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 monotone association between two variables (Lehmann and D'Abrera
1998). The coefficient is commonly denoted 
, while 
is also used for its sample value. It can be used to give
an R-estimate and is useful when the distribution
of the data makes Pearson's correlation coefficient
undesirable or misleading.
For a sample of 
pairs, let 
and 
be the statistical ranks
of the two components of pair 
, with respective means 
and 
. The sample Spearman rank correlation coefficient is the
Pearson correlation coefficient of these
ranks,
 |
(1)
|
If there are no tied statistical ranks and 
,
this simplifies exactly to
 |
(2)
|
When ties occur, the Pearson correlation coefficient of the statistical ranks remains the definition, while the shortcut formula must be replaced by a tie-corrected form.
Under the null hypothesis that the two rank orderings are independent, with no ties, 
has mean 0. Its variance, kurtosis excess, and odd standardized
moments include
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
Student was the first to obtain the variance.
