### Rank statistics

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, this is known as a weak order or total preorder of objects. It is not necessarily a total order 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, 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.

## 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 shorthand 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").

### 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. 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).

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").

Here's an example: Suppose you have the data set 1 1 2 3 3 4 5 5 5 There are 5 different numbers, so there would be five different ranks. If 1 and 1 were actually different numbers, they would occupy ranks 1 and 2. Since they are the same number, you find their rank by finding the average as follows : (rank) 1 + (rank) 2 / 2 numbers total = 1.5 (average rank). The next number in the data set, 2, is thus assigned the rank of 3 (the average takes up 1 and 2 in the first two 1's). The two 3's in the set would occupy ranks 4 and 5 if they were different numbers, so the average rank would be computed as follows: (4 + 5) / 2 = 4.5. 4 would get the rank of 6 (because your average took into account rank 4 and 5 in the average). there are 3 5's in the data set. Their average rank is computed as (7+8+9)/3 = 8

Your ranks would be: 1.5 1.5 3 4.5 4.5 6 8 8 8

## Ranking in statistics

In statistics, "ranking" refers to the data transformation in which numerical 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, which consists of the original dataset rearranged into ascending order.

Some kinds of statistical tests employ calculations based on ranks. Examples include:

Some ranks can have non-integer values for tied data values. For example, when there is an even number of copies of the same data value, the above described fractional statistical rank of the tied data ends in Â˝.

### Rank function in Excel

The rank function in Microsoft Excel 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.[1]

## Examples of ranking

• In politics, 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 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.e., its seniority in the company's capital structure. 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.
• Search engines rank web pages by their expected relevance 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, PageRank and TrustRank. 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 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 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".
• Academic journals are sometimes ranked according to impact factor; the number of later articles that cite articles in a given journal.