The ultimate test is to assess the cumulative
probability associated with the empirical rank
sums Rgroup1 or Rgroup2. This means that we are
not satisfied by concluding that Rgroup1 or Rgroup2
is equal to or smaller or larger than certain tabulated
critical rank sums: we want to assess the
exact cumulative probability associated with the
empirical values Rgroup1 or Rgroup2. To achieve
this, all possible rank sums and the associated
probabilities have to be calculated.
The possible values which can be taken by the
rank sums follow from the fact that two samples
with sizes n1 and n2 provide a total number of
n1n21 rank sums which in the smaller group can
take the values: