# Scale level

The scale level or measurement level or the scale dignity (rarely scale quality ) is an important property of characteristics or variables in empirical research .

## Systematics of the scales

Different levels of scalability can be distinguished depending on the type of feature or depending on which regulations can be complied with when measuring it :

Scale level logical / math operations Measurable properties example [additional] location parameters
Nominal scale = / ≠ frequency Postcodes, genders mode
Ordinal scale = / ≠; </> Frequency, ranking School grades (“very good” to “unsatisfactory”), table position in the Bundesliga Median
Cardinal scale
Interval scale = / ≠; </>; -; + (Feature + feature difference) Frequency, ranking, distance Time scale (date), intelligence quotient, temperature (in degrees Celsius) Arithmetic mean
Ratio scale = / ≠; </>; +/-; ÷ (returns a number without a unit); x (number x feature) Frequency, order of precedence, distance, natural zero Age (in years), sales (in euros), temperature (in Kelvin) Geometric mean
Scale levels in comparison; red: The newly added properties at the respective scale level. Nominal: only frequencies, ordinal: order, interval: distances, scaled to the ratio: zero point

The interval and ratio scales are combined to form the cardinal scale. Features on this scale are then referred to as metric . Nominal or ordinal scaled features (see below) are also referred to as categorical .

The scale level determines

• which (mathematical) operations are permitted with a correspondingly scaled variable. Operations that are permitted for variables of a certain scale level can in principle also be carried out on variables of all higher scale levels. A feature that is scalable at a certain level can be represented at all lower scale levels, but not vice versa.
• which transformations can be carried out with appropriately scaled variables without losing or changing information .
• which information is provided by the corresponding characteristic, which interpretations allow the corresponding characteristic to be expressed.

The scale level does not provide any information about

• whether a variable is discrete (categorical) or continuous (see main article characteristic ). Only in the case of nominal scaling is the feature basically not continuous, but rather discrete.

"Although the scale level and number of possible expressions represent independent concepts, in practice nominal and ordinal scaled features are mostly discrete and metrically scaled features are mostly continuous."

## History of the division

“Scales can be classified according to which transformations are permissible for them.” However, this classification of scales is not undisputed, criticism of this can be found primarily in Prytulak (1975) and Duncan (see Michell). “Since there are an infinite number of permissible transformations on a certain scale, an infinite number of different scale levels could in principle also be distinguished. The most commonly used classification goes back to Stevens (1946). This differentiates between nominal, ordinal, interval and ratio scales ”. “A more detailed classification, for example by Narens and Luce (1986) or by Orth (1974), usually contains a 'log interval scale' between the interval and the ratio scale. With a log interval scale, power transformations (x '= s * x r ; with s and r greater than 0) are permitted. "

Marks (1974) tries to systematically capture the possibilities of different scale levels. He proposes a general transformation function in which three constants can be freely selected. The constants can each be either positive (+) or zero (0). Zero indicates that a scale transformation would lead to a loss of information here. A plus sign indicates that such a transformation would be possible without loss of information. The general formula he proposed is:
x '= (a + 1) x (b + 1) + c

For example, for an interval scale, the constants a would have to be positive, b zero, c positive. This results in the linear transformation as a generally permissible transformation rule for an interval scale:
x '= ax + b

Accordingly, Marks comes to the following 8 scales, whereby it can be seen that the significance increases, while in the opposite direction the transformation possibilities decrease without loss of information:

(English) scale designation a b c Number of allowed transformations Expressiveness
Ordinal + + + 3 0
Hyperordinal 0 + + 2 1
Interval + 0 + 2 1
Log interval + + 0 2 1
Difference 0 0 + 1 2
power 0 + 0 1 2
Ratio + 0 0 1 2
absolute 0 0 0 0 3

## Nominal scale

Lowest scale level. For different objects or appearances, a comparison is only used to make a decision about the equality or inequality of the characteristic values ​​(e.g. x ≠ y ≠ z). It is therefore only a question of qualitative characteristics (e.g. blood groups or gender). The equality relation applies, so you can decide whether two characteristics are the same or not. However, the values ​​cannot be sorted according to size, in the sense of “is greater than” or “better than”.

## Ordinal scale

For an ordinally scalable feature, there are rankings of the type “larger”, “smaller”, “more”, “less”, “stronger”, “weaker” between two different characteristic values ​​(e.g. x> y> z). However, nothing is said about the distances between these neighboring judgment classes. Usually these are qualitative features, such as B. the "highest attainable educational qualification" sought in the question. Another example is the school grades: Grade 1 is better than Grade 2, but it is extremely doubtful whether the difference between Grade 1 and 2 is the same as that between Grade 3 and Grade 4.

A special form of the ordinal scale is the rank scale . Each value can only be assigned once. Examples of this are the achievement of ranks in sport, just like in other performance comparisons , or the natural order, as it often occurs in the animal kingdom in living beings that live in social groups such as z. B. Chicken birds. Their order is therefore also called the hack order .

## Interval scale

The order of the characteristic values ​​is fixed, and the size of the distance between two values ​​can be substantiated. As a metric scale, it makes statements about the amount of differences between two classes. The inequality of the characteristic values ​​can be quantified by forming the difference (e.g. for the date, the result could be “three years earlier”). The zero point (“after the birth of Christ”) and the distance between the classes (years or moons) are, however, determined arbitrarily. Note: The metric scales distinguish between discrete and continuous features.

## Ratio scale (also ratio scale)

The ratio scale has the highest level. It is also a metric scale, but unlike the interval scale, there is an absolute zero point (e.g. blood pressure, absolute temperature , age, length measures ). Only at this scale level are multiplication and division meaningful and permitted. Relationships of characteristic values ​​can therefore be formed (e.g. x = y · z).

## Gray areas between the scale levels

There are features that cannot be assigned to exactly one scale level. So could z. For example, it is not possible to reliably prove that a feature is interval-scaled, but one is certain that it is more than ordinally scaled. In such a case one could try an interpretation on an interval scale, but take this assumption into account in the interpretation and proceed accordingly cautiously. An example of this is the formation of averages for school grades coded as digits, which actually represent an ordinal-scaled feature because they are defined in fixed terms, for example from very good to insufficient.

Other examples are times without specifying the date (circadian dates) or cardinal points. Here, values ​​can be arranged and distances measured within sub-areas, and with a corresponding restriction for the size of the distances, any number of distances can be added meaningfully (more precisely: 'unambiguously'). Without a restriction, this no longer applies (“Is 2:00 a.m. before or after 10:00 p.m.?” - “Both and”).

## Problems with scaling

In individual cases, natural orders can occur, which in principle can be described with a certain scale, but sometimes contain individual deviations. An example are placements at sporting events (scaled up), where each athlete actually only takes one place (first, second, third, etc.), but has to share his place with another athlete if he has achieved exactly the same measured value. Depending on the regulations, a higher or lower rank cannot be assigned, so that the scale has a blank that does not otherwise exist (silver medal not awarded for two first places). Strictly speaking, this is an ordinal scale that is scaled to rank.

In the animal kingdom, rank scales are sometimes not stringent, so that within an ascending pecking order, especially in the lower range of the scale, there are intermediate triplets or multiplets that “chop” each other according to the scheme A> B> C> A. One speaks of intransitivity . Such a phenomenon cannot be exhaustively described by converting it to the ordinal scale level and requires a complete representation in a matrix or the use of a further feature, e.g. B. Success in feed dispute in terms of feed weight eaten, provided that higher-ranking animals always eat more than lower-ranking animals, which is often not the case. The matrix display is therefore preferred to scaling in such cases, although it is more difficult to visually detect and statistically more complex to use.

## Individual evidence

1. Herbert Büning, Götz Trenkler: Non-parametric statistical methods . Walter de Gruyter, 1994, ISBN 978-3-11-016351-3 , p. 8 ( limited preview in Google Book search).
2. a b Gerhard Tutz: The analysis of categorical data: application-oriented introduction to logit modeling and categorical regression . Oldenbourg Wissenschaftsverlag, Munich 2000, ISBN 3-486-25405-7 , p. 3 ( limited preview in Google Book Search).
3. Wolfgang Brachinger: Multivariate statistical methods . Walter de Gruyter, Berlin 1996, ISBN 3-11-013806-9 , p. 11 ( limited preview in Google Book search).
4. a b c d Rainer Schnell, Paul Hill, Elke Esser: Methods of empirical social research. 2nd Edition. R. Oldenbourg, Munich / Vienna 1989, ISBN 3-486-21463-2 , p. 137 (8th edition online)
5. LS Prytulak: Critique of SS Stevens' Theory of Measurement Scales classification. In: Perceptual and Motor Skills. 1975 (41, 3, 28).
6. probably OD Duncan: Notes on Social Measurement. Historical and Critical. Russell Sage Foundation, New York 1984, ISBN 0-87154-219-6 , pp. 119-156
7. J. Michell: Measurement scales and statistics: a clash of paradigms . In: Psychological Bulletin . 100, No. 3, 1986, pp. 398-407. doi : 10.1037 / 0033-2909.100.3.398 .
8. ^ SS Stevens: On the Theory of Scales of Measurement. In: Science. 1946, 103, pp. 677-680. (on-line)
9. L. Narens, RD Luce ,: Measurement: The Theory of Numerical Assignments. In: Psychological Bulletin. 1986, 99, 2, pp. 166-180. (online) (PDF; 1.6 MB)
10. B. Orth: Introduction to the theory of measurement. Kohlhammer, Stuttgart / Berlin / Cologne / Mainz 1974, ISBN 3-17-002055-2 .
11. Lawrence E. Marks: Sensory Processes: The New Psychophysics. Academic Press, New York 1974, ISBN 0-12-472950-9 , pp. 247-249.

## literature

• L. Fahrmeir , A. Hamerle, G. Tutz (Ed.): Multivariate statistical methods. 2nd, revised edition. Walter de Gruyter, Berlin / New York 1996, ISBN 3-11-013806-9 .
• L. Fahrmeir, R. Artist, I. Pigeot, G. Tutz: Statistics. The way to data analysis. Springer, Berlin / Heidelberg / New York 1999, ISBN 3-540-67826-3 .
• SH Kan: Metrics and Models in Software Quality Engineering. 2nd Edition. Pearson Education, Boston 2003, ISBN 0-201-63339-6 .
• K. Backhaus , B. Erichson, W. Plinke, R. Weiber: Multivariate analysis methods. An application-oriented introduction. 11th edition. Springer-Verlag, Berlin a. a. 2005, ISBN 3-540-27870-2 , pp. 4-6.
• Jürgen Bortz: Statistics for human and social scientists . 6th edition. Springer, 2005, ISBN 3-540-21271-X , pp. 15-27 .