Downloaded from https://www.everyspec.com AMCP 706-198 A heuristic (and rigorously erroneous, but A table of X 2,P values is given in Table ver* useful) description of the source of the 2-9; other tables are given in Ref. 30. statistic is helpful (see Ref. 24, Chapter 30 for detailed derivations). Example No. 10 illustrates the applica- tlon of the Weibull distribution. The number in each cell has a binomial distribution, wlch depends on 2 param- 2-4.2 THE KOLMOGOROV-SMIRNOV eters: the total number in the sample, and (KS) TEST 7' the probability of a value falling i, that cell.
As the s-expected number in the cell The Kolmogorov-Smirnov test is another X becomes very large, the binomial distribu- analytic procedure for testing goodness-of- tion turns into a s.normal distribution with fit. The procedure compares the observed the same moan n, and a standard deviation distribution with a completely-specified equal to the square root of the mean Vn 4 . hypothesized-distribution and finds the The number ir each cell is converted to a maximum deviation between the Cdf's for standard s-normal variate by subtracting the the two. ThI's deviation is then compared mean and dividing by the standard devia- with a critical value that depends on a pre- tion.
The sum of the squares of such van- selected level of s-confidence (Refs. 7, 19, ates has a chi-square distribution with k - 1 24, and 25); see Table 2-12. degrees of freedom; each term is of the form The Kolmogorov-Smirnov test is distribu- tion-free; it can be used regardless of the failure distribution that the data are assum- """(245) ed to follow, provided the random variable is continuous. The discriminating ability of the e:st depends on the sample size; larger The I degree of freedom is lost because the sizes discriminate better.
If the random last variate Is not s-independent, Le., it can variable is discrete,.the s-confidence levei be calculated from the previous data be- will be greater than that shcwn in Table cause the total number in the sample is 2-12 (Ref. 24). known. The test is good regardless of sample size. Conventionally, only a 1-tail test is used, Most discussions of the discriminating ability i.e,, the calculated value of X 2 ought not of the test arm from a statistical rather than be too large. But one ought to be equally an engineering viewpoint. suspicious of too-smail values; if the usual variability is not there, someone may have The steps in a Kolmogorov-Smirnov test tar.,pered with the data.
ConvL ,ional wis- are as follows. dorm suggests that there ousht to be at least 5 data points in each cell and at lcast I. Completely specify the hypothetical 30 total data roints. But the usual eng&- distribution to be tested, Fky, (x). If it neer will do the best he can with the data has several parameters, a value for each of he has. those parameteor must be specified. If any of the paramueters were estimated from the If the cahmulrted value of X1 is greater data, step 4 must be modified as specified than the tabulated value for x-, 2 (St a In step 4. particular i-fIgnificanie level) reject the assumed distribution. 2. At ech sample point x.: [text continhues on i.age 2-691