# Ordinal scale

An ordinal scale sorts variables with characteristics between which there is a ranking . Ordinally scaled variables contain nominal information and also information about the order (order) of the variable values. Observations on a trait with an ordinal measurement level can be grouped with regard to that trait and ordered according to their size.

If the characteristic values ( categories ) are designated with (ranking) numbers (ordinal numbers), these are selected so that the ranking of the numbers corresponds to the ranking of the characteristics. This means that an observation or an object with a higher rank also has a higher value on the characteristic under consideration than an observation with a lower rank. About the size of the feature difference between the objects, i. H. about the distances between the ranks, but no statement can be made.

## Formal conditions

In addition to the conditions for constructing a nominal scale, constructing an ordinal scale requires:

Trichotomy
Either a is greater than b, or b is greater than a, or a is equal to b.
Transitivity
If a is greater than b and b is greater than c, then a must be greater than c.

## Examples

The ordinal scale is only about establishing a sequence or ranking. In a car race a first, second and third place can be awarded. It doesn't matter whether the winner was an hour or a minute faster than the second.

The table below provides additional examples of ordinal-scaled features.

feature Categories
Pressure ulcer risk low to high risk on the Norton scale
Satisfaction with a product very satisfied> rather satisfied> rather unsatisfied> very unsatisfied
Self-classification of income 1 high> medium> low
School performance 2 very good> good> satisfactory> sufficient> unsatisfactory> unsatisfactory
Rank in the military General> Major> Lieutenant> Sergeant> NCO> Private

1 If the income is divided into classes (e.g. 0 to 999 euros, 1000 to 2000 euros, over 2000 euros), it is an ordinally scaled characteristic. If, on the other hand, the exact amount is collected and statistically processed, there is a metric characteristic. Since the willingness to provide information is less when it comes to specifying the exact income, many surveys use a question of the income bracket.

2 School grades are often used as if they were scaled metrically, e.g. B. the average is calculated. It becomes problematic when such use has serious consequences, e.g. B. when assessing different teaching methods.

Another example of the consequences of the restriction to the ordinal measurement level can be found under Arrow's theorem .

## Possible operations

Even if categories are coded by numbers, mathematical operations with these numbers do not make sense because they do not represent a numerical value, but a category (e.g. satisfied). For example, a “satisfied / unsatisfied” division makes little sense. Since school grades are usually ordinal-scaled features, the formation of average grades is actually not useful, but is carried out regularly in educational institutions. Qualitative comparisons (“greater than / less than”) can, however, be carried out.

It is also possible to determine the frequency of occurrences of the categories in a set of examination objects (or to determine the frequency of occurrence of characteristic values ​​smaller or larger than a certain category). The central value, which halves the sample, the so-called median value, serves as the location parameter .

## Allowed transformations

All transformations using (strictly) monotonically increasing functions are permissible.

## Mathematical interpretation

From a mathematical point of view, an ordinal scale is a set that has: ${\ displaystyle S}$

1. There is an equivalence relation , namely the identity relation to : .${\ displaystyle E \ subseteq S \ times S}$${\ displaystyle S}$${\ displaystyle E = \ operatorname {id} _ {S} = \ left \ {\, (m, m) \ mid m \ in S \, \ right \}}$
2. There is a linear order relation .${\ displaystyle O \ subseteq S \ times S}$

Each element is called expression of . ${\ displaystyle m \ in S}$${\ displaystyle S}$

Each ordinal scale is, as a subset, a nominal scale .