# Scale (empiricism)

The terms system of relations ( relative for short ), empirical relative and numerical relative are technical terms used in empirical science for the abstract description of the construction of a scale in the sense of a measurement specification. The concrete development of a measurement specification is the operationalization . Depending on the type of variable that is to be measured using a scale, a distinction is made between different scale levels .

In the empirical sciences, it is assumed that the objects under observation (objects, people, works, events, ...) have properties independent of the observation, and a distinction is therefore made between properties and measurement results.

## Example "temperature"

If one deals more closely with temperature effects and tries to divide the area between given fixed points (e.g. melting and boiling point of water) evenly, one recognizes that different measuring principles (expansion of liquids and gases, electrical resistance of or contact voltage between metals, position of the Equilibrium of chemical reactions, ...) lead to inconsistent results. The assumption that not one of the measurement methods defines the property "temperature" and the others provide different results, but that all methods represent only approximations to a true property T , was confirmed by physical discoveries, such as the existence of a lower limit for temperature ( T  = 0) or the T 4 form of the temperature dependence of the radiant power of the thermal radiation, if temperature ratios are defined via the maximum efficiency of heat engines , see absolute temperature .

## Definitions

"A relative or a system of relations is understood to mean a set of objects and one or more relations with which the type of relationship between the objects is characterized."

A set of objects to be observed and the relationships between them with regard to a property is an empirical relative , a set of numbers used to define a relation is a numerical relative .

If one understands by empirical relations “that they are the direct result of pair comparison experiments”, there are also considerations to regard the empirical relative rather as “theoretical abstraction of empirical observations” due to the observation that assumptions are sometimes violated.

The relationship between an empirical relative and a numerical relative is determined by a mapping function. Mapping functions that do not disrupt the structure of the empirically observed relations have particular practical value, see homomorphism . The entirety

• an empirical relative,
• a numerical relative
• and a homomorphic mapping between the two (assignment rule, measurement rule, see operationalization )

is a scale .

“Formally, a scale is defined as the ordered triple of an empirical relational system A, the numerical relative N and the morphism (the assignment function) f: A → N, hence the triple (A, N, f). Expressed in more detail, scale = [(A; R 1 , ... R n ), (N; S 1 , ... S n ) f]. A denotes a set of empirical objects for which the relations R i apply, N a subset of the real numbers with the relations S i and f the mapping rule of morphism. "

Scales can be varied. For a system, see scale level .

## Individual evidence

1. Jürgen Bortz, Christof Schuster: Statistics for human and social scientists . 7th edition. Springer, Berlin 2010, ISBN 978-3-642-12769-4 , pp. 15–16 ( limited preview in Google Book search).
2. Only numbers as scale values ​​are general enough, because in the case of a nominal scale they are also suitable as mere identifiers.
3. ^ Rolf Steyer, Michael Eid : Measuring and testing. With exercises and solutions . 2nd Edition. Springer, Berlin 2009, ISBN 978-3-486-70242-2 , pp. 94–97 (here p. 94) ( limited preview in Google Book search).
4. ^ Rolf Steyer, Michael Eid: Measuring and testing. With exercises and solutions . 2nd Edition. Springer, Berlin 2009, ISBN 978-3-486-70242-2 , pp. 94–97 (here p. 96) ( limited preview in Google Book search).
5. ^ Siegfried Schumann: Representative survey. Practice-oriented introduction to empirical methods and statistical analysis processes . 5th edition. Oldenbourg Wissenschaftsverlag, Munich 2011, ISBN 978-3-486-70242-2 , p. 20 ( limited preview in Google Book search).
6. ^ Jürgen Bortz, Nicola Döring: Research methods and evaluation for human and social scientists . 3. Edition. Springer, Heidelberg 2005, ISBN 3-540-41940-3 , pp. 69–70 ( limited preview in Google Book search).
7. Wolfgang J. Koschnick: Management: Enzyklopädisches Lexikon . 1st edition. Walter de Gruyter, Berlin 1995, ISBN 978-3-11-012847-5 , p. 564 ( limited preview in Google Book search).