By evaluating each item in the collection in such a way that any two items can then be compared to see which should come higher in the ranking. In mathematical terms, this is known as a weak order or total preorder of objects. In Mathematics, especially Order theory, a strict weak ordering is a Binary relation S that is a strict partial order It is not necessarily a total order of objects because two different objects can have the same ranking. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation The rankings themselves are totally ordered. For example, materials are totally preordered by hardness, while degrees of hardness are totally ordered. Hardness refers to various properties of Matter in the Solid phase that give it high resistance to various kinds of shape change when Force

By reducing detailed measures to a sequence of ordinal numbers, rankings make it possible to evaluate complex information according to certain criteria. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. Thus, for example, an Internet search engine may rank the pages it finds according to an evaluation of their relevance, making it possible for the user quickly to select the pages they are likely to want to see.

Ranking is a technique commonly used in non-parametric statistics. Non-parametric statistics is a branch of Statistics concerned with non-parametric Statistical models and non-parametric inference, including non-parametric

## Examples of ranking

• In many sports, individuals or teams are given rankings, generally by the sport's governing body
• In relation to credit standing, the ranking of a security refers to where that particular security would stand in a wind up of the issuing company, i. The snooker world rankings are the official system of ranking professional Snooker players to determine automatic qualification and seeding for tournaments The IIHF World Ranking is a ranking of the national teams of member countries of the International Ice Hockey Federation. The Official World Golf Rankings is a system for rating the performance level of male professional Golfers. Credit is the provision of resources (such as granting a Loan) by one party to another party where that second party does not reimburse the first party immediately thereby generating e. , its seniority in the company's capital structure. In Finance, seniority refers to the order of repayment in the event of Bankruptcy. For instance, capital notes are subordinated securities; they would rank behind senior debt in a wind up. In other words the holders of senior debt would be paid out before subordinated debt holders received any funds. In Finance, senior debt, frequently issued the form of a senior note, is debt that takes priority over other debt securities sold by the issuer In Finance, subordinated debt (also known as subordinated loan, subordinated bond, subordinated debenture or junior debt) is debt
• Some sites propose the user to rank whatever exists (what's the best restaurant, the best city, the most popular dog, the top scorer in international air-guitar, . . . ), like Rankingfever, the most well-known in Europe, ItsRanked!, or Rankrz.
• Search engines rank web pages depending on their relevance to a user's query. See HITS algorithm, PageRank, TrustRank. Hyperlink-Induced Topic Search (HITS is a Link analysis Algorithm that rates Web pages developed by Jon Kleinberg. PageRank is a link analysis algorithm that assigns a numerical weighting to each element of a Hyperlinked set of documents such as the World Wide Web, TrustRank is a Link analysis technique described in a paper by Stanford University and Yahoo! researchers for semi-automatically separating useful
• In video gaming, players may be given a ranking. To "rank up" is to achieve a higher ranking relative to other players, especially with strategies that do not depend on the player's skill. In video gaming Slang, and especially in Online multiplayer games to rank up (sometimes abbreviated
• A bibliogram ranks common noun phrases in a piece of text. A bibliogram is a verbal construct made when Noun phrases from extended stretches of text are ranked high to low by their frequency of Co-occurrence with
• In language, the status of an item (usually through what is known as "downranking" or "rank-shifting") in relation to the uppermost rank in a clause; for example, in the sentence "I want to eat the cake you made today", "eat" is on the uppermost rank, but "made" is downranked as part of the nominal group "the cake you made today"; this nominal group behaves as though it were a single noun (i. e. , I want to eat it), and thus the verb within it ("made") is ranked differently from "eat".
See also: ordinal measurement. The level of Measurement of a Variable in Mathematics and Statistics is a classification that is used to describe the nature of information . .

## Strategies for assigning rankings

It is not always possible to assign rankings uniquely. For example, in a race or competition two (or more) entrants might tie for a place in the ranking. When computing an ordinal measurement, two (or more) of the quantities being ranked might measure equal. The level of Measurement of a Variable in Mathematics and Statistics is a classification that is used to describe the nature of information In these cases, one of the strategies shown below for assigning the rankings may be adopted.

A common short-hand way to distinguish these ranking strategies is by the ranking numbers that would be produced for four items, with the first item ranked ahead of the second and third (which compare equal) which are both ranked ahead of the fourth. These names are also shown below.

### Standard competition ranking ("1224" ranking)

In competition ranking, items that compare equal receive the same ranking number, and then a gap is left in the ranking numbers. The number of ranking numbers that are left out in this gap is one less than the number of items that compared equal. Equivalently, each item's ranking number is 1 plus the number of items ranked above it. This ranking strategy is frequently adopted for competitions, as it means that if two (or more) competitors tie for a position in the ranking, the position of all those ranked below them is unaffected (ie, a competitor only comes second if exactly one person scores better than them, third if exactly two people score better than them, fourth if exactly three people score better than them, etc).

Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 2 ("joint second"), C also gets ranking number 2 ("joint second") and D gets ranking number 4 ("fourth"). In this case, nobody would get ranking number 3 ("third") and that would be left as a gap.

### Modified competition ranking ("1334" ranking)

Sometimes, competition ranking is done by leaving the gaps in the ranking numbers before the sets of equal-ranking items (rather than after them as in standard competition ranking). The number of ranking numbers that are left out in this gap remains one less than the number of items that compared equal. Equivalently, each item's ranking number is equal to the number of items ranked equal to it or above it. This ranking ensures that a competitor only comes second if they score higher than all but one of their opponents, third if they score higher than all but two of their opponents, etc.

Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 3 ("joint third"), C also gets ranking number 3 ("joint third") and D gets ranking number 4 ("fourth"). In this case, nobody would get ranking number 2 ("second") and that would be left as a gap.

### Dense ranking ("1223" ranking)

In dense ranking, items that compare equal receive the same ranking number, and the next item(s) receive the immediately following ranking number. Equivalently, each item's ranking number is 1 plus the number of items ranked above it that are distinct with respect to the ranking order.

Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 2 ("joint second"), C also gets ranking number 2 ("joint second") and D gets ranking number 3 ("third").

### Ordinal ranking ("1234" ranking)

In ordinal ranking, all items receive distinct ordinal numbers, including items that compare equal. The assignment of distinct ordinal numbers to items that compare equal can be done at random, or arbitrarily, but it is generally preferable to use a system that is arbitrary but consistent, as this gives stable results if the ranking is done multiple times. An example of an arbitrary but consistent system would be to incorporate other attributes into the ranking order (such as alphabetical ordering of the competitor's name) to ensure that no two items exactly match.

With this strategy, if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first") and D gets ranking number 4 ("fourth"), and either B gets ranking number 2 ("second") and C gets ranking number 3 ("third") or C gets ranking number 2 ("second") and B gets ranking number 3 ("third").

In computer data processing, ordinal ranking is also referred to as "row numbering". . . .

### Fractional ranking ("1 2. 5 2. 5 4" ranking)

Items that compare equal receive the same ranking number, which is the mean of what they would have under ordinal rankings. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean Equivalently, the ranking number of 1 plus the number of items ranked above it plus half the number of items equal to it. This strategy has the property that the sum of the ranking numbers is the same as under ordinal ranking. For this reason, it is used in computing Borda counts and in statistical tests (see below). The Borda count is a single-winner election method in which voters rank candidates in order of preference

Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B and C each get ranking number 2. 5 (average of "joint second/third") and D gets ranking number 4 ("fourth").

## Ranking in statistics

Some kinds of statistical tests employ calculations based on ranks. A statistical hypothesis test is a method of making statistical decisions using experimental data Examples:

Ranks can have non-integer values for tied data values. The Friedman test is a non-parametric Statistical test developed by the U In Statistics, the Mann-Whitney U test (also called the Mann-Whitney-Wilcoxon ( MWW) Wilcoxon rank-sum test, or Wilcoxon-Mann-Whitney The Wilcoxon signed-rank test is a non-parametric alternative to the paired Student's t-test for the case of two related samples or repeated measurements on a single In Statistics, the Kruskal-Wallis one-way analysis of variance by ranks (named after William Kruskal and W The rank product is a biologically motivated test for the detection of differentially expressed genes in replicated microarray experiments When there is an even number of the same data value, the statistical rank (being the median rank of the tied data) ends in ½. In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution

Some related statistical tests employ the use of u-scores, which are computed as the number of inferior minus the number of superior items. Examples:

For univariate data, tests ranks and u-scores are equivalent (Example: The "Wilcoxon/Mann-Whitney test"). In Statistics, the sign test can be used to test the hypothesis that there is "no difference" between the continuous distributions of two Random variables In Statistics, the Mann-Whitney U test (also called the Mann-Whitney-Wilcoxon ( MWW) Wilcoxon rank-sum test, or Wilcoxon-Mann-Whitney For multivariate data, however, generalizations of ranks (Kalbfleisch and Prentice 1973) and u-scores (Hoeffding 1948) can differ.

### Rank function in Excel

The rank function in Microsoft Excel assigns competition ranks ("1224") as described above. In Computing, Microsoft Excel (full name Microsoft Office Excel) consists of a proprietary Spreadsheet -application written and distributed For some statistical purposes, that is not the desired result - for instance, it means that the sum of ranks for a list of a given length changes depending on the number of ties. Pottel has described a user defined ranking function which assigns fractional ranks to ties to keep the sum consistent. [1]