Illustration of two-sample Kolmogorov-Smirnov test:
 
Data from Table 5.7 p180 in text
(Salivation Rates With/Without Feedback.)
 
Sample 1 (Xs, m=10, Xbar=4.215)
  -0.15   8.60   5.00   3.71   4.29   7.74   2.48   3.25  -1.15   8.38
 
Sample 2 (Ys, n=10, Ybar=1.892)
   2.55  12.07   0.46   0.35   2.69  -0.94   1.73   0.73  -0.35  -0.37
 
After sorting each sample:
Sample 1
  -1.15  -0.15   2.48   3.25   3.71   4.29   5.00   7.74   8.38   8.60
Sample 2
  -0.94  -0.37  -0.35   0.35   0.46   0.73   1.73   2.55   2.69  12.07
 
Samples sorted together (3rd column is Group: X=0, Y=1)
Columns are Ordinal,Value,Group
    1    -1.15    0
    2    -0.94    1
    3    -0.37    1
    4    -0.35    1
    5    -0.15    0
    6     0.35    1
    7     0.46    1
    8     0.73    1
    9     1.73    1
   10     2.48    0
   11     2.55    1
   12     2.69    1
   13     3.25    0
   14     3.71    0
   15     4.29    0
   16     5.00    0
   17     7.74    0
   18     8.38    0
   19     8.60    0
   20    12.07    1
  
Calculation of maximum absolute difference
   of empirical distribution functions:
     J2 = max_t |F_{m,X}(t) - F_{n,Y}(t)|
  
Increment for X: xadd=0.10  for Y: yadd=-0.10
Multiplied by LCM=10:  xad2=1  for yad2=-1f
 
Table used to calculate J = LCM*J2:  Columns are
  Ordinal,Value,Group,  Term,Diffsum,Maxdif, New?
    1   -1.15  0      1     1    1   1
    2   -0.94  1     -1     0    1   0
    3   -0.37  1     -1    -1    1   0
    4   -0.35  1     -1    -2    2   1
    5   -0.15  0      1    -1    2   0
    6    0.35  1     -1    -2    2   0
    7    0.46  1     -1    -3    3   1
    8    0.73  1     -1    -4    4   1
    9    1.73  1     -1    -5    5   1
   10    2.48  0      1    -4    5   0
   11    2.55  1     -1    -5    5   0
   12    2.69  1     -1    -6    6   1
   13    3.25  0      1    -5    6   0
   14    3.71  0      1    -4    6   0
   15    4.29  0      1    -3    6   0
   16    5.00  0      1    -2    6   0
   17    7.74  0      1    -1    6   0
   18    8.38  0      1     0    6   0
   19    8.60  0      1     1    6   0
   20   12.07  1     -1     0    6   0
  
Maxdif=JJ for Table A10, J2=J/LCM = Maxdif of empirical distributions,
  Jstar for large-sample approximation:

JJ_J2_Jstar =

    6.0000    0.6000    1.3416

P-value for large-sample approximation by summing an infinite series:
Infinite sum stopped after 3 terms
(Intrinsically two-sided) approximate P-value = 0.0546
 
Doing nsims permutations to estimate the P-value:

nsims =

      100000

Number greater-than-or-equal and number of simulations:
 
        5265      100000

95% confidence interval for simulated 2-sided P-value
  bracketed by estimated P-value:
 
    0.0513    0.0527    0.0540

 
Compare with result of Wilcoxon Rank-Sum Test:
Data with midranks:  Columns are
  Ordinal,Value,Group, Midrank,Tiegroups
    1.0000   -1.1500         0    1.0000         0
    2.0000   -0.9400    1.0000    2.0000         0
    3.0000   -0.3700    1.0000    3.0000         0
    4.0000   -0.3500    1.0000    4.0000         0
    5.0000   -0.1500         0    5.0000         0
    6.0000    0.3500    1.0000    6.0000         0
    7.0000    0.4600    1.0000    7.0000         0
    8.0000    0.7300    1.0000    8.0000         0
    9.0000    1.7300    1.0000    9.0000         0
   10.0000    2.4800         0   10.0000         0
   11.0000    2.5500    1.0000   11.0000         0
   12.0000    2.6900    1.0000   12.0000         0
   13.0000    3.2500         0   13.0000         0
   14.0000    3.7100         0   14.0000         0
   15.0000    4.2900         0   15.0000         0
   16.0000    5.0000         0   16.0000         0
   17.0000    7.7400         0   17.0000         0
   18.0000    8.3800         0   18.0000         0
   19.0000    8.6000         0   19.0000         0
   20.0000   12.0700    1.0000   20.0000         0


Wx_Wy_Sum_ExpWx_ExpWy =

   128    82   210   105   105


Wilcoxon_RankSum_Zval_NormPval =

    1.7386    0.0821