By inspection, it can be seen that distribution function derived from 

 the chi-squared probability law is reasonably close to the empirical 

 distribution functions although there are systematic differences. In 

 general, the empirical curves tend to lie below the theoretical curve for 

 most argument values. The Kolmogorov confidence interval for the dif- 

 ference between the true distribution function and the empirical one is 

 drawn in on record 6878. The chi-squared curve exceeds the upper boundary 

 in the midranges but the two distribution curves are in fair agreement in 

 the vicinity of the 5th and 95th percentiles. This is probably why the 

 probability intervals agree fairly well. It should be noted that record 

 6878 is one of the more extreme cases and most of the other records 

 attain closer agreement between the two curves. 



A numerical error was made in the earlier report (Borgman, 1972, Figs. 

 7 and 8) relative to the chi-squared related distribution function and 

 probability densities. The theoretical curves in those figures should be 

 ignored. 



XII. THE OUTLIER SPECTRAL LINES 



In 7 of the 12 records, one or more spectral lines loom high above 

 the others. These are ordinarily associated with the largest spectral 

 density values and determine where the peak of the spectral density will 

 occur in most cases. The spectral density decreases appreciably if these 

 extreme or outlier spectral lines are deleted from the averaging process 

 in the density determination. 



A list of all the outliers is given in Table 3. Spectral lines which 

 exceeded 500 are shown and one value which went to 477 is included. The 

 spectral density value at that same frequency is tabled as well as the 

 spectral density which results if the outlier spectral lines for that 

 record were all deleted from the averaging. The spectral density without 

 the lines is usually about 60 percent of what the density is with the lines 

 included in the averaging. 



Two other measures of the "extremeness" of the lines are listed in 

 Table 3. The first is the ratio of the spectral line value to the density 

 computed with the outliers deleted. If chi-squared theory holds, this 



ratio should behave like a Xt / 2 random variable. The 99.5 percentile 



for a Xo / 2 random variable is 5.3. Six of the 14 outliers listed 



exceed 5.3 in value. However, it is difficult to interpret this. One is 



examining the larger members of 300 lines. Hence, extremal statistical 



theory needs to be introduced. Straightforward application of the theory 



of extremes would say that the probability that the largest such ratio 

 in a record would be less than 5.3 is: 



(0.955]"^*^° = 0.22 , (66) 



providing the chi-squared interrelation and the independence assumptions 



78 



