# Interval scale

The interval scale (one of three cardinal scales ) is a scale level in statistics . It is part of the metric measurement level , since the characteristics of this scale level can be represented quantitatively using numbers. In particular, this also means that rank differences and the distance between values ​​can be measured; that is, quantitative characteristics go beyond ordinal or even nominal characteristics in their requirements .

## description

In the case of interval-scaled features, in addition to the properties of the ordinal scale, the distances between the various features can be precisely determined.

However, there is no natural zero point for the scale. Arbitrarily defined zero points, such as B. in the degree Celsius temperature scale, while the zero point of the Kelvin temperature scale, which corresponds to absolute zero , is a natural zero.

The difference can be seen from the fact that 20 ° C does not mean twice as much as 10 ° C (e.g. twice as much heat). In the case of Kelvin, on the other hand, the numerical values ​​are actually in relation : 20 Kelvin also means twice as much energy as 10 Kelvin.

If two data pairs (a, b) and (c, d) are equivalent (see below), then the quotient of the differences (a − b) / (c − d) is always the same for interval scales .

Permissible statements for interval scales can be illustrated using the following example. In a second step, two interval scales are set in relation to one another ( ratio scale ). This corresponds to a further data processing of the interval scale: We know the temperatures of day A, day B and day C. Now we form the ratio of the differences: (A − B) / (A − C). Assuming the ratio is 2. Then a valid statement would be: "The temperature difference between days A and B is twice as great as the temperature difference between days A and C."

Each interval scale is such that the ranking of the difference between numbers is equal to the ranking of the feature differences between the corresponding objects.

## Examples

Examples of interval-scaled features with a mathematical pair formation from the scale are: ${\ displaystyle S \ times S}$${\ displaystyle S}$

• Temperature on the Celsius scale ${\ displaystyle ({\ text {temperature}} _ {12 \ colon 00 {\ text {clock}}}, {\ text {temperature}} _ {6 \ colon 00 {\ text {clock}}}) \ in S \ times S}$
• Years with ${\ displaystyle ({\ text {year of death}}, {\ text {year of birth}}) \ in S \ times S}$
• Points in time ${\ displaystyle ({\ text {finish time}}, {\ text {start time}}) \ in S \ times S}$

## Possible operations

In addition to size comparisons, differences and sums from interval-scaled characteristics are useful, since the distances between the individual characteristics are precisely defined here. This means that average values ​​can also be calculated here. Due to the missing zero point, multiplication is not a useful operation for interval-scaled features.

An example:

If it was 10 degrees Celsius yesterday , and today it is 20 degrees, you can say: “It is ten degrees Celsius warmer”, but not: “It is twice as warm as yesterday”. This becomes particularly clear when converting Celsius to Kelvin or degrees Fahrenheit .

## Allowed transformations

Positive- linear transformations of the type${\ displaystyle y = \ alpha x + \ beta}$

## Mathematical interpretation

From a mathematical point of view, an interval scale is a set for which the following applies: ${\ displaystyle S}$

1. There is an equivalence relation with (set of pairs from ). (Nominal scale property). With reference to the example , all pairs are combined into an equivalence class that required the same period of time, e.g. B. and are in an equivalence class (formally:) , because both data pairs and needed the same time between start and finish. See also difference function.${\ displaystyle E \ subseteq P \ times P}$${\ displaystyle P: = S \ times S}$${\ displaystyle S}$${\ displaystyle E = \ left \ {\ left (m, n \ right) \ vert m = (m_ {1}, m_ {2}) \ in P \ wedge n = (n_ {1}, n_ {2} ) \ in P \ wedge m_ {1} -m_ {2} = n_ {1} -n_ {2} \ right \}}$${\ displaystyle ({\ text {finish time}}, {\ text {start time}}) \ in P = S \ times S}$${\ displaystyle ({\ text {finish time}}, {\ text {start time}})}$${\ displaystyle m = (m_ {1}, m_ {2}) = (20.7)}$${\ displaystyle n = (n_ {1}, n_ {2}) = (30.17)}$${\ displaystyle (m, n) \ in E}$${\ displaystyle m}$${\ displaystyle n}$
2. There is a linear order relation with (ordinal scale property). . The relationship can e.g. B. can also be replaced by another order relation on the difference in (e.g. ) if the mathematical properties of the order relation are retained. Based on the example , all pairs are ordered based on the time difference, e.g. B..with and would be ( is less than ) because it took less time between start and finish than . An order relation is defined on the set of pairs of numbers in via the difference between the components of or (see the following definition of the difference function on ).${\ displaystyle O \ subseteq P \ times P}$${\ displaystyle P: = S \ times S}$${\ displaystyle O = \ left \ {\ left (m, n \ right) \ vert m = (m_ {1}, m_ {2}) \ in P = S \ times S \ wedge n = (n_ {1} , n_ {2}) \ in P = S \ times S \ wedge m_ {1} -m_ {2} \ leq n_ {1} -n_ {2} \ right \}}$${\ displaystyle \ leq}$${\ displaystyle D}$${\ displaystyle \ geq}$${\ displaystyle ({\ text {finish time}}, {\ text {start time}}) \ in P = S \ times S}$${\ displaystyle ({\ text {finish time}}, {\ text {start time}})}$${\ displaystyle m = (m_ {1}, m_ {2}) = (40.37)}$${\ displaystyle n = (n_ {1}, n_ {2}) = (30.7)}$${\ displaystyle (m, n) \ in O}$${\ displaystyle m}$${\ displaystyle n}$${\ displaystyle m}$${\ displaystyle n}$${\ displaystyle P}$${\ displaystyle m}$${\ displaystyle n}$${\ displaystyle S \ times S}$
3. Interval scale property:
1. There is a function (difference function ) (you can form differences , e.g. it is assigned to the duration ).${\ displaystyle \ Box - \ Box: S \ times S \ longrightarrow D}$${\ displaystyle ({\ text {finish time}}, {\ text {start time}}) \ in S \ times S}$${\ displaystyle d = {\ text {finish time}} - {\ text {start time}} \ in D}$
2. There is a function (the differences can again be added to the values ​​of ), for which the following also applies: ${\ displaystyle \ Box + \ Box: S \ times D \ longrightarrow S}$${\ displaystyle S}$
1. ${\ displaystyle \ forall \ left (m \ in S \ right): \ left (m + 0 = m \ right)}$ (Adding zero does not change anything)
2. ${\ displaystyle \ forall \ left (m_ {0} \ in S \ right): \ forall \ left (m_ {1} \ in S \ right): \ left (m_ {0} + \ left (m_ {1} -m_ {0} \ right) = m_ {1} \ right)}$ (Difference formation is consistent with addition).
3. ${\ displaystyle \ forall \ left (d_ {0} \ in D \ right): \ forall \ left (d_ {1} \ in D \ right): \ forall \ left (m \ in S \ right): \ left (\ left (m + d_ {0} \ right) + d_ {1} = m + \ left (d_ {0} + d_ {1} \ right) \ right)}$(a kind of one-sided associative law )
3. The set of differences is similar to real numbers in the following ways: ${\ displaystyle D}$
1. ${\ displaystyle \ left (D, \ Box + \ Box \ right)}$is a sub-monoid of ( real numbers with addition ).${\ displaystyle \ left (\ mathbb {R}, \ Box + \ Box \ right)}$

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

Each interval scale is an ordinal scale with a difference function on it . ${\ displaystyle (S, -)}$ ${\ displaystyle S}$${\ displaystyle -}$${\ displaystyle S \ times S}$