Illustrating nonparametric two-sample methods:
(Wilcoxon rank sum and allied procedures - January 31, 2007.)
(This is  twosamps_ranks.txt  on the Ma408 Web site.)

Sample I (m=9):
  23  21  22  14  23  19  15  15  16
Sample II (n=14):
  27  27  22  21  27  28  19  20  31  24  21  23  23  21

Sorted data from both samples:
 Samp  Value  Tgrp  Ord  Midrank
  X     14     A    1    1.0
  X     15     B    2    2.5
  X     15     B    3    2.5
  X     16     C    4    4.0
  X     19     D    5    5.5
  Y     19     D    6    5.5
  Y     20     E    7    7.0
  X     21     F    8    9.5
  Y     21     F    9    9.5
  Y     21     F   10    9.5
  Y     21     F   11    9.5
  Y     22     G   12   12.5
  X     22     G   13   12.5
  X     23     H   14   15.5
  Y     23     H   15   15.5
  Y     23     H   16   15.5
  X     23     H   17   15.5
  Y     24     I   18   18.0
  Y     27     J   19   20.0
  Y     27     J   20   20.0
  Y     27     J   21   20.0
  Y     28     K   22   22.0
  Y     31     L   23   23.0

12 tiegroups
6 tiegroups with more than one tied value
Check: End reassortment:  i=9 m=9   j=14 n=14

The ranks/midranks in each sample:
Sample I (m=9) (Rank sum=68.5):
  1  2.5  2.5  4  5.5  9.5  12.5  15.5  15.5
Sample II (n=14) (Rank sum=207.5):
  5.5  7  9.5  9.5  9.5  12.5  15.5  15.5  18  20  20  20  22  23
Check: Rsumx+Rsumy=276:  MN*(MN+1)/2 = 23*24/2 = 276

Observed Wilcoxon rank-sum statistic based on Sample II:
  W=207.5   E(W)=168   W-E(W)=39.5
  Normal approx. W/O tie corr.:  Z=2.488  P=0.0128 (two-sided)
  12 tiegroups   Tiesum: 162   Tiesum correction: 0.0133
  Correct Z-score with tie correction:
  Normal approximation:  Z=2.505  P=0.0122 (two-sided)

Finding Hodges-Lehmann estimator of Y-X:
Collecting and sorting  N = 9*14 = 126  Mann-Whitney differences:
HLEst for ``E(Y)-E(X)'' = Median of MW differences
  HLEst: 5.00   Ybar-Xbar = 5.19

Using normal approximation to find indices in (1,126)
  for HL-associated 95% confidence interval:
  C offsets (31,94)   95% CI: (1, 8):
The 126 sorted Mann-Whitney differences:
(The starred values for median or 95% CI)
  -4  -4  -3  -3  -3  -2  -2  -2  -2  -2  -2  -2  -2  -1  -1  -1
  -1  -1  -1  0  0  0  0  0  0  0  0  0  1  1  1  1(*)
  1  1  2  2  2  2  2  2  3  3  3  4  4  4  4  4
  4  4  4  4  4  4  5  5  5  5  5  5  5  5  5(*)  5(*)
  5  5  6  6  6  6  6  6  6  6  6  6  6  6  7  7
  7  7  7  7  7  7  8  8  8  8  8  8  8  8  8(*)  8
  8  9  9  9  9  9  9  10  10  11  11  11  12  12  12  12
  12  12  12  12  13  13  13  13  13  14  15  16  16  17

Summary:
           Test:Z   P-value   EstDiff    95% CI
WRS:        2.505    0.0122    5.00    (1.00, 8.00)
Student-t:  3.353    0.0030    5.19    (1.97, 8.41)   df=21