U.S. Department of Transportation
Federal Highway Administration
1200 New Jersey Avenue, SE
Washington, DC 20590
2023664000
Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
This report is an archived publication and may contain dated technical, contact, and link information 

Publication Number: FHWARD03041 
The KolmogorovSmirnov test procedure involves the comparison between the experimental cumulative frequency and an assumed theoretical distribution function. If the discrepancy is large compared to what is normally expected from a given sample size, the theoretical model is rejected.
The KolmogorovSmirnov test procedure involves the following steps:
Figure 94: Equation. Cumulative frequencies definition.
Where x_{k} is a layer thickness value from sample of n layer thickness measurements sorted in the ascending order by thickness value. The k  index indicates the order of layer thickness observation in the sorted layer thickness array.
Figure 95: Equation. Dmax statistic definition.
8. Select level of significance α = 1 percent
9. Compute the critical value _{ } based on selected value of α. Based on value of n, _{ } is found as following
Figure 96: Equation. Critical value _{} definition.
10. The KolmogorovSmirnov test determines whether, for specified level of significance α, the proposed distribution is an acceptable representation of the field data.
Figure 97: Equation. KolmogorovSmirnov test evaluation criteria.
The following figure 98 demonstrates the results of the KolmogorovSmirnov test for a layer that did not pass the test of normality.
Figure 98: Chart. Example of KolmogorovSmirnov normal distribution goodnessoffit test (DGAB layer SPS1 LTPP section 01_0101).
The layer thickness measurements taken along the SPS LTPP sections for the structural layers were tested to determine how well the distribution of layer thickness measurements taken along the LTPP section follow selected theoretical distribution. The following table 69 provides the description of the layer and material types used in the SPS experiments. The table also provides information about layer thickness measurement sample sizes available in the LTPP database.
LayerMaterial Type 
Total number of samples 
Number of samples with the following number of observations 


1 
5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
60 or more 

AC_SURFACE_COURSE 
133 
4 
0 
0 
1 
1 
7 
0 
0 
0 
0 
3 
117 
0 
BINDER_COURSE 
50 
1 
3 
0 
0 
1 
3 
0 
0 
0 
0 
4 
38 
0 
DENSE_GRADE_AGG_BASE 
220 
1 
0 
2 
5 
0 
3 
15 
0 
1 
8 
1 
174 
10 
DENSE_GRD_ASPH_TREAT_BASE 
97 
0 
0 
1 
0 
0 
0 
0 
0 
0 
2 
2 
92 
0 
LEAN_CONCRETE 
48 
0 
0 
0 
0 
0 
0 
8 
0 
0 
0 
0 
35 
5 
PCC_SURFACE 
178 
1 
0 
1 
0 
0 
2 
40 
1 
0 
2 
3 
112 
16 
PERM_ASPH_TREAT_BASE 
130 
1 
0 
2 
0 
0 
1 
9 
0 
0 
1 
1 
111 
4 
AC_SURFACE_AND_BINDER 
191 
0 
0 
2 
0 
0 
0 
0 
1 
0 
4 
4 
177 
3 
One data sample represents a group of measurements taken along the LTPP section for a specific layer and material type. There are 1,047 layers with thickness measurements along the LTPP section available in the LTPP database for the surface and base courses. The number of thickness measurements per layer and material type taken along the LTPP section ranges from 1 to 60. About 85 percent of all layers have at least 55 observations.
A total of 1034 pavement layers were tested to determine how well variability in layer thickness data along the LTPP section could be described using normal distribution. KolmogorovSmirnov goodnessoffit test evaluated for level of significance alpha equal to 1 percent are summarized in table 70.
The results did not show as strong an indication of layer thickness distribution normality as the results of combined skewness and kurtosis test. This could be explained by lower power of KolmogorovSmirnov goodnessoffit test compared to the combined skewness and kurtosis test. Low power indicates high probability of failing to reject the false null hypothesis.
Experiment 
Number of layers 
Not rejected (Normal) 
Rejected (Not normal) 
AC_SURFACE_COURSE 

SPS5 
93 
34 (36.6 %) 
59 (63.4 %) 
SPS6 
36 
12 (33.3 %) 
24 (66.7 %) 
SURFACE_AND_BINDER 

SPS1 
167 
61 (36.5 %) 
106 (63.5 %) 
SPS8 
22 
14 (63.6 %) 
8 (36.4 %) 
PERM_ASPH_TREAT_BASE 

SPS1 
83 
46 (55.4 %) 
37 (44.6 %) 
SPS2 
46 
28 (60.9 %) 
18 (39.1 %) 
PCC_SURFACE 

SPS2 
139 
70 (50.4 %) 
69 (49.6 %) 
SPS7 
24 
21 (87.5 %) 
3 (12.5 %) 
SPS8 
14 
9 (64.3 %) 
5 (35.7 %) 
LEAN_CONCRETE 

SPS2 
48 
26 (54.2 %) 
22 (45.8 %) 
DENSE_GRD_ASPH_TREAT_BASE 

SPS1 
97 
45 (46.4 %) 
52 (53.6 %) 
DENSE_GRADE_AGG_BASE 

SPS1 
97 
63 (64.9 %) 
34 (35.1 %) 
SPS2 
84 
53 (63.1 %) 
31 (36.9 %) 
SPS8 
38 
30 (78.9 %) 
8 (21.1 %) 
BINDER_COURSE 

SPS5 
33 
11 (33.3 %) 
22 (66.7 %) 
SPS6 
13 
7 (53.8 %) 
6 (46.2 %) 