# Measurement Scales

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The values random variables take can differ distinctively:

Symbol Variable Sample space
$X$ Age (rounded to years) $\left\{0,1,2,\ldots \right\}$ $S$ Sex {female, male}
$T$ Marital status {single,married, divorced}
$Y$ Monthly income $\left[0,\infty \right)$ They can be classified into quantitative, i.e. numerically valued (age and income) and qualitative, i.e. categorical, (sex, marital status) variables. As numerical values are usually assigned to observations of qualitative variables, they may appear quantitative. Yet such synthetic assignments aren’t of the same quality as numerical measurements that naturally arise in observing a phenomenon. The crucial distinction between quantitative and qualitative variables lies in the properties of the actual scale of measurement, which in turn is crucial to the applicability of statistical methods. In developing new tools statisticians make assumptions about permissible measurement scales. A measurement is a numerical assignment to an observation. Some measurements appear more natural than others. By measuring the height of persons, for example, we apply a yardstick that ensures comparability between observations up to almost any desired precision—regardless of the units (such as inches or centimeters). School grades, on the other hand, represent a relatively rough classification indicating a certain ranking, yet putting many pupils into the same category. The values assigned to qualitative statements like ‘very good’, ‘average’ etc. are an arbitrary yet practical short cut in assessing people’s achievements. As there is no conceptual reasoning behind a school grade scale, one should not try to interpret the ’distances’ between grades. Clearly, height measurements convey more information than school marks, as distances between measurements can consistently be compared. Statements such as ‘Tom is twice as tall as his son’ or ‘Manuela is 35 centimeters smaller than her partner’ are permissible. As statistical methods are developed in mathematical terms, the applicable scales are also defined in terms of mathematical concepts. These are the transformations that can be imposed on them without loss of information. The wider the range of permissible transformations, the less information the scale can convey. The following table lists common measurement scales in increasing order of information content. Scales carrying more information can always be transformed into less informative scales.

Variable Measurement Scale Statements Permissible Transformations
Qualitative Nominal Scale equivalence any equivalence preserving mapping
Categorical Ordinal Scale equivalence, order any order preserving mapping
Quantitative Interval Scale equivalence, order, $y=\alpha x+\beta ,\alpha >0$ Metric distance
Ratio Scale equivalence, order, $y=\alpha x,\alpha >0$ distance, ratio
Absolute Scale equivalence, order, identity function
distance, ratio,
absolute level