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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.
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 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. [4] These names are also shown below.
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").
This method is called "Low" by IBM SPSS [5] and "min" by the R programming language [6] in their methods to handle ties.
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.
This method is called "High" by IBM SPSS [5] and "max" by the R programming language [6] in their methods to handle ties.
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").
This method is called "Sequential" by IBM SPSS [5] and "dense" by the R programming language [7] in their methods to handle ties.
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".
This method corresponds to the "first", "last", and "random" methods in the R programming language [6] to handle ties.
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
This method is called "Mean" by IBM SPSS [5] and "average" by the R programming language [6] in their methods to handle ties.
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League tables are used to compare the academic achievements of different institutions. College and university rankings order institutions in higher education by combinations of factors. In addition to entire institutions, specific programs, departments, and schools are ranked. These rankings usually are conducted by magazines, newspapers, governments and academics. For example, league tables of British universities are published annually by The Guardian , The Independent , The Sunday Times , and The Times . The primary aim of these rankings is to inform potential applicants about British universities based on a range of criteria. Similarly, in countries like India, league tables are being developed and a popular magazine, Education World, published them based on data from TheLearningPoint.net. [ citation needed ]
It is complained that the ranking of England's schools to rigid guidelines that fail to take into account wider social conditions actually makes failing schools even worse. This is because the most involved parents will then avoid such schools, leaving only the children of non-ambitious parents to attend. [8]
In business, league tables list the leaders in the business activity within a specific industry, ranking companies based on different criteria including revenue, earnings, and other relevant key performance indicators (such as market share and meeting customer expectations) enabling people to quickly analyze significant data. [9]
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 include the 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 ten indicators that are synthesized to produce the final rank. Each indicator is composed of sub-indicators; for instance, the Registering Property Indicator is composed of four sub-indicators measuring time, procedures, costs, and quality of the land registration system. 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. [10]
A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, that is, a candidate preferred by more voters than any others, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.
Copeland's method is a ranked voting method based on a scoring system of pairwise "wins", "losses", and "ties". The method has a long history:
Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as is parametric statistics. Nonparametric statistics can be used for descriptive statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated.
In the social sciences, scaling is the process of measuring or ordering entities with respect to quantitative attributes or traits. For example, a scaling technique might involve estimating individuals' levels of extraversion, or the perceived quality of products. Certain methods of scaling permit estimation of magnitudes on a continuum, while other methods provide only for relative ordering of the entities.
In statistics, the Mann–Whitney U test is a nonparametric test of the null hypothesis that, for randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X.
A Likert scale is a psychometric scale named after its inventor, American social psychologist Rensis Likert, which is commonly used in research questionnaires. It is the most widely used approach to scaling responses in survey research, such that the term is often used interchangeably with rating scale, although there are other types of rating scales.
Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement originated in psychology and has since had a complex history, being adopted and extended in some disciplines and by some scholars, and criticized or rejected by others. Other classifications include those by Mosteller and Tukey, and by Chrisman.
The Kruskal–Wallis test by ranks, Kruskal–Wallis H test, or one-way ANOVA on ranks is a non-parametric method for testing whether samples originate from the same distribution. It is used for comparing two or more independent samples of equal or different sample sizes. It extends the Mann–Whitney U test, which is used for comparing only two groups. The parametric equivalent of the Kruskal–Wallis test is the one-way analysis of variance (ANOVA).
Positional voting is a ranked voting electoral system in which the options or candidates receive points based on their rank position on each ballot and the one with the most points overall wins. The lower-ranked preference in any adjacent pair is generally of less value than the higher-ranked one. Although it may sometimes be weighted the same, it is never worth more. A valid progression of points or weightings may be chosen at will or it may form a mathematical sequence such as an arithmetic progression, a geometric one or a harmonic one. The set of weightings employed in an election heavily influences the rank ordering of the candidates. The steeper the initial decline in preference values with descending rank, the more polarised and less consensual the positional voting system becomes.
The Friedman test is a non-parametric statistical test developed by 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 complete block designs, it is thus a special case of the Durbin test.
In statistics, a rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them. For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.
Kendall's W is a non-parametric statistic for rank correlation. It is a normalization of the statistic of the Friedman test, and can be used for assessing agreement among raters and in particular inter-rater reliability. Kendall's W ranges from 0 to 1.
In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient, is a statistic used to measure the ordinal association between two measured quantities. A τ test is a non-parametric hypothesis test for statistical dependence based on the τ coefficient. It is a measure of rank correlation: the similarity of the orderings of the data when ranked by each of the quantities. It is named after Maurice Kendall, who developed it in 1938, though Gustav Fechner had proposed a similar measure in the context of time series in 1897.
Discounted cumulative gain (DCG) is a measure of ranking quality which is often presented in its normalized and more easily interpretable form, known as Normalized DCG (nDCG or NDCG). In information retrieval, NDCG is often used to measure effectiveness of search engine algorithms and related applications. Using a graded relevance scale of documents in a search-engine result set, DCG measures the usefulness, or gain, of a document based on its position in the result list. To compute NDCG, the gain is accumulated from the top of the result list to the bottom, with the gain of each result discounted at lower ranks. The normalized version corrects for the fact that different queries may have different numbers of relevant results.
In multivariate statistics, exploratory factor analysis (EFA) is a statistical method used to uncover the underlying structure of a relatively large set of variables. EFA is a technique within factor analysis whose overarching goal is to identify the underlying relationships between measured variables. It is commonly used by researchers when developing a scale and serves to identify a set of latent constructs underlying a battery of measured variables. It should be used when the researcher has no a priori hypothesis about factors or patterns of measured variables. Measured variables are any one of several attributes of people that may be observed and measured. Examples of measured variables could be the physical height, weight, and pulse rate of a human being. Usually, researchers would have a large number of measured variables, which are assumed to be related to a smaller number of "unobserved" factors. Researchers must carefully consider the number of measured variables to include in the analysis. EFA procedures are more accurate when each factor is represented by multiple measured variables in the analysis.
Learning to rank or machine-learned ranking (MLR) is the application of machine learning, typically supervised, semi-supervised or reinforcement learning, in the construction of ranking models for information retrieval systems. Training data may, for example, consist of lists of items with some partial order specified between items in each list. This order is typically induced by giving a numerical or ordinal score or a binary judgment for each item. The goal of constructing the ranking model is to rank new, unseen lists in a similar way to rankings in the training data.
Cardinal voting refers to any electoral system which allows the voter to give each candidate an independent evaluation, typically a rating or grade. These are also referred to as "rated", "evaluative", "graded", or "absolute" voting systems. Cardinal methods and ordinal methods are two main categories of modern voting systems, along with plurality voting.
Fair random assignment is a kind of a fair division problem.
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an ordinal scale, one of four levels of measurement described by S. S. Stevens in 1946. The ordinal scale is distinguished from the nominal scale by having a ranking. It also differs from the interval scale and ratio scale by not having category widths that represent equal increments of the underlying attribute.
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. Ranks are related to the indexed list of order statistics, which consists of the original dataset rearranged into ascending order.
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