Illustration of two-sample Kolmogorov-Smirnov test:
 
Data from textbook Table 4.1 p110 (2nd edn)
Water Diffusion across human chorioaminon tissue
  X: At term (approx 39weeks, Control)    Y: 12-26 weeks
 
Sample 1 (Xs, m=10, Xbar=1.313)
   0.80   0.83   1.89   1.04   1.45   1.38   1.91   1.64   0.73   1.46
 
Sample 2 (Ys, n=5, Ybar=0.976)
   1.15   0.88   0.90   0.74   1.21
 
After sorting each sample:
Sample 1
   0.73   0.80   0.83   1.04   1.38   1.45   1.46   1.64   1.89   1.91
Sample 2
   0.74   0.88   0.90   1.15   1.21
 
Samples sorted together (3rd column is Group: X=0, Y=1)
Columns are Ordinal,Value,Group
    1     0.73    0
    2     0.74    1
    3     0.80    0
    4     0.83    0
    5     0.88    1
    6     0.90    1
    7     1.04    0
    8     1.15    1
    9     1.21    1
   10     1.38    0
   11     1.45    0
   12     1.46    0
   13     1.64    0
   14     1.89    0
   15     1.91    0
  
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=-0.20
Multiplied by LCM=10:  xad2=1  for yad2=-2f
 
Table used to calculate J = LCM*J2:  Columns are
  Ordinal,Value,Group,  Term,Diffsum,Maxdif, New?
    1    0.73  0      1     1    1   1
    2    0.74  1     -2    -1    1   0
    3    0.80  0      1     0    1   0
    4    0.83  0      1     1    1   0
    5    0.88  1     -2    -1    1   0
    6    0.90  1     -2    -3    3   1
    7    1.04  0      1    -2    3   0
    8    1.15  1     -2    -4    4   1
    9    1.21  1     -2    -6    6   1
   10    1.38  0      1    -5    6   0
   11    1.45  0      1    -4    6   0
   12    1.46  0      1    -3    6   0
   13    1.64  0      1    -2    6   0
   14    1.89  0      1    -1    6   0
   15    1.91  0      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    1.0954

P-value for large-sample approximation by summing an infinite series:
Infinite sum stopped after 3 terms
(Intrinsically two-sided) approximate P-value = 0.1813
 
Doing nsims permutations to estimate the P-value:

nsims =

       10000

Number greater-than-or-equal and number of simulations:
 
        1660       10000

95% confidence interval for simulated 2-sided P-value
  bracketed by estimated P-value:
 
    0.1587    0.1660    0.1733

 
Compare with result of Wilcoxon Rank-Sum Test:
Data with midranks:  Columns are
  Ordinal,Value,Group, Midrank,Tiegroups
    1.0000    0.7300         0    1.0000         0
    2.0000    0.7400    1.0000    2.0000         0
    3.0000    0.8000         0    3.0000         0
    4.0000    0.8300         0    4.0000         0
    5.0000    0.8800    1.0000    5.0000         0
    6.0000    0.9000    1.0000    6.0000         0
    7.0000    1.0400         0    7.0000         0
    8.0000    1.1500    1.0000    8.0000         0
    9.0000    1.2100    1.0000    9.0000         0
   10.0000    1.3800         0   10.0000         0
   11.0000    1.4500         0   11.0000         0
   12.0000    1.4600         0   12.0000         0
   13.0000    1.6400         0   13.0000         0
   14.0000    1.8900         0   14.0000         0
   15.0000    1.9100         0   15.0000         0


Wx_Wy_Sum_ExpWx_ExpWy =

    90    30   120    80    80


Wilcoxon_RankSum_Zval_NormPval =

    1.2247    0.2207