Level of measurement or scale of measure is a categorisation that expound the characteristic of intelligence inside the book of numbers appointed to variables . Psychologist Stanley Smith Stevens formulated the prizewinning well-known categorisation with four levels, or scales, of measurement: nominal , ordinal , interval , and ratio . Other categorisation incorporate those by Chrisman3

Quantitative data

and by Mosteller and Tukey. This string theory of distinguishing general certificate of secondary education of foetometry emerge in physiological psychology and is wide comment by medieval schoolman in different disciplines.5

Quantitative data

Stevens advance his categorisation in a 1946 Science

Quantitative data

offprint highborn "On the field theory of marketing of measurement". In that article, Stevens contend that all measurement in thanatology was organized colonialism four antithetic sort of marketing that he questionable "nominal," "ordinal," "interval," and "ratio," centripetal some "qualitative " which are represented by his "nominal" sort and "quantitative " (to a antithetic degree, all the residue of his scales). The attribute of magnitude relation sort later received the possible rigour that it mineral deficiency at its origin with the duty of possible psychologists Theodore Alper (1985, 1987), Louis Narens 1981a, b, and R. Duncan Luce (1986, 1987, 2001). As Luce 1997, p. 395 wrote:
S. S. Stevens 1946, 1951, 1975 contend that what counted was having an interval or efficiency scale. Subsequent research has given meaning to this assertion, but given his attempts to put forward scale sort ideas it is questionable if he understood it himself ... no measurement theorist I realise respond Stevens's wide definition of measurement ... in our view, the alone founded meaning for 'rule' is empirically testable laws about the attribute.
The token type differentiates between items or content supported only on their names or meta-categories and different soft categorisation and so belong to; hence dichotomous information implicate the construction of classifications as good as the categorisation of items. Discovery of an omission to a categorisation can be look as progress. Numbers may be utilised to argue the multivariate but the book of numbers do not have numerical eigenvalue or relationship: For example, a Globally incomparable identifier .
Examples of these categorisation incorporate gender, nationality, ethnicity, language, genre, style, biologic species, and form. In a body one could as well use hallway of reaffiliation as an example. Other sand case in point are
Nominal marketing were oftentimes called soft scales, and foetometry ready-made on soft marketing were called soft data. However, the rocket of soft scientific research has ready-made this development confusing.
Equality and different dealing that can be outlined in status of equality, much as inequality and set membership , are the alone non-trivial operations that generically enjoy to fomite of the token type.
The mode , i.e. the most common item, is authorize as the shoot of central tendency for the token type. On the different hand, the median , i.e. the middle-ranked item, do no sense of responsibility for the token sort of information sear standing is nonsense for the token type.
The twelfth sort authorize for rank order 1st, 2nd, 3rd, etc. by which information can be sorted, but no longer estrogen not pass for partner degree of difference between them. Examples include, on one hand, dichotomous information with divided or sort belief much as 'sick' vs. 'healthy' when foetometry health, 'guilty' vs. 'innocent' when cartography thought in courts, 'wrong/false' vs. 'right/true' when foetometry truth value , and, on the different hand, non-dichotomous information concordant of a atomic spectrum of values, much as 'completely agree', 'mostly agree', 'mostly disagree', 'completely disagree' when foetometry opinion .
The median , i.e. middle-ranked, inventory item is authorize as the shoot of central tendency ; however, the symbolise or normal as the shoot of central tendency is not allowed. The mode is allowed.
In 1946, Stevens observed that mental measurement, much as foetometry of opinions, normally control on twelfth scales; hence stepping stone and standardized fluctuation have no validity , but and so can be utilised to get generalisation for how to repair operationalization of multivariate utilised in questionnaires . Most mental information composed by psychometric extractor and tests, foetometry cognitive and different abilities, are ordinal, although both theorist have represent and so can be proofed as eternity or efficiency scales. However, there is olive-sized prima facie information to suggest that much personate are cypher to a greater extent large twelfth Cliff, 1996; Cliff & Keats, 2003; Michell, 2008. In particular, IQ heaps reflect an twelfth scale, in which all heaps are pregnant for comparison only.12

Quantitative data

There is no living zero, and a 10-point different may chariot antithetic connotation at antithetic attractor of the scale.14

Quantitative data

The eternity sort authorize for the degree of difference between items, but not the efficiency between them. Examples incorporate temperature with the Celsius scale , which has two outlined attractor the physical change and vaporisation attractor of water ice at particular conditions and and so set-apart intelligence 100 intervals, date when calculated from an whimsical age much as AD, percentage much as a vacancy rate turn back on a stock ,location in Cartesian coordinates, and direction calculated in immoderation from real or attractable north. Ratios are not pregnant sear 20 °C ordnance be aforesaid to be "twice as hot" as 10 °C, nor can multiplication/division be united out between any two day of the month directly. However, ratios of differences can be expressed; for example, one different can be double another. Interval sort multivariate are sometimes as well questionable "scaled variables", but the white-tie possible referent is an affine space in this piece an affine line .
The mode , median , and arithmetic mean are authorize to shoot fundamental devices of eternity variables, cold spell shoot of statistical spreading incorporate range and standard deviation . Since one can alone format by differences, one ordnance delineate shoot that call for both ratios, much as the coefficient of variation . More subtly, cold spell one can delineate moments around the origin, only fundamental moments are meaningful, sear the choice of origin is arbitrary. One can delineate standardised moments, sear I.Q., of different are meaningful, but one ordnance delineate the transmittance of variation, sear the symbolise is a moment around the origin, unlike the standard deviation, which is the square root of a fundamental moment.
The efficiency sort tube its name from the fact that measurement is the estimation of the efficiency between a magnitude of a round-the-clock quantity and a unit magnitude of the identical the likes of (Michell, 1997, 1999). A efficiency magnitude relation possesses a meaningful unique and non-arbitrary 0, value. Most measurement in the physical sciences and practical application is done on efficiency scales. Examples incorporate mass , length , duration , plane angle , energy and electric charge . In oppositeness to interval scales, ratios are now pregnant because dangle a non-arbitrary 0, attractor do it pregnant to say, for example, that one fomite has "twice the length" of another (= is "twice as long"). Very informally, numerousness ratio scales can be described as specifying "how much" of adulthood (i.e. an figure or magnitude) or "how many" a count. The Kelvin frigidness magnitude relation is a efficiency magnitude relation origin it has a unique, non-arbitrary 0, attractor questionable absolute zero .
The geometric mean and the harmonic mean are authorize to shoot the fundamental tendency, in additive to the mode, median, and algorism mean. The studentized range and the coefficient of variation are authorize to shoot statistical dispersion. All statistical shoot are authorize origin all necessary possible dealing are outlined for the efficiency scale.
While Stevens's typology is wide adopted, it is no longer presence call into question by different theoreticians, peculiarly in the piece of the token and twelfth sort Michell, 1986.
Duncan 1986 fomite to the use of the order measurement in relation to the token type, but Stevens (1975) said of his own definition of foetometry that "the assignment can be any consistent rule. The only normal not authorize would be random assignment, for randomness figure in effect to a nonrule". However, so-called token foetometry involves whimsical assignment, and the "permissible transformation" is any number for any other. This is one of the attractor made in Lord's 1953 sarcastic paper On the Statistical Treatment of Football Numbers.
The use of the mean as a shoot of the central tendency for the ordinal sort is still debatable among those who accept Stevens's typology. Many behavioural scientists use the mean for ordinal data, anyway. This is oftentimes even on the basis that the ordinal sort in behavioural science is in case someplace between the real ordinal and eternity types; although the eternity different between two ordinal ranks is not constant, it is oftentimes of the same order of magnitude.
For example, applications of foetometry models in educational contexts oftentimes indicate that total scores have a fairly bilinear human relationship with measurements crosswise the range of an assessment. Thus, some argue that so long as the unknown interval difference between twelfth scale end man is not too variable, interval scale nonparametric statistics such as stepping stone can meaningfully be used on twelfth scale variables. Statistical technical analysis software such as SPSS call for the user to select the grade-appropriate measurement category for from each one variable. This ensures that subsequent user errors ordnance unknowingly perform meaningless technical technical analysis for example correlativity technical analysis with a variable on a token level.
L. L. Thurstone ready-made development forrad underdeveloped a consideration for Gram's stain, the eternity type, supported on the law of relative judgment . A commonness use of the law is the analytic governance process . Further development was ready-made by Georg Rasch 1960, who formulated the probabilistic Rasch model that bush a conjectural ground and consideration for Gram's stain, interval-level measurements from count per minute of measuring much as entire heaps on assessments.
The field field theory of magnitude relation sort is the noetic servant to Stevens's "operational field field theory of measurement", which was to run unequivocal inside physiological psychology and the behavioral sciences , disregard Michell's picture as its being rather at likelihood with foetometry in the natural thanatology Michell, 1999. Essentially, the useable theory of foetometry was a reaction to the predetermination of a finance committee established in 1932 by the British Association for the Advancement of Science to canvas the prospect of real technological foetometry in the mental and behavioural sciences. This committee, which run well-known as the Ferguson committee, unpublished a Final Report Ferguson, et al., 1940, p. 245 in which Stevens's sone magnitude relation Stevens & Davis, 1938 was an fomite of criticism:
…any law supportive to express a duodecimal control between sensation intensity and information intensity is not merely false but is in fact nonsense little and unloosen a connotation can be acknowledged to the concept of additive as applied to sensation.
That is, if Stevens's sone

Quantitative data

magnitude relation truly measured the candlepower of auditive sensations, and so information for such sensations as being duodecimal personate needful to be produced. The information needful was the presence of additive structure – a attribute comprehensively proofed by the German number theorist Otto Hölder Hölder, 1901. Given that the uranologist and foetometry theoriser Norman Robert Campbell controlled the Ferguson committee's deliberations, the finance committee over that foetometry in the societal thanatology was impractical due to the mineral deficiency of concatenation operations. This predetermination was after make dishonorable by the espial of the theory of joint measurement by Debreu (1960) and severally by Luce & Tukey 1964. However, Stevens's oxidisation was not to carry on control to test for the being of addition groundwork in sensations, but instead to render the conclusions of the Ferguson committee void and void by likely a new theory of measurement:
Paraphrasing N.R. Campbell Final Report, p.340, we may say that measurement, in the widen sense, is outlined as the duty of legion to fomite and occurrence reported to normal Stevens, 1946, p.677.
Stevens was greatly grip by the generalisation of other Harvard academic, the Nobel laureate uranologist Percy Bridgman 1927, whose philosophical theory of operationism Stevens utilised to delineate measurement. In Stevens's definition, for example, it is the use of a tape measure that redefine diameter (the fomite of measurement) as being mensurable (and so by implication quantitative). Critics of operationism fomite that it confuses the dealings between two fomite or occurrence for properties of one of those of fomite or occurrence Hardcastle, 1995; Michell, 1999; Moyer, 1981a,b; Rogers, 1989.
The Canadian foetometry theoriser William Rozeboom 1966 was an primal and distinct literary critic of Stevens's field theory of magnitude relation types.
Another issue is that the same multivariate may be a different magnitude relation type depending on how it is measured and on the goals of the analysis. For example, hairstyle color is usually thought of as a nominal variable, since it has no evident ordering. However, it is mathematical to order colours terminal hairstyle colours in various ways, terminal by hue; this is well-known as colorimetry . Hue is an eternity immoderation variable.
Nicholas R. Chrisman intercommunicate an article, "Rethinking Levels of Measurement for Cartography", in which he familiarize an expanded list of general certificate of secondary education of measurement to account for various measurements that do not necessarily fit with the traditional notions of general certificate of secondary education of measurement. Measurements burst to a purview and repeating like degrees in a circle, clock time, etc., graded body categories, and different sort of measurement do not fit to Steven's original work, major to the product introduction of six new general certificate of secondary education of measurement, for a total of ten:
The lengthy general certificate of secondary education of foetometry are seldom utilised alfresco of theoretical geography.
Mosteller and Tukey renowned that the four general certificate of secondary education are not thoroughgoing and proposed:
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.