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 introduction to the field and includes some of the most basic definitions. For a quick, 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 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 permanent 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, 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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Strategies for assigning rankings
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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 The "levels of measurement", or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled "On the theory of scales of measurement". In this article Stevens claimed that all measurement in, 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 mean There are other statistical measures that use samples that some people confuse with averages - including 'median' and 'mode'. Other simple statistical analyses use measures of spread, such as range, interquartile range, or standard deviation. For a real-valued random variable X, the mean is the expectation of X. Note that not every probability 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 counts 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 votes have been counted the candidate with the most 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
See also: Rank test, Order statistics In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference, Mann-Whitney U test, U statistic, and Level of measurement The "levels of measurement", or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled "On the theory of scales of measurement". In this article Stevens claimed that all measurement inIn statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments, "ranking" refers to the data transformation in which numerical A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number. In addition to their use in counting and measuring, numerals are often used for labels , or ordinal 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 In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference, which consists of the original dataset rearranged into ascending order.
Some kinds of statistical tests A statistical hypothesis test is a method of making statistical decisions using experimental data. In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. The phrase "test of significance" was coined by Ronald Fisher: "Critical tests of this kind may be called tests of significance, employ calculations based on ranks. Examples include:
- 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 considering the values of ranks by columns. Applicable to
- Kruskal-Wallis test
- Rank products
- 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. The test is named for Frank Wilcoxon who, in a single paper,
Ranks can sometimes have non-integer values for tied data values. Thus, in one way of treating tied data values[citation needed], when there is an even number of copies of the same data value, the statistical rank (being the median In probability theory and statistics, a median is described as the numeric 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 highest value and picking the middle one. If there is rank of the tied data) can end in ½.
Some related statistical tests employ the use of Mann-Whitney U statistics, which are computed as the number of inferior minus the number of superior items:
- 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, some generalizations of ranks differ from the U statistics of Hoeffding Wassily Hoeffding was an American statistician, and one of the founding fathers of the nonparametric statistics 1948.[1]
Rank function in Excel
The rank function in Microsoft Excel Microsoft 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 (Visual Basic for Applications). It has been the most widely used spreadsheet application available for these platforms since version 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.[2]
Examples of ranking
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- In politics Politics , is a process by which groups of people make collective decisions. The term is generally applied to behavior within civil governments, but politics has been observed in other group interactions, including corporate, academic, and religious institutions. It consists of "social relations involving authority or power" and refers, rankings focus on the comparison of economic, social, environmental and governance performance of countries, see List of international rankings Categories: International relations | 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. It is widely considered to be the most popular sport in the world national teams are ranked in the 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. The rankings were introduced in 1992, and six teams (Argentina, and, unofficially, in the World Football Elo Ratings The 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. Elo should not be confused with the FIFA World Rankings, which is more prevalent because it is the rating system.
- In the Olympic Games The Olympic Games are a major international event featuring summer and winter sports, in which thousands of athletes participate in a variety of competitions. The Games are currently held every two years in even-numbered years, with Summer and Winter Olympic Games alternating, although they occur every four years within their respective seasonal, each member country (NOC 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. They may nominate cities within their respective areas as candidates for future Olympic Games. NOCs also) is ranked based upon gold, silver and bronze medal counts in the Olympic medal rankings The Olympic medal table is a method of sorting the medal placements of countries in the modern day Olympics. Officially, the International Olympic Committee does not recognize a ranking of participating countries at the Olympic Games. Nevertheless, the IOC does publish medal tables for informational purposes, showing the total number of medals.
- In snooker Snooker is a cue sport that is played on a large 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 12 × 6 ft (3.7 × 1.8 m). It is played using a cue and snooker balls: one white cue ball, 15 red balls worth one point each, and six balls of different colours:, players are ranked using the Snooker world rankings The 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. As of 22 January 2009, the number one-ranked player is Ronnie O'Sullivan
- 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 physical sport. Ice hockey is most popular in areas that are sufficiently cold for natural reliable seasonal ice cover, such as Canada, the Czech Republic, Latvia, the Nordic countries (especially Finland and, national teams are ranked in the IIHF World Ranking The system was approved by the IIHF congress in September 2003. According to IIHF President René Fasel, the system was designed to be simple to understand and "reflect the long-term quality of all national hockey programs and their commitment to international hockey."
- 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. Instead, the game is played on golf "courses", each of, the top male golfers are ranked using the Official World Golf Rankings The 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 six major professional tours, five of which are charter members of the International Federation of PGA Tours:
- 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 (or material(s) of equal value) at a later date. It is any form of deferred payment. The first party is called a standing, the ranking of a security refers to where that particular security would stand in a wind up 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. The process of liquidation also arises when customs, an authority or of the issuing company, i.e., its seniority In finance, seniority refers to the order of repayment in the event of bankruptcy. Senior debt must be repaid before subordinated debt is repaid. Bonds that have the same seniority in a company's capital structure are described as being pari passu in the company's 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 billion in debt is said to be 20% equity-financed and 80%. For instance, capital notes 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. Investors who buy the Capital Notes are the first in line to bear risk if the cash flows from the SIV's assets are insufficient to cover promised payments to all investors are subordinated securities; they would rank behind senior debt in a wind up. In other words the holders of senior debt In finance, senior debt, frequently issued in the form of senior notes or referred to as senior loans, is debt that takes priority over other unsecured or otherwise more "junior" debt owed by the issuer. Senior debt has greater seniority in the issuer's capital structure than subordinated debt. In the event the issuer goes bankrupt, would be paid out before subordinated debt In finance, subordinated debt is debt which ranks after other debts should a company fall into receivership or be closed holders received any funds.
- Search engines A web search engine is designed to search for information on the World Wide Web. The search results are generally presented in a list of results and are often called hits. The information may consist of web pages, images, information and other types of files. Some search engines also mine data available in databases or open directories. Unlike Web rank web pages by their expected 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 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 Hyperlink-Induced Topic Search (also known as Hubs and authorities) is a link analysis algorithm that rates Web pages, developed by Jon Kleinberg. It determines two values for a page: its authority, which estimates the value of the content of the page, and its hub value, which estimates the value of its links to other pages, PageRank PageRank 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. The algorithm may be applied to any collection and 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. 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 heuristics Heuristic is an adjective for experience-based techniques that help in problem solving, learning and discovery. Archimedes is said to have shouted "Heureka" (later converted to "Eureka") after discovering the principle of displacement in his bath. A heuristic method is used to come to a solution rapidly that is hoped to be that consider the number and locations of matches of the various query words on the page itself, in the URL In computing, a Uniform Resource Locator is a Uniform Resource Identifier (URI) that specifies where an identified resource is available and the mechanism for retrieving it. In popular usage and in many technical documents and verbal discussions it is often incorrectly used as a synonym for URI,. The best-known example of a URL is the " or in any anchor text referring to the page.
- In Webometrics 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
- 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.
- The TrueSkill ranking system is a skill based ranking system for Xbox Live developed at Microsoft Research
- A bibliogram 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 journals
References
- ^ Kalbfleisch and Prentice 1973
- ^ Hans Pottel. Statistical flaws in Excel
External links
| Look up ranking in Wiktionary, the free dictionary. |
- Ronen Perry, The Relative Value of American Law Reviews: A Critical Appraisal of Ranking Methods
- Ronen Perry, The Relative Value of American Law Reviews: Refinement and Implementation
- A MATLAB Toolbox for computing rankings using five different methodologies
- TrueSkill Ranking System
Categories: Non-parametric statistics
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Wed, 28 Jul 2010 05:56:26 GMT+00:00
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Sports Illustrated has published their annual Top 20 Highest International Sports earners and naturally MotoGP superstar Valentino Rossi is on the list, even if
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Asked by drock95 - Sun Jun 28 12:32:39 2009 - - 3 Answers - 0 Comments
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Answered by FrankJ - Sun Jun 28 13:07:02 2009
