gives Spearman's rank correlation coefficient for the vectors v1 and v2.


gives Spearman's rank correlation coefficient for the matrix m.


gives Spearman's rank correlation coefficient for the matrices m1 and m2.


gives Spearman's rank correlation matrix for the multivariate symbolic distribution dist.


gives the ^(th) Spearman rank correlation for the multivariate symbolic distribution dist.


  • SpearmanRho[v1,v2] gives Spearman's rank correlation coefficient between v1 and v2.
  • Spearman's is a measure of association based on the rank differences between two lists which indicates how well a monotonic function describes their relationship.
  • Spearman's is given by , where is equal to Length[xlist], r_(i) is the rank difference between and , is the correction term for ties in v1, and is the correction term for ties in v2.
  • SpearmanRho[{v11,v12,},{v21,v22,}] is equivalent to Correlation[{r11,r12,},{r21,r22,}] where rij is the tie-corrected ranking corresponding to vij.
  • The arguments v1 and v2 can be any realvalued vectors of equal length.
  • For a matrix m with columns SpearmanRho[m] is a × matrix of the rank correlations between columns of m.
  • For an × matrix m1 and an × matrix m2 SpearmanRho[m1,m2] is a × matrix of the rank correlations between columns of m1 and columns of m2.
  • SpearmanRho[dist,i,j] is 12 Expectation[F[x]G[y],{x,y}disti,j]-3 where F[x] and G[y] are the CDFs of the i^(th) and j^(th) marginals of dist, respectively, and disti,j is the ^(th) marginal of dist.
  • SpearmanRho[dist] gives a matrix where the ^(th) entry is given by SpearmanRho[dist,i,j].


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Basic Examples  (4)

Spearman's for two vectors:

Spearman's for a matrix:

Spearman's for two matrices:

Compute Spearman's for a bivariate distribution:

Compare to a simulated value:

Scope  (7)

Data  (4)

Exact input yields exact output:

Approximate input yields approximate output:

Works with large arrays:

SparseArray data can be used:

Distributions and Processes  (3)

Spearman's for a continuous multivariate distribution:

Spearman's for derived distributions:

Data distribution:

Spearman's for a random process at times s and t:

Applications  (4)

Spearman's is typically used to detect linear dependence between two vectors:

The absolute magnitude of tends to 1 given strong linear dependence:

The value tends to 0 for linearly independent vectors:

Spearman's can be used to measure linear association:

Spearman's can only detect monotonic relationships:

HoeffdingD can be used to detect a variety of dependence structures:

A collection of measurements were taken from a representative sample of new cars in 1993. Because some of the variables are measured at an ordinal scale, Spearman's is more appropriate than Correlation for measuring monotonic association:

A scatter plot matrix of the various metrics:

Spearman's corresponding to the scatter plot matrix:

SpearmanRankTest suggests that vehicles with higher horsepower are more costly:

Higher fuel economy meant lower prices in 1993:

Properties & Relations  (10)

Spearman's ranges from -1 to 1 for high negative and high positive association, respectively:

Spearman's is Correlation applied to ranks:

With no ties, ranks can be computed using ordering:

Spearman's matrix is symmetric:

The diagonal elements of Spearman's matrix are 1:

Spearman's is related to KendallTau:

KendallTau tends to be about of given weak linear association:

Spearman's will attain 1 or -1 if the variables are perfectly monotonically related:

This is in contrast to Correlation, which strictly measures linear association:

Spearman's is less sensitive to outliers than Correlation:

With outliers:

Without outliers:

Use SpearmanRankTest to test for independence:

Alternatively, use IndependenceTest to automatically select an appropriate test:

Use CorrelationTest to test a particular value of Spearman's :

Test against a value of :

Spearman's for a continuous bivariate distribution:

Introduced in 2012