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Wilcoxon Rank-Sum test for two samples:

X:  23  21  22  14  23  19  15  15  16
Y:  27  27  22  21  27  28  19  20  31  24  21  23  23  21 

SampleSizes_X_Y_Both =

     9    14    23

Columns are Ordinal, Group number, and Values:

dd =

     1     1    23
     2     1    21
     3     1    22
     4     1    14
     5     1    23
     6     1    19
     7     1    15
     8     1    15
     9     1    16
    10     2    27
    11     2    27
    12     2    22
    13     2    21
    14     2    27
    15     2    28
    16     2    19
    17     2    20
    18     2    31
    19     2    24
    20     2    21
    21     2    23
    22     2    23
    23     2    21

The function  getmidranks()  was used to generate midranks
  and tie-group sizes
Tie-group sizes for tie-groups with more than one value:

Number_TieGroups_of_Size_ge_2 =

     6

TieGroup Sizes of Size ge 2:
  
     2     2     4     2     4     3

Columns are Ordinal, Group numbers, Values, and Midranks:
Note: Directly displaying matrix dd would have all columns
  to 4 places beyond decimal point because of midranks.
Instead, we use fprintf() on the rows of dd so that we can
  determine the format.
 
    1.   1   23    15.5
    2.   1   21     9.5
    3.   1   22    12.5
    4.   1   14     1.0
    5.   1   23    15.5
    6.   1   19     5.5
    7.   1   15     2.5
    8.   1   15     2.5
    9.   1   16     4.0
   10.   2   27    20.0
   11.   2   27    20.0
   12.   2   22    12.5
   13.   2   21     9.5
   14.   2   27    20.0
   15.   2   28    22.0
   16.   2   19     5.5
   17.   2   20     7.0
   18.   2   31    23.0
   19.   2   24    18.0
   20.   2   21     9.5
   21.   2   23    15.5
   22.   2   23    15.5
   23.   2   21     9.5
 
Data sorted by Z-value to show tie groups
(Sorted on Z-value then Group number then Ordinal as tie-breakers)
Columns are Ordinal, Group number, Value, and Midrank
 
    4.   1   14     1.0
    7.   1   15     2.5
    8.   1   15     2.5
    9.   1   16     4.0
    6.   1   19     5.5
   16.   2   19     5.5
   17.   2   20     7.0
    2.   1   21     9.5
   13.   2   21     9.5
   20.   2   21     9.5
   23.   2   21     9.5
    3.   1   22    12.5
   12.   2   22    12.5
    1.   1   23    15.5
    5.   1   23    15.5
   21.   2   23    15.5
   22.   2   23    15.5
   19.   2   24    18.0
   10.   2   27    20.0
   11.   2   27    20.0
   14.   2   27    20.0
   15.   2   28    22.0
   18.   2   31    23.0
  
WILCOXON RANK SUM SCORES:
Xscore:  Observed:   68.50    Expected:  108.00
Yscore:  Observed:  207.50    Expected:  168.00
Difference in both cases:  -39.50

Wstd_Zval_Pval_NOTIECORR =

   15.8745   -2.4883    0.0128


Tiesum_TieCorr =

  162.0000    0.0133


Wstd_Zval_Pval_TIECORR =

   15.7683   -2.5050    0.0122

SIMULATING THE TRUE Wilcoxon Signed-Rank P-VALUE:
  
Using NSIMS random permutations:

Nsims =

       10000

INITIALIZING the random-number generator at

Wseed =

  2.0709e+005

ASSUMING ObsWscoreX is on the LOWER TAIL:
(That is, ObsWscoreX should be LESS than WExpScX)

ObsWscoreX_ExpWscoreX =

   68.5000  108.0000

 
SIMULATED value for true 2-sided P-value:
NLessEq=57  Nsims=10000   Pvalue=0.01140 (2-sided)
 
95%% Confidence Interval for true P-value (with Estimate in middle):

Plow_PEst_Phigh =

    0.0084    0.0114    0.0144

 
The Hodges-Lehmann estimator for theta for Y=X+theta
  is the median of the Mann-Whitney differences W_k=Y_i-X_j:

Mann-Whitney differences (n=23, nmwd=126, nrows=13):
     4     4    -1    -2     4     5    -4    -3     8     1
    -2     0     0    -2     6     6     1     0     6     7
    -2    -1    10     3     0     2     2     0     5     5
     0    -1     5     6    -3    -2     9     2    -1     1
     1    -1    13    13     8     7    13    14     5     6
    17    10     7     9     9     7     4     4    -1    -2
     4     5    -4    -3     8     1    -2     0     0    -2
     8     8     3     2     8     9     0     1    12     5
     2     4     4     2    12    12     7     6    12    13
     4     5    16     9     6     8     8     6    12    12
     7     6    12    13     4     5    16     9     6     8
     8     6    11    11     6     5    11    12     3     4
    15     8     5     7     7     5

SORTED M-W differences (n=23, nmwd=126, nrows=13):
 (  1)    -4    -4    -3    -3    -3    -2    -2    -2    -2    -2
 ( 11)    -2    -2    -2    -1    -1    -1    -1    -1    -1     0
 ( 21)     0     0     0     0     0     0     0     0     1     1
 ( 31)     1     1     1     1     2     2     2     2     2     2
 ( 41)     3     3     3     4     4     4     4     4     4     4
 ( 51)     4     4     4     4     5     5     5     5     5     5
 ( 61)     5     5     5     5     5     5     6     6     6     6
 ( 71)     6     6     6     6     6     6     6     6     7     7
 ( 81)     7     7     7     7     7     7     8     8     8     8
 ( 91)     8     8     8     8     8     8     8     9     9     9
 (101)     9     9     9    10    10    11    11    11    12    12
 (111)    12    12    12    12    12    12    13    13    13    13
 (121)    13    14    15    16    16    17

YXmeandiff_MWMedian =

    5.1905    5.0000

APPROXIMATE EXACT CONFIDENCE INTERVALS
  based on the Wilcoxon rank-sum statistic:

Estimator: 5

Normal confidence intervals for theta (Mean diff=5.19048):
  ( 2.16,  8.22)  95% CI
  ( 1.20,  9.18)  99% CI
 
NONPARAMETRIC approximate exact CIs for theta
  based on the Wilcoxon rank-sum statistic:
  ( 1,  8)  Offsets r1=31  r2= 96  95% CI
  ( 0, 10)  Offsets r1=22  r2=105  99% CI