the lowest measurement level  used from a statistical point of view.

====Mathematical operations====
[[Equality (mathematics)|Equality]] and other operations that can be defined in terms of equality, such as [[inequality (mathematics)|inequality]] and [[set membership]], are the only [[non-trivial]] [[operation (mathematics)|operation]]s that generically apply to objects of the nominal type.

====Central tendency====
The [[mode (statistics)|mode]], i.e. the ''most common'' item, is allowed as the measure of [[central tendency]] for the nominal type. On the other hand, the [[median]], i.e. the ''middle-ranked'' item, makes no sense for the nominal type of data since ranking is meaningless for the nominal type.<ref>{{cite journal|last1=Manikandan|first1=S.|title=Measures of central tendency: Median and mode|journal=Journal of Pharmacology and Pharmacotherapeutics|date=2011|volume=2|issue=3|pages=214–5|doi=10.4103/0976-500X.83300|pmc=3157145|pmid=21897729 |doi-access=free }}</ref>

===Ordinal scale===
{{anchor |Ordinal scale}}<!-- This section is linked from [[IQ classification]] and other articles -->
{{further|Ordinal data}}

The ordinal type allows for [[rank order]] (1st, 2nd, 3rd, etc.) by which data can be sorted but still does not allow for a relative ''degree of difference'' between them. Examples include, on one hand, '''dichotomous''' data with dichotomous (or dichotomized) values such as "sick" vs. "healthy" when measuring health, "guilty" vs. "not-guilty" when making judgments in courts, "wrong/false" vs. "right/true" when measuring [[truth value]], and, on the other hand, '''non-dichotomous''' data consisting of a spectrum of values, such as "completely agree", "mostly agree", "mostly disagree", "completely disagree" when measuring [[opinion]].

The ordinal scale places events in order, but there is no attempt to make the intervals of the scale equal in terms of some rule. Rank orders represent ordinal scales and are frequently used in research relating to qualitative phenomena. A student's rank in his graduation class involves the use of an ordinal scale. One has to be very careful in making a statement about scores based on ordinal scales. For instance, if Devi's position in his class is 10th and Ganga's position is 40th, it cannot be said that Devi's position is four times as good as that of Ganga.
Ordinal scales only permit the ranking of items from highest to lowest. Ordinal measures have no absolute values, and the real differences between adjacent ranks may not be equal. All that can be said is that one person is higher or lower on the scale than another, but more precise comparisons cannot be made. Thus, the use of an ordinal scale implies a statement of "greater than" or "less than" (an equality statement is also acceptable) without our being able to state how much greater or less. The real difference between ranks 1 and 2, for instance, may be more or less than the difference between ranks 5 and 6. Since the numbers of this scale have only a rank meaning, the appropriate measure of central tendency is the median. A percentile or quartile measure is used for measuring dispersion. Correlations are restricted to various rank order methods. Measures of statistical significance are restricted to the non-parametric methods (R. M. Kothari, 2004).

==== Central tendency ====
The [[median]], i.e. ''middle-ranked'', item is allowed as the measure of [[central tendency]]; however, the mean (or average) as the measure of [[central tendency]] is not allowed. The [[mode (statistics)|mode]] is allowed.

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 (logic)|validity]], but they can be used to get ideas for how to improve [[operationalization]] of variables used in [[questionnaire]]s. 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).<ref>*{{Cite book |title=Statistical Theories of Mental Test Scores |last1=Lord |first1=Frederic M. |last2=Novick |first2=Melvin R. |last3=Birnbaum |first3=Allan |year=1968 |publisher=Addison-Wesley |location=Reading, MA |lccn=68011394 |page=21 |quote=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.}}
*{{cite journal |author=William W. Rozeboom |title=Reviewed Work: ''Statistical Theories of Mental Test Scores'' |journal=American Educational Research Journal |volume=6 |issue=1 |date=January 1969 |pages=112–116 |jstor=1162101}}</ref> In particular,<ref>{{cite book |title=Handbook of Parametric and Nonparametric Statistical Procedures |last=Sheskin |first=David J. |year=2007 |edition=Fourth |publisher=Chapman & Hall/CRC |location=Boca Raton |isbn=978-1-58488-814-7 |page=3 |quote=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. }}</ref> IQ scores reflect an ordinal scale, in which all scores are meaningful for comparison only.<ref>{{cite book |title=Psychology: An Introduction |last=Mussen |first=Paul Henry |year=1973 |publisher=Heath |location=Lexington (MA) |isbn=978-0-669-61382-7 |page=[https://archive.org/details/psychologyintrod00muss/page/363 363] |quote=The I.Q. is essentially a rank; there are no true "units" of intellectual ability. |url=https://archive.org/details/psychologyintrod00muss/page/363 }}</ref><ref>{{cite book |title=The WISC-III Companion: A Guide to Interpretation and Educational Intervention |last=Truch |first=Steve |year=1993 |publisher=Pro-Ed |location=Austin (TX) |isbn=978-0-89079-585-9 |page=35 |quote=An IQ score is not an equal-interval score, as is evident in Table A.4 in the WISC-III manual. }}</ref><ref>{{cite book |title=Measuring Intelligence: Facts and Fallacies |url=https://archive.org/details/measuringintelli00bart |url-access=registration |last=Bartholomew |first=David J. |author-link=D.J. Bartholomew |year=2004 |publisher=Cambridge University Press |location=Cambridge |isbn=978-0-521-54478-8 |quote=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. |page=[https://archive.org/details/measuringintelli00bart/page/n65 50] }}</ref> There is no absolute zero, and a 10-point difference may carry different meanings at different points of the scale.<ref>{{cite book |author=Eysenck, Hans |title=Intelligence: A New Look |location=New Brunswick (NJ) |publisher=[[Transaction Publishers]] |isbn=978-1-56000-360-1 |year=1998 |pages=24–25 |quote=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. }}</ref><ref>{{cite book |title=IQ and Human Intelligence |last=Mackintosh |first=N. J. |author-link=Nicholas Mackintosh |year=1998 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-852367-3 |pages=[https://archive.org/details/iqhumanintellige00mack/page/30 30–31] |quote=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 |url=https://archive.org/details/iqhumanintellige00mack/page/30 }}</ref>

===Interval scale===
The interval type allows for defining the ''degree of difference'' between measurements, but not the ratio between measurements. Examples include ''[[temperature scale]]s'' with the [[degree Celsius|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), ''location'' in Cartesian coordinates, and ''direction'' measured in degrees from true or magnetic north. Ratios are not meaningful since 20&nbsp;°C cannot be said to be "twice as hot" as 10&nbsp;°C (unlike temperature in [[kelvin]]s), 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; for example, the ten-degree difference between 15&nbsp;°C and 25&nbsp;°C is twice the five-degree difference between 17&nbsp;°C and 22&nbsp;°C. 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====
The [[mode (statistics)|mode]], [[median]], and [[arithmetic mean]] are allowed to measure central tendency of interval variables, while measures of statistical dispersion include [[range (statistics)|range]] and [[standard deviation]]. Since one can only divide by ''differences'', one cannot define measures that require some ratios, such as <!--- the studentized range or. --- Error: studentized range is a ratio of a difference (range) to a root-mean-square difference (standard DEVIATION from the mean ---> the [[coefficient of variation]]. More subtly, while one can define [[Moment (mathematics)|moments]] about the [[Origin (mathematics)|origin]], only central moments are meaningful, since the choice of origin is arbitrary. One can define [[standardized moment]]s, since ratios of differences are meaningful, but one cannot define the 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 ===
:''See also'': {{section link|Positive real numbers#Ratio scale}}

The ratio 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 of measurement]] of the same kind (Michell, 1997, 1999). Most measurement in the physical sciences and engineering is done on ratio scales. Examples include [[mass]], [[length]], [[Time|duration]], [[plane angle]], [[energy]] and [[electric charge]]. In contrast to interval scales, ratios can be compared using [[division (mathematics)|division]]. Very informally, many ratio scales can be described as specifying "how much" of something (i.e. an amount or magnitude). Ratio scales are often used to express an [[order of magnitude]] such as for temperature in [[Orders of magnitude (temperature)]].

==== Central tendency and statistical dispersion ====
The [[geometric mean]] and the [[harmonic mean]] are allowed to measure the central tendency, in addition to the mode, median, and arithmetic mean. The [[studentized range]] and the [[coefficient of variation]] are allowed to measure statistical dispersion. All statistical measures are allowed because all necessary mathematical operations are defined for the ratio scale.