Characterized

Table A.1 The most important scale types

Scale type	Possible operations	Examples
nominal	test for equality	sex
(categorical, qualitative)		blood type
ordinal	test for equality	school grade
(rank scale, comparative)	greater/less than	wind strength
metric (numerical)	test for equality	length
(interval scale, quantitative)	greater/less than difference	weight time
	maybe ratio	temperature

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   Descriptive Statistics 305
   

Within nominal scales, one sometimes distinguishes according to the number of possible values. Attributes with only two values are called dichotomous, alterna- tives or binary, while attributes with more than two values are called polytomous. Within metric scales, one distinguishes whether only differences (temperature, cal- ender time) are meaningful or whether it makes sense to compute ratios (length, weight, duration). One calls the former case interval scale and the latter ratio scale. In the following, however, we will not make much use of these additional distinc- tions.

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From the above explanations of notions and expressions it already follows that a data set is the joint statement of attribute values for the objects or cases of a sample. The number of attributes that is used to describe the sample is called its dimension. One-dimensional data sets will be denoted by lowercase letters from the end of the alphabet, that is, for example x, y, z. These letters denote the attribute that is used to describe the objects or cases. The elements of the data set (the sample values are denoted by the same lowercase letter, with an index that states their position in the data set. For instance, we write x (x₁, x₂,. , x_n) for a sample of size n.

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(A data set is written as a vector and not as a set, since several objects or cases may have the same sample value.) Multidimensional data sets are written as vectors of lowercase letters from the end of the alphabet. The elements of such data sets are vectors themselves. For example, a two-dimensional data set is written as (x, y) ((x₁, y₁), (x₂, y₂), , (x_n, y_n)), where x and y are the two attributes by which the
sample is described.

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