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
mathematicsOrder 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...
, this is known as a of objects. It is not necessarily a
total orderIn mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
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, while degrees of hardness are totally ordered.
By reducing detailed measures to a sequence of ordinal numbers, rankings make it possible to evaluate complex information according to certain criteria.
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
mathematicsOrder 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...
, this is known as a of objects. It is not necessarily a
total orderIn mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
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, while degrees of hardness are totally ordered.
By reducing detailed measures to a sequence of ordinal numbers, 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
relevanceRelevance 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...
, 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 statisticsIn statistics, the term non-parametric statistics covers a range of topics:*distribution free methods which do not rely on assumptions that the data are drawn from a given probability distribution. As such it is the opposite of parametric statistics...
.
Examples of ranking
- In politics
Politics is a process by which groups of people make decisions. The term is generally applied to behavior within civil governments, but politics has been observed in all human group interactions, including corporate, academic and religious institutions...
, rankings focus on the comparison of economic, social, environmental and governance performance of countries, see List of international rankings
- In many sports, individuals or teams are given rankings, generally by the sport's governing body
- In football (soccer)
Association football, more commonly known as football or soccer, is a team sport played between two teams of eleven players using a spherical ball...
national teams are ranked in the FIFA World RankingsThe FIFA World Rankings is a ranking system for men's national teams in Association football, currently led by Brazil. The teams of the member nations of FIFA , football's world governing body, are ranked based on their game results with the most successful teams being ranked highest...
and, unofficially, in the World Football Elo RatingsThe World Football Elo Ratings is a ranking system for men's national teams in football. The method used to rank teams is based upon the Elo rating system method but modified to take various football-specific variables into account...
.
- In the Olympic Games
The Olympic Games are a major international event of summer and winter sports, in which thousands of athletes compete in a wide variety of events. The Games are currently held every two years, with Summer and Winter Olympic Games alternating. Originally, the ancient Olympic Games were held in...
, each member country (NOCNational Olympic Committees are the national constituents of the worldwide Olympic movement. Subject to the controls of the International Olympic Committee, they are responsible for organizing their people's participation in the Olympic Games...
) is ranked based upon gold, silver and bronze medal counts in the Olympic medal rankings.
- In snooker
Snooker is a cue sport that is played on a large baize-covered table with pockets in each of the four corners and in the middle of each of the long side cushions. A regulation table is . It is played using a cue and snooker balls: one white , 15 worth one point each, and six balls of different ...
, players are ranked using the Snooker world rankingsThe snooker world rankings are the official system of ranking professional snooker players to determine automatic qualification and seeding for tournaments. They are maintained by the sport's governing body, the WPBSA...
- In ice hockey
Ice Hockey is a team sport played on ice, in which skaters use sticks to direct a puck into the opposing team's goal. It is a fast-paced and physical sport...
, national teams are ranked in the IIHF World RankingThe IIHF World Ranking is a ranking of the national teams of member countries of the International Ice Hockey Federation.The system was approved by the IIHF congress in September 2003...
- In golf
Golf is a precision club-and-ball sport, in which competing players , using many types of clubs, attempt to hit balls into each hole on a golf course while employing the fewest number of strokes. Golf is one of the few ball games that does not require a standardized playing area...
, the top male golfers are ranked using the Official World Golf RankingsThe Official World Golf Rankings is a system for rating the performance level of male professional golfers. They were introduced in 1986 and are endorsed by the four major championships and the six professional tours which make up the International Federation of PGA Tours, namely the PGA Tour, the...
- In relation to credit
Credit is the provision of resources by one party to another party where that second party does not reimburse the first party immediately, thereby generating a debt, and instead arranges either to repay or return those resources at a later date. It is any form of deferred payment...
standing, the ranking of a security refers to where that particular security would stand in a wind upIn law, liquidation is the process by which a company is brought to an end, and the assets and property of the company redistributed. Liquidation can also be referred to as winding-up or dissolution, although dissolution technically refers to the last stage of liquidation...
of the issuing company, i.e., its seniority in the company's capital structureIn finance, capital structure refers to the way a corporation finances its assets through some combination of equity, debt, or hybrid securities. A firm's capital structure is then the composition or 'structure' of its liabilities. For example, a firm that sells $20 billion in equity and $80...
. For instance, capital noteIn structured finance the Capital note is the most junior security issued by a Structured investment vehicle. It is comparable to the Equity Tranche of a CDO...
s 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 debtIn finance, subordinated debt is debt which ranks after other debts should a company fall into receivership or be closed....
holders received any funds.
- Search engine
A search engine is an information retrieval system designed to help find information stored on a computer system. The search results are usually presented in a list and are commonly called hits. Search engines help to minimize the time required to find information and the amount of information...
s rank web pages by their expected relevanceRelevance 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...
to a user's query using a combination of query-dependent and query-independent methods. Query-independent methods attempt to measure the estimated importance of a page, independent of any consideration of how well it matches the specific query. Query-independent ranking is usually based on link analysis; examples include the HITS algorithmHyperlink-Induced Topic Search is a link analysis algorithm that rates Web pages, developed by Jon Kleinberg...
, PageRankPageRank is a link analysis algorithm, named after Larry Page, used by the Google Internet search engine that assigns a numerical weighting to each element of a hyperlinked set of documents, such as the World Wide Web, with the purpose of "measuring" its relative importance within the set...
and TrustRankTrustRank is a link analysis technique described in a paper by Stanford University and Yahoo! researchers for semi-automatically separating useful webpages from spam.Many Web spam pages are created only with the intention of misleading search engines...
. Query-dependent methods attempt to measure the degree to which a page matches a specific query, independent of the importance of the page. Query-dependent ranking is usually based on heuristicHeuristic is an adjective for experience-based techniques that help in problem solving, learning and discovery. A heuristic method is particularly used to rapidly come to a solution that is hoped to be close to the best possible answer, or 'optimal solution'...
s that consider the number and locations of matches of the various query words on the page itself, in the URL or in any anchor textThe anchor text, link label or link title is the visible, clickable text in a hyperlink. The words contained in the anchor text can determine the ranking that the page will receive by search engines. Since 1998, some web browsers have added the ability to show a tooltip for a hyperlink before it is...
referring to the page.
- In video gaming, players may be given a ranking. To "rank up
In video gaming slang, and especially in online multiplayer games, to rank up is to achieve a higher ranking relative to other players...
" is to achieve a higher ranking relative to other players, especially with strategies that do not depend on the player's skill.
- The TrueSkill
TrueSkill is a Bayesian ranking algorithm developed by Microsoft Research and used in the Xbox matchmaking system built to address some perceived flaws in the Elo rating system....
ranking system is a skill based ranking system for Xbox Live developed at Microsoft Research
- A bibliogram
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 one or more user-supplied seed terms...
ranks common noun phrases in a piece of text.
- 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".
- In Academic journal
An academic journal is a peer-reviewed periodical in which scholarship relating to a particular academic discipline is published. Academic journals serve as forums for the introduction and presentation for scrutiny of new research, and the critique of existing research. Content typically takes the...
s
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. 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
meanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
of what they would have under ordinal rankings. 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 countThe Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...
s and in statistical tests (see below).
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. Examples:
- Friedman test
The Friedman test is a non-parametric statistical test developed by the U.S. economist Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking each row together, then...
- Kruskal-Wallis test
- Rank product
The rank product is a biologically motivated test for the detection of differentially expressed genes in replicated microarray experiments.It is a simple non-parametric statistical method based on ranks of fold changes...
s
- Wilcoxon rank-sum test
- Wilcoxon signed-rank test
The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test for the case of two related samples or repeated measurements on a single sample. It can be used as an alternative to the paired Student's t-test when the population cannot be assumed to be normally distributed...
Ranks can have non-integer values for tied data values. When there is an even number of the same data value, the statistical rank (being the
medianIn probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest...
rank of the tied data) ends in ½.
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:
- Mann-Whitney U test
- Sign 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 X and Y. Formally:...
For univariate data, tests ranks and u-scores are equivalent (Example: The "Wilcoxon/Mann-Whitney test"). 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 ExcelMicrosoft Excel is a spreadsheet-application written and distributed by Microsoft for Microsoft Windows and Mac OS X. It features calculation, graphing tools, pivot tables and a macro programming language called VBA...
assigns competition ranks ("1224") as described above. 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.
External links