Ranking
Encyclopedia
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
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

, this is known as a weak order or total preorder of objects. It is not necessarily a total order
Total order
In 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 relevance
Relevance
-Introduction:The concept of relevance is studied in many different fields, including cognitive sciences, logic and library and information science. Most fundamentally, however, it is studied in epistemology...

, 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 statistics
In statistics, the term non-parametric statistics has at least two different meanings:The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:...

.

## 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 (i.e., 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
Mean
In 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 count
Borda count
The 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

In statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, "ranking" refers to the data transformation
Data transformation (statistics)
In statistics, data transformation refers to the application of a deterministic mathematical function to each point in a data set — that is, each data point zi is replaced with the transformed value yi = f, where f is a function...

in which numerical or ordinal
Ordinal
Ordinal may refer to:* Ordinal number , a word representing the rank of a number* Ordinal scale, ranking things that are not necessarily numbers* Ordinal indicator, the sign adjacent to a numeral denoting that it is an ordinal number...

values are replaced by their rank when the data are sorted. For example, the numerical data 3.4, 5.1, 2.6, 7.3 are observed, the ranks of these data items would be 2, 3, 1 and 4 respectively. For example, the ordinal data hot, cold, warm would be replaced by 3, 1, 2. In these examples, the ranks are assigned to values in ascending order. (In some other cases, descending ranks are used.) Ranks are related to the indexed list of order statistics, which consists of the original dataset rearranged into ascending order.

Some kinds of statistical tests employ calculations based on ranks. Examples include:
• Friedman test
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
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
• Spearman's rank correlation coefficient
Spearman's rank correlation coefficient
In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho or as r_s, is a non-parametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can...

• Wilcoxon rank-sum test
• Wilcoxon signed-rank test
Wilcoxon signed-rank test
The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used when comparing two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used...

Ranks can sometimes have non-integer values for tied data values. Thus, in one way of treating tied data values, when there is an even number of copies of the same data value, the statistical rank (being the median
Median
In probability theory and statistics, a median is described as the numerical value 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...

rank of the tied data) can end in ½.

### Rank function in Excel

The rank function in Microsoft Excel
Microsoft Excel
Microsoft Excel is a proprietary commercial 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 Visual Basic for Applications...

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.

## Examples of ranking

• In politics
Politics
Politics is a process by which groups of people make collective decisions. The term is generally applied to the art or science of running governmental or state affairs, including behavior within civil governments, but also applies to institutions, fields, and special interest groups such as the...

, 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)
Football (soccer)
Association football, more commonly known as football or soccer, is a sport played between two teams of eleven players with a spherical ball...

national teams are ranked in the FIFA World Rankings
FIFA World Rankings
The FIFA World Rankings is a ranking system for men's national teams in association football, currently led by Spain. 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 Ratings
World Football Elo Ratings
The World Football Elo Ratings is a ranking system for men's national teams in association 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
Olympic Games
The Olympic Games is a major international event featuring summer and winter sports, in which thousands of athletes participate in a variety of competitions. The Olympic Games have come to be regarded as the world’s foremost sports competition where more than 200 nations participate...

, each member country (NOC
National Olympic Committee
National 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
Snooker is a cue sport that is played on a green baize-covered table with pockets in each of the four corners and in the middle of each of the long side cushions. A regular 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 rankings
Snooker world rankings
The snooker world rankings are the official system of ranking professional snooker players to determine automatic qualification and seeding for tournaments on the World Snooker Tour. They are maintained by the sport's governing body, the World Professional Billiards and Snooker Association...

• In ice hockey
Ice hockey
Ice hockey, often referred to as hockey, is a team sport played on ice, in which skaters use wooden or composite sticks to shoot a hard rubber puck into their opponent's net. The game is played between two teams of six players each. Five members of each team skate up and down the ice trying to take...

, national teams are ranked in the IIHF World Ranking
IIHF World Ranking
The 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
Golf is a precision club and ball sport, in which competing players use many types of clubs to hit balls into a series of holes on a golf course using the fewest number of strokes....

, the top male golfers are ranked using the Official World Golf Rankings
Official World Golf Rankings
The Official World Golf Ranking is a system for rating the performance level of male professional golfers...

• In relation to credit
Credit (finance)
Credit is the trust which allows one party to provide resources to another party where that second party does not reimburse the first party immediately , but instead arranges either to repay or return those resources at a later date. The resources provided may be financial Credit is the trust...

standing, the ranking of a security refers to where that particular security would stand in a wind up
Liquidation
In law, liquidation is the process by which a company is brought to an end, and the assets and property of the company redistributed. Liquidation is also sometimes 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 structure
Capital structure
In 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 note
Capital note
In 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 debt
Subordinated debt
In finance, subordinated debt is debt which ranks after other debts should a company fall into receivership or bankruptcy....

• Search engine
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 relevance
Relevance
-Introduction:The concept of relevance is studied in many different fields, including cognitive sciences, logic and library and information science. Most fundamentally, however, it is studied in epistemology...

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 algorithm
HITS algorithm
Hyperlink-Induced Topic Search is a link analysis algorithm that rates Web pages, developed by Jon Kleinberg. It was a precursor to PageRank...

, PageRank
PageRank
PageRank is a link analysis algorithm, named after Larry Page and 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 TrustRank
TrustRank
TrustRank 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 heuristic
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

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 text
Anchor text
The anchor text, link label or link text 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 Webometrics
Webometrics
The science of webometrics tries to measure the World Wide Web to get knowledge about the number and types of hyperlinks, structure of the World Wide Web and usage patterns...

it is possible to rank institutions according to their presence in the web (number of webpages) and the impact of these contents (external inlinks=site citations), such as the Webometrics Ranking of World Universities
Webometrics Ranking of World Universities
The Webometrics Ranking of World Universities, also known as Ranking Web of World Universities, is ranking system for the world's universities based on a composite indicator that takes into account both the volume of the Web contents and the visibility and impact of these web publications...

• In video gaming, players may be given a ranking. To "rank up
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
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
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".
• Academic journals are sometimes ranked according to impact factor
Impact factor
The impact factor, often abbreviated IF, is a measure reflecting the average number of citations to articles published in science and social science journals. It is frequently used as a proxy for the relative importance of a journal within its field, with journals with higher impact factors deemed...

; the number of later articles that cite articles in a given journal.