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Spearman Rank Correlation Coefficient

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.

The Spearman rank correlation coefficient is defined by

r'\equiv 1-6\sum {d^2\over N(N^2-1)},
\end{displaymath} (1)

where $d$ is the difference in Rank of corresponding variables, and is an approximation to the exact Correlation Coefficient
r\equiv {\sum xy\over \sqrt{\sum x^2\sum y^2}}
\end{displaymath} (2)

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

The Variance, Kurtosis, and higher order Moments are

$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle {1\over N-1}$ (3)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle -{114\over 25N}-{6\over 5N^2}-\ldots$ (4)
$\displaystyle \gamma_3$ $\textstyle =$ $\displaystyle \gamma_5=\ldots=0.$ (5)

Student was the first to obtain the Variance.

See also Correlation Coefficient, Least Squares Fitting, Linear Regression, Rank (Statistics)


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.

© 1996-9 Eric W. Weisstein