Editing Level of measurement (section)

===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>