In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, a rank correlation is the relationship between different ranking

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....

s of the same set of items. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess its significance

Statistical significance

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....

.

Correlation coefficients

Some of the more popular rank correlation

Correlation

In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

 statistics include

1. Spearman's ρ
   Spearman's rank correlation coefficient
   In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho or as r_s, is a non-parametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can...
   
   
2. Kendall's τ
3. Goodman and Kruskal's γ
   Gamma test (statistics)
   In statistics, a gamma test tests the strength of association of the cross tabulated data when both variables are measured at the ordinal level. It makes no adjustment for either table size or ties. Values range from −1 to +1...
   
   

An increasing rank correlation coefficient

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

 implies increasing agreement between rankings. The coefficient is inside the interval [−1, 1] and assumes the value:

• −1 if the disagreement between the two rankings is perfect; one ranking is the reverse of the other.
• 0 if the rankings are completely independent.
• 1 if the agreement between the two rankings is perfect; the two rankings are the same.

Following , a ranking can be seen as a permutation

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

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

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

. We can then introduce a metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

, making the symmetric group into a metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

. Different metrics will correspond to different rank correlations.