

MILHDBK338B: Electronic Reliability Design Handbook 
 

8.3.2.6.1 KolmogorovSmirnov (KS) GoodnessofFit Test (also called “d” test)
8.3.2.6.1 KolmogorovSmirnov (KS) GoodnessofFit Test (also
called “d” test)
This test is based upon the fact
that the observed cumulative distribution of a sample is expected to be fairly
close to the true cumulative distribution. The goodnessoffit is measured by
finding the point at which the sample and the population are farthest apart
and comparing this distance with the entry in a table of critical values,
Table 8.313, which will then indicate whether such a large distance is likely
to occur. If the distance is too large, the chance that the observations
actually come from a population with the specified distribution is very small.
This is evidence that the specified distribution is not the correct
one.
1. When to
Use
When failure times from a sample
have been observed and it is desired to determine the underlying distribution
of failure times.
2. Conditions for
Use
(a) 

Usually historical data or engineering judgment suggest
that item failure times of interest are from a given statistical
failure distribution. This test then follows the step of assuming a
given failure distribution and is useful to determine if empirical
data disprove this hypothesis.

(b) 

The KolmogorovSmirnov test for
goodnessoffit is distribution free and can therefore be used
regardless of the failure distribution that the data are assumed to
follow.

(c) 

The discriminating ability of the
statistical test is dependent on sample size; the larger the sample
size, the more reliable the results. When large sample sizes are
available, the c^{2} Test for
Goodnessof Fit is more powerful but requires additional manipulation
of the data. Where sample sizes are small, the KolmogorovSmirnov test
provides limited information but is a better choice than the
c^{2}
alternative.

(d) 

Strictly speaking, this test method
requires prior knowledge of the parameters. If the parameters are
estimated from the sample the exact error risks are
unknown.

(e) 

A KolmogorovSmirnov table is required
(see Table
8.313). 
TABLE 8.313: CRITICAL VALUES d_{a;n} OF THE MAXIMUM ABSOLUTE DIFFERENCE BETWEEN SAMPLE AND
POPULATION RELIABILITY FUNCTIONS
3. Graphic Method
(Example Using Exponential Distribution)
Fortyeight samples of an
equipment’s timetofailure are acquired. Based upon the assumption of an
exponential distribution of timetofailure, the point estimate of MTBF is
calculated to be 1546 hours.
We would like to test the
hypothesis that the sample came from a population where timetofailure
followed an exponential distribution with an MTBF of 1546 hours (see Figure
8.318).
(a) 

Draw the curve (dashed line) for the theoretical
distribution of R(t) which is assumed to be an exponential with an
MTBF = 1546 hours.

(b) 

Find the value, d, using Table 8.313
which corresponds to sample size, n = 48, and level of significance,
a = 0.05: d =
(1.36/ = 0.196).

(c) 

Draw curves at a distance d = 0.196 above
and below the theoretical curve drawn in step (a), providing upper and
lower boundaries as shown in Figure 8.318.

(d) 

On the same graph draw the observed
cumulative function (solid line).

(e) 

If the observed function falls outside
the confidence band drawn in step (c), there would be a five percent
chance that the sample came from an exponential population with a mean
life of 1546 hours.

(f) 

If the observed function remains inside
the band, as it does in the example, this does not prove that the
assumed distribution is exactly right, but only that it might be
correct and that it is not unreasonable to assume that it
is. 
FIGURE 8.318: EXAMPLE OF THE APPLICATION OF THE "d"
TEST
This example could have also been solved analytically by
calculating the difference between the theoretical cumulative distribution
function (CDF) and the actual CDF at each data point, finding the maximum
deviation and comparing it with the value derived from Table 8.313 (d =
0.196). If the maximum deviation is less than 0.196, we accept the hypothesis
(at the .05 significance level) that the time to failure is exponentially
distributed with an MTBF of 1546 hours.

Analytical
Method 
Example
(Weibull Distribution) 
a. 
Observe and record part
failure times 

a. 
Given the following 20 failure times in hours
92 130 233 260 320 325 420 430 465 518


640 700 710 770 830 1010 1020 1280 1330 1690  
b. 
Assume a distribution of failure
times based on historical information or on engineering
judgment 

b. 
Assume failure times are distributed
according to the twoparameter Weibull distribution. 
c. 
Estimate the parameters of the
assumed distribution from the observed data. 

c. 
By the graphic method or the method
of least squares, find the Weibull parameters. The Weibull shape
parameter b equals 1.50 and the Weibull scale parameter
a equals 28400. 
d. 
Calculate the probability of failure
for each observation from the cumulative failure function for the
assumed distribution. 

d. 
For the Weibull distribution the cumulative failure function
is
"1">
where X = observed
failure time, b = 1.5 = Weibull shape parameter,
a =
28400 = Weibull scale parameter, (X) = probability of
failure at or before time X.
For the 20 observations
of this example, the probability of failure at the respective
times is:
X 

(X) 
92 130 233 260 320 325 420 430 465 518 640 700 710 770 830 1010 1020 1280 1330 1690 

.03 .05 .12 .14 .18 .19 .26 .27 .30 .34 .43 .48 .49 .53 .57 .68 .68 .80 .82 .91  
e. 
Calculate the percentile for each of
(i) failure times by the relationship . Subtract
those of Step d. above. Record the absolute value of the
difference. 

e. 
For gives the following results:
(x) 

F(i) 

ï(x)F(i)
ï 
.03 .05 .12 .14 .18 .19 .26 .27 .30 .34 .43 .48 .49 .53 .57 .68 .68 .80 .82 .91


.05 .10 .14 .19 .24 .29 .33 .38 .43 .48 .52 .57
.62 .67 .71 .76 .81 .86 .90 .95 

.02 .05 .02 .05 .06 .10 .07 .11 .13 .14 .09 .09 .13 .14 .14 .08 .13 .06 .08 .04  
f. 
Compare the largest difference from
step e with a value at the desired significance level in the
KolmogorovSmirnov tables to test for goodnessoffit. If the
tabled value is not exceeded then it is not possible to reject the
hypothesis that the failure times are from the assumed
distribution. 

f. 
The largest difference in Step e. was
.14. From the KolmogorovSmirnov table for a significance of .05
and for a sample of size 20 a difference of greater than .294 must
be observed before it can be said that the data could not have
come from a Weibull distribution with b
= 1.5, a =
28400.  




 
 