# Spearman coefficient of rank correlation

A measure of the dependence of two random variables and , based on the rankings of the 's and 's in independent pairs of observations . If is the rank of corresponding to that pair for which the rank of is equal to , then the Spearman coefficient of rank correlation is defined by the formula

or, equivalently, by

where is the difference between the ranks of and . The value of lies between and ; when the rank sequences completely coincide, i.e. , ; and when the rank sequences are completely opposite, i.e. , . This coefficient, like any other rank statistic, is applied to test the hypothesis of independence of two variables. If the variables are independent, then , and . Thus, the amount of deviation of from zero gives information about the dependence or independence of the variables. To construct the corresponding test one computes the distribution of for independent variables and . When one can use tables of the exact distribution (see [2], [4]), and when one can take advantage, for example, of the fact that as the random variable is asymptotically distributed as a standard normal distribution. In the latter case the hypothesis of independence is rejected if , where is the root of the equation and is the standard normal distribution function.

Under the assumption that and have a joint normal distribution with (ordinary) correlation coefficient ,

as , and therefore the variable can be used as an estimator for .

The Spearman coefficient of rank correlation was named in honour of the psychologist C. Spearman (1904), who used it in research on psychology in place of the ordinary correlation coefficient. The tests based on the Spearman coefficient of rank correlation and on the Kendall coefficient of rank correlation are asymptotically equivalent (when , the corresponding rank statistics coincide).

#### References

[1] | C. Spearman, "The proof and measurement of association between two rings" Amer. J. Psychol. , 15 (1904) pp. 72–101 |

[2] | M.G. Kendall, "Rank correlation methods" , Griffin (1962) |

[3] | B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) |

[4] | L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) |

#### Comments

#### References

[a1] | J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967) |

[a2] | M. Hollander, D.A. Wolfe, "Nonparametric statistical methods" , Wiley (1973) |

**How to Cite This Entry:**

Spearman coefficient of rank correlation. A.V. Prokhorov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Spearman_coefficient_of_rank_correlation&oldid=15078