# Ranking

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.[1] 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. If two items are the same in rank it is considered a tie.

By reducing detailed measures to a sequence of ordinal numbers, rankings make it possible to evaluate complex information according to certain criteria.[2] 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.[3] 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).[where?] 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 equally receive the same ranking number, and the next items 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 is an example: Suppose you have the data set 1.0, 1.0, 2.0, 3.0, 3.0, 4.0, 5.0, 5.0, 5.0.

The ordinal ranks are 1, 2, 3, 4, 5, 6, 7, 8, 9.

For v = 1.0, the fractional rank is the average of the ordinal ranks: (1 + 2) / 2 = 1.5. In a similar manner, for v = 5.0, the fractional rank is (7 + 8 + 9) / 3 = 8.0.

Thus the fractional ranks are: 1.5, 1.5, 3.0, 4.5, 4.5, 6.0, 8.0, 8.0, 8.0

## Ranking in statistics

In statistics, ranking is 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:

The distribution of values in decreasing order of rank is often of interest when values vary widely in scale; this is the rank-size distribution (or rank-frequency distribution), for example for city sizes or word frequencies. These often follow a power law.

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 ½. Percentile rank is another type of statistical ranking.

### Rank function in Excel

Microsoft Excel provides two ranking functions, the Rank.EQ function which assigns competition ranks ("1224") and the Rank.AVG function which assigns fractional ranks ("1 2.5 2.5 4") as described above. The functions have the order argument,[4] which is by default is set to descending, i.e. the largest number will have a rank 1. This is generally uncommon for statistics where the ranking is usually in ascending order, where the smallest number has a rank 1.

### Comparison of rankings

A rank correlation can be used to compare two rankings for the same set of objects. For example, Spearman's rank correlation coefficient is useful to measure the statistical dependence between the rankings of athletes in two tournaments. And the Kendall rank correlation coefficient is another approach. Alternatively, intersection/overlap-based approaches offer additional flexibility. One example is the "Rank–rank hypergeometric overlap" approach,[5] which is designed to compare ranking of the genes that are at the "top" of two ordered lists of differentially expressed genes. A similar approach is taken by the "Rank Biased Overlap (RBO)",[6] which also implements an adjustable probability, p, to customize the weight assigned at a desired depth of ranking. These approaches have the advantages of addressing disjoint sets, sets of different sizes, and top-weightedness (taking into account the absolute ranking position, which may be ignored in standard non-weighted rank correlation approaches).

### Definition

This definition is due to Vaart, Chpater 13 [7] Let ${\displaystyle X_{1},..X_{n}}$ be a set of random variables. By sorting them into order, we have defined their order statistics ${\displaystyle X_{n,(1)}\leq ...\leq X_{n,(n)}}$ If all the values are unique, the rank of variable number ${\displaystyle i}$ is the unique solution ${\displaystyle R_{n,i}}$ to the equation ${\displaystyle X_{i}=X_{N,(R_{n,i})}}$. In the precense of ties, we may either use a midrank (corresponding to the "Fractional Rank" defined above), defined as the average of all indices ${\displaystyle i}$ such that ${\displaystyle X_{j}=X_{N,(R_{n,j})}}$, or the uprank (corresponding to the "Modified Competition Ranking" above) defined by ${\displaystyle \sum _{j=1}^{n}1\{X_{j}\leq X_{i}\}}$.

## Applications

### Ranking and socio-economic evaluation

The rank methodology based on some specific indices is one of the most common systems used by policy makers and international organizations in order to assess the socio-economic context of the countries. Some notable examples are: Human Development Index (United Nations), Doing Business Index (World Bank), Corruption Perceptions Index (Transparency International) and Index of Economic Freedom (the Heritage Foundation). For instance, the Doing Business Indicator of the World Bank measures business regulations and their enforcement in 190 countries. Countries are ranked according to 10 indicators that are synthetized to produce the final rank. Each indicator is composed of sub-indicators; for instance, the Registering Property Indicator is composed of 4 sub-indicators measuring time, procedures, costs and quality of the land registration system. Obviously, these kinds of ranks are based on subjective criteria for assigning the score. Sometimes, the adopted parameters may produce discrepancies with the empirical observations, therefore potential biases and paradox may emerge from the application of these criteria.[8]

### Ranking as a social game

Being competitive is the very nature of human beings. The desire to achieve a higher social rank can be perceived as a driving force for human beings. In simple terms, we want to know who is the richest, the cleverest, the most handsome or prettiest. We are also sometimes ranked by others: our supervisors, our neighbors, and compare our status in society with that of the others. An inevitable question is how objective or subjective these rankings are? Many ranked lists are based on subjective categorization. We can even pose the question: do we always want to be seen objectively, or rather do not mind having a better image than we deserve? There are certainly specific difficulties in measuring society. In order to find our place in real and virtual communities we need to understand the issues emerging when navigating between objectivity and subjectivity by combining human and artificial intelligence. The set of subjects to treat this topics include comparison, ranking, rating, choices, laws, ranking games, struggle for reputation, etc (see Péter Érdi).[9][10]

### Other examples

2. ^ Malara, Zbigniew; Miśko, Rafał; Sulich, Adam. "Wroclaw University of Technology graduates' career paths". {{cite journal}}: Cite journal requires |journal= (help)
3. ^ Sulich, Adam. "The young people's labour market and crisis of integration in European Union". Retrieved 2017-03-04. {{cite journal}}: Cite journal requires |journal= (help)CS1 maint: url-status (link)