TeilGanzVerhältnisse

TeilGanzVerhältnis is a term used in German-language mathematics and statistics to describe the ratio of a subset or part (Teil) to the whole (Ganz) within a given universe. It is commonly used to express how large a part is relative to the total, and can be written as a fraction, decimal, or percentage.

Formal definition: In a finite set W with size |W| > 0 and a subset T ⊆ W with |T| = k, the TeilGanzVerhältnis is k/|W|. In a measure-theoretic setting, for a measurable space (W, μ) and a measurable subset T ⊆ W, the ratio μ(T)/μ(W) generalizes the concept. If W is divided into disjoint parts, each part has its own TeilGanzVerhältnis equal to its measure divided by the total measure.

Examples: If a class of 100 students has 25 enrolled in a club, the TeilGanzVerhältnis for the club is 25/100 = 0.25 (25%). If a city district has area 40 km² in a country of 200 km², its TeilGanzVerhältnis is 0.2.

Relation to other concepts: It is closely related to fractions, proportions and percentages, and in probability, P(T) corresponds to μ(T)/μ(W) when W is the sample space. The concept is also compatible with normalization in data analysis and with measure theory in formal mathematics.

See also: Fraction, Proportion, Percentage, Probability, Measure theory.