Editing Level of measurement (section)

=== 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]]
|}