# Rank correlation

**ordinal associationrank correlation coefficientrank regression**

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.wikipedia

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### Ranking

**rankrankedrankings**

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 can be used to compare two rankings for the same set of objects.

### Correlation and dependence

**correlationcorrelatedcorrelations**

Rank correlation coefficients, such as Spearman's rank correlation coefficient and Kendall's rank correlation coefficient measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship.

### Mann–Whitney U test

**Mann–Whitney UWilcoxon rank-sum testMann-Whitney U test**

For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.

A method of reporting the effect size for the Mann–Whitney U test is with a measure of rank correlation known as the rank-biserial correlation.

### Spearman's rank correlation coefficient

**Spearman's rank correlationrank correlation coefficientSpearman**

which is exactly Spearman's rank correlation coefficient \rho.

In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho (rho) or as r_s, is a nonparametric measure of rank correlation (statistical dependence between the rankings of two variables).

### Goodman and Kruskal's gamma

**Gamma test (statistics)Yule's ''Qgamma**

In statistics, Goodman and Kruskal's gamma is a measure of rank correlation, i.e., the similarity of the orderings of the data when ranked by each of the quantities.

### Somers' D

**d yx /d xy Somers' ''D**

In statistics, Somers’ D, sometimes incorrectly referred to as Somer’s D, is a measure of ordinal association between two possibly dependent random variables X and Y. Somers’ D takes values between -1 when all pairs of the variables disagree and 1 when all pairs of the variables agree.

### Kendall rank correlation coefficient

**Kendall tau rank correlation coefficientKendall's tauKendall's τ**

The sum is the number of concordant pairs minus the number of discordant pairs (see Kendall tau rank correlation coefficient).

In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's tau coefficient (after the Greek letter τ), is a statistic used to measure the ordinal association between two measured quantities.

### Statistics

**statisticalstatistical analysisstatistician**

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.

### Ordinal data

**ordinalordinal variableordered categorical data**

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.

### Statistical significance

**statistically significantsignificantsignificance level**

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.

### Nonparametric statistics

**non-parametricnon-parametric statisticsnonparametric**

For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.

### Wilcoxon signed-rank test

**Wilcoxon Signed Rank Testsigned-ranksWilcoxon**

For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.

### Contingency table

**cross tabulationcontingency tablescrosstab**

As another example, in a contingency table with low income, medium income, and high income in the row variable and educational level—no high school, high school, university—in the column variable), a rank correlation measures the relationship between income and educational level.

### Coefficient

**coefficientsleading coefficientfactor**

An increasing rank correlation coefficient implies increasing agreement between rankings.

### Permutation

**permutationscycle notationpermuted**

Following, a ranking can be seen as a permutation of a set of objects.

### Set (mathematics)

**setsetsmathematical set**

Following, a ranking can be seen as a permutation of a set of objects.

### Symmetric group

**symmetricinfinite symmetric groupS 4**

Thus we can look at observed rankings as data obtained when the sample space is (identified with) a symmetric group.

### Metric (mathematics)

**metricdistance functionmetrics**

We can then introduce a metric, making the symmetric group into a metric space.

### Metric space

**metricmetric spacesmetric geometry**

We can then introduce a metric, making the symmetric group into a metric space.

### Frobenius inner product

**FrobeniusFrobenius inner productsmatrix inner product**

where is the Frobenius inner product and the Frobenius norm.

### Matrix norm

**Frobenius normnormspectral norm**

where is the Frobenius inner product and the Frobenius norm.

### Stochastic empirical loading and dilution model

In SELDM, these three treatment variables are modeled by using the trapezoidal distribution and the rank correlation with the associated highway-runoff variables.

### Maurice Kendall

**M. G. KendallMaurice George KendallM.G. Kendall**

During this period he also began work on the rank correlation coefficient which currently bears his name (Kendall's tau), which eventually led to a monograph on Rank Correlation in 1948.

### Bivariate analysis

**bivariate**

If both variables are ordinal, meaning they are ranked in a sequence as first, second, etc., then a rank correlation coefficient can be computed.

### Jonckheere's trend test

**JonckheereJonckheere test for ordered alternatives**

The test can be seen as a special case of Maurice Kendall’s more general method of rank correlation and makes use of the Kendall’s S statistic.