A ranking is a relationship between a set of items such that, for any two items, the first is either 'ranked higher than', 'ranked lower than' or 'ranked equal to' the second. In mathematics Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. This article gives a detailed introduction to the field and includes some of the most basic definitions. For, this is known as a weak order or total preorder In mathematics, especially order theory, a strict weak ordering is a binary relation < on a set S that is a strict partial order in which the relation "neither a < b nor b < a" is transitive of objects. It is not necessarily a total order In mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain of objects because two different objects can have the same ranking. The rankings themselves are totally ordered. For example, materials are totally preordered by hardness Hardness refers to various properties of matter in the solid phase that give it high resistance to various kinds of shape change when force is applied. Hard matter is contrasted with soft matter, while degrees of hardness are totally ordered.
By reducing detailed measures to a sequence of ordinal numbers In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. The finite, rankings make it possible to evaluate complex information according to certain criteria. Thus, for example, an Internet search engine may rank the pages it finds according to an estimation of their relevance Relevance is a term used to describe how pertinent, connected, or applicable something is to a given matter. A thing is relevant if it serves as a means to a given purpose. Imagine a patient suffering a well-defined disease such as scurvy caused by lack of vitamin C. The relevant medical treatment for him would be doses of tablets containing, making it possible for the user quickly to select the pages they are likely to want to see.
Analysis of data obtained by ranking commonly requires non-parametric statistics Non-parametric methods are widely used for studying populations that take on a ranked order . The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences; in terms of levels of measurement, for data on an ordinal scale.
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