Statistics

===Types of data===
{{main|Statistical data type||Levels of measurement}}

Various attempts have been made to produce a taxonomy of [[level of measurement|levels of measurement]]. The psychophysicist [[Stanley Smith Stevens]] defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one (injective) transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with [[longitude]] and [[temperature]] measurements in [[Celsius]] or [[Fahrenheit]]), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different measurements defined, and permit any rescaling transformation.

Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as [[categorical variable]]s, whereas ratio and interval measurements are grouped together as [[Variable (mathematics)#Applied statistics|quantitative variables]], which can be either [[Probability distribution#Discrete probability distribution|discrete]] or [[Probability distribution#Continuous probability distribution|continuous]], due to their numerical nature. Such distinctions can often be loosely correlated with [[data type]] in computer science, in that dichotomous categorical variables may be represented with the [[Boolean data type]], polytomous categorical variables with arbitrarily assigned [[integer]]s in the [[integer (computer science)|integral data type]], and continuous variables with the [[real data type]] involving [[floating-point arithmetic]]. But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented.

Other categorizations have been proposed. For example, Mosteller and Tukey (1977)<ref>{{cite book | last1 = Mosteller | first1 = F. | author-link1 = Frederick Mosteller | last2 = Tukey | first2 = J.W | author-link2 = John Tukey | year = 1977 | title = Data analysis and regression | location = Boston | publisher = Addison-Wesley}}</ref> distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990)<ref>[[John Nelder|Nelder, J.A.]] (1990). The knowledge needed to computerise the analysis and interpretation of statistical information. In ''Expert systems and artificial intelligence: the need for information about data''. Library Association Report, London, March, 23–27.</ref> described continuous counts, continuous ratios, count ratios, and categorical modes of data. (See also: Chrisman (1998),<ref>{{cite journal | last1 = Chrisman | first1 = Nicholas R | year = 1998 | title = Rethinking Levels of Measurement for Cartography | journal = Cartography and Geographic Information Science | volume = 25 | issue = 4| pages = 231–242 | doi=10.1559/152304098782383043| bibcode = 1998CGISy..25..231C }}</ref> van den Berg (1991).<ref>van den Berg, G. (1991). ''Choosing an analysis method''. Leiden: DSWO Press</ref>)

The issue of whether or not it is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement procedures is complicated by issues concerning the transformation of variables and the precise interpretation of research questions. "The relationship between the data and what they describe merely reflects the fact that certain kinds of statistical statements may have truth values which are not invariant under some transformations. Whether or not a transformation is sensible to contemplate depends on the question one is trying to answer."<ref>Hand, D.J. (2004). ''Measurement theory and practice: The world through quantification.'' London: Arnold.</ref>{{rp|82}}