are plotted versus frequency for all 12 data records. The average over 

 the 10 frequencies for each record is plotted on the right of the graph 

 above the frequency value, 4.0 sec." . For the chi-squared distribution 

 the corresponding values are theoretically. 



95th percentile p(fni) / p(fm) = ^^ 



16 



^16,0.95/ 

 5th percentile i(f^) / p(fj = xf^^g.Os/ ^^ " ^ 



1.64 



50 



(50) 

 (51) 



These values are shown as dashlines in the figure. 



On the average the two types of probability intervals agree fairly 

 well. 



XI. COMPARISON OF THE EMPIRICAL DISTRIBUTION FUNCTION OF THE SYMMETRI- 

 CALLY NORMED RESIDUALS WITH THE CHI-SQUARED VERSION 



If the chi-squared distribution is reasonably valid for p(fin)j it 

 may also hold for the spectral lines, at least approximately. The theore- 

 tical relation to be checked for validity is: 



P(fm3 /P(fm) = X^ / 2 , 

 The chi-squared analogy to a^ in equation (38) is: 



theoretical a^ = E 



P(fm) - P(fm)} 

 P(fJ 



By exact analogy: 



theoretical a^ 



{ 



P(fm) > P(fm) 



P'(fm) 



P^(fm) 



Pffm) 



X2 < 2 



> 1 



Since the probability density for Xt random variable is: 



(w) 



^-w/2 

 



for w > 

 for w < 



(52) 



(53) 



(54) 



(55) 



75 



