Level of measurement

The scale of measurement or measurement level or Skalendignität (rarely scale quality ) is an important property of empirical characteristics or variables. Depending on the type of a feature or depending on which rules are to be adhered to in its measurement, different levels of scalability can be distinguished:

Interval and ratio scale are combined to cardinal scale. Features on this scale are referred to as metric. Nominal or ordinal features are also called categorical.

Determines the scale level

• The (mathematical) operations that are allowed in accordance with a scaled variable. It can perform operations that are permitted for variables of a certain scale levels, are generally performed on variables of all the higher scale levels. Scalable at a certain level feature can be displayed on all the underlying scale levels, but not vice versa.

• What transformations can be performed with appropriately scaled variables without losing information or change.

• Which information provides the corresponding feature, which allow interpretations characteristics of the corresponding feature.

The scale of measurement does not tell you,

• Whether a variable is discrete ( categorical) or is continuous. ( see main article feature ) Only in the case of nominal scale one can safely say that the feature is not continuous but discrete.

" Although scale level and the number of possible values ​​represent independent concepts, nominal and ordinal features usually discrete and metric scaled features are in practice usually steadily. "

History of classification

" Scales can then be classified, what transformations are allowed for them." But this classification of economies of scale is not without controversy, criticism this, you find mainly in Prytulak (1975) and Duncan. "There are an infinite number of admissible transformations of a certain scale, many different levels of scale could in principle also be distinguished infinite. The most widely used classification goes back to STEVENS ( 1946). This differs nominal, ordinal, interval and ratio scales. " " A more detailed classification, for example, of Narens and Luce ( 1986) and Orth (1974 ) usually contain an even, log- interval scale ' between the interval and ratio scale. In a log - interval scale are still power transformations (x '= s * xr, with s and r is greater than 0 ) only ".

Marks (1974 ) tried the features of different scale levels to systematically record. He proposes a general transformation function in the three constants can be chosen freely. The constants may be each either positively ( ) or zero (0). Zero indicates that a scale transformation here would lead to a loss of information. A plus sign indicates that such a transformation would be possible without loss of information. His proposed general formula is: x ' = (a 1 ) x ( b 1 ) c

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

Accordingly, Marks takes on the following 8 scales, it will be seen that the explanatory power increases, while the opposite direction decrease the possibilities of transformation without loss of information:

Nominal scale

Lowest scale level. For different objects or phenomena merely a decision on equality or inequality of dimension values ​​is by using a comparer taken (eg, x ≠ y ≠ z). It is therefore only qualitative characteristics (eg blood groups or gender). It is the equality relation, so you can decide whether two expressions are equal or unequal. However, the values ​​can not be sorted by size, in the sense of " greater than " or " better than".

Ordinal scale

Exist for a scalable ordinal rankings feature the kind of " greater ", "less ", " more ", " less ", " more ", " less " between two different feature values ​​( eg, x > y > z). However, nothing is said about the distances between these adjacent judgment classes. Most are qualitative characteristics, such as looking into the question of " highest attainable level of education ". Another example are the school grades: Grade 1 is better than grade 2, but I have no information as to 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 is the ranking scale. Here, each value can be assigned only once. Examples include the achievement of ranks in the sport, as well as with other benchmarking or the natural order, as often happens in the animal kingdom in animals that live in social groups such as gallinaceous birds. Your order is therefore also called pecking order.

Interval scale

The order of the feature values ​​is set, and the magnitude of the distance between two values ​​can be justified objectively. As a metric scale makes statements about the magnitude of the differences between two classes. The inequality of characteristic values ​​can be quantified by difference (eg in date could be the result, " three years earlier "). The zero point ( " birth of Christ " ), and the distance between the classes ( years or moons ), however, are set arbitrarily. Note: The metric scale, a distinction is discrete and continuous features.

Ratio scale (even ratio scale)

Has the highest scale level. For her, it also is a metric scale, but in contrast to the interval scale exists an absolute zero point (eg, blood pressure, absolute temperature, age, length dimensions). Only at this scale level multiplication and division are useful and allowed. Ratios of characteristic values ​​may thus be formed (eg, x = y · z).

Gray areas between the scale levels

There are features that can not be precisely assigned to a scale level. This could, for example, in a characteristic can not be confirm with absolute certainty that it is interval scaled, but it is certain that there is more than ordinally scaled. In such a case, you could try an interpretation on an interval scale, but consider this assumption in the interpretation and there proceed with appropriate care. One example is the formation of averages in school grades is encoded as numbers would otherwise constitute a ordinalskaliertes feature because they are defined in fixed expressions such Z. Germany from very good to very poor.

Other examples are times without indication of date ( circadian data) or the compass. Here can be sorted within subregions values ​​and measure distances and with a corresponding limitation on the size of intervals can be even as many useful distances (more precisely: ' clear ') to add. Without limitation is no longer ( " is 2:00 clock before or after 22:00 clock? " - "Both ").

Problems in the scaling

In individual cases, natural systems can occur, although they can be in principle described by a certain scale, but sometimes contain individual deviations. An example are placements at sporting events ( senior scale ), where every athlete actually only occupies a place (first, second, third, etc. ), but has to share his space with another athlete, when it has reached exactly the same reading. Depending on the regulations may be higher or lower rank situated not be forgiven, so that the scale has a gap that does not exist otherwise (not awarded silver medal at the first two places). This is strictly reprimanded before a ranked ordinal scale.

In the animal kingdom rank scales are sometimes not stringent, so that there is within an ascending pecking order, especially in the lower part is interposed triplets or multiplets, which are mutually " hack " according to the scheme A> B> C> A. This is referred to intransitivity. Such a phenomenon can not be described exhaustively by conversion to ordinal scale and requires a complete representation in a matrix or the aid of an additional feature, such as success in lining dispute in uneaten feed weight unless rank higher animals always eat more than lower-ranking, but this often it is not. The matrix representation is therefore preferred in such cases, the scale, though it is visually detectable heavy and expensive to use randomly.