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Spearman Rank Correlation Coefficient


The Spearman rank correlation coefficient, also known as Spearman's rho, is a nonparametric (distribution-free) rank statistic proposed by Spearman in 1904 as a measure of the strength of the associations between two variables (Lehmann and D'Abrera 1998). The Spearman rank correlation coefficient can be used to give an R-estimate, and is a measure of monotone association that is used when the distribution of the data make Pearson's correlation coefficient undesirable or misleading.

The Spearman rank correlation coefficient is defined by

 r^'=1-6sum(d^2)/(N(N^2-1)),
(1)

where d is the difference in statistical rank of corresponding variables, and is an approximation to the exact correlation coefficient

 r=(sumxy)/(sqrt(sumx^2sumy^2))
(2)

computed from the original data. Because it uses ranks, the Spearman rank correlation coefficient is much easier to compute.

The variance, kurtosis excess, and higher-order moments are

sigma^2=1/(N-1)
(3)
gamma_2=-(114)/(25N)-6/(5N^2)-...
(4)
gamma_3=gamma_5=...=0.
(5)

Student was the first to obtain the variance.


See also

Correlation Coefficient, Least Squares Fitting, Linear Regression, Statistical Rank

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References

Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 5th ed. New York: Macmillan, pp. 338 and 400, 1995.Lehmann, E. L. and D'Abrera, H. J. M. Nonparametrics: Statistical Methods Based on Ranks, rev. ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 292, 300, and 323, 1998.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:Cambridge University Press, pp. 634-637, 1992.

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Spearman Rank Correlation Coefficient

Cite this as:

Weisstein, Eric W. "Spearman Rank Correlation Coefficient." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html

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