PROC CAPABILITY and General Statements
Signed Rank Statistic
The signed rank statistic S is computed as
![\[ S =\sum _{ i:x_ i > \mu _0} r_ i^+ - \frac{n (n+1)}{4} \]](images/qcug_capability0071.png){kind=small} |
where 
is the rank of 
after discarding values of 
, and n is the number of 
values not equal to 
. Average ranks are used for tied values.
If 
, the significance of S is computed from the exact distribution of S, where the distribution is a convolution of scaled binomial distributions. When 
, the significance of S is computed by treating
![\[ S \sqrt { \frac{n - 1}{nV -S^2} } \]](images/qcug_capability0078.png){kind=small} |
as a Student t variate with 
degrees of freedom. V is computed as
![\[ V = \frac{1}{24} n(n+1)(2n+1) - \frac{1}{48} \sum t_ i(t_ i+1)(t_ i-1) \]](images/qcug_capability0079.png){kind=small} |
where the sum is over groups tied in absolute value and where 
is the number of values in the ith group (Iman 1974, Conover 1980). The null hypothesis tested is that the mean (or median) is , assuming that the distribution is symmetric. Refer to Lehmann (1998).