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BELL SYSTEM TECHNICAL JOURNAL 



in Table I and Fig. 1. Case No. 2, it is interesting to note, is the 

 familiar "Student" formula; Case No. 3's outstanding characteristic 

 is that all values of h are a priori equally likely. The errors in the 

 mean have here been greatly reduced by merely changing the existence 

 probability distribution of the precision constant. 



(c) Again, the experienced analyst is quite likely to assume willingly 

 that the distribution of the precision constant (and likewise the 

 standard deviation) is of a unimodal form having its maximum value 

 not greatly distant from the figure determined in the sample. Cases 

 Nos. 4 to 7 inclusive typify this kind of assumption while, at the same 

 time, all values of the true mean are held a priori, equally likely. 



Z 5 



0.1 0.2 0.3 0.4 0.5 0.6 OTi 



h = PRECISION CONSTANT 



Fig. 3 — Typical a priori frequency distributions of the precision constant. 



I— c = 3, a = C52 = 13.1922. 

 II— c = Z,a = . . ., = 23.5467- 



III— c = i,a = 



1 - .15^ 



cs- 



= 9.1629. 



1 + As' 

 IV— c = 6, a = cj2 = 26.3845. 



The constants for Cases Nos. 4 and 7 have been so selected as to 

 bring the modal value of li at that found from the sample, that is, 

 that value of h has been made most likely a priori which will make 

 the probability of occurrence of the particular value (1/25-) calculated 

 from the observations, a maximum. Case No. 7 is a considerably 

 more peaked distribution than Case No. 4 indicating more faith in 

 the modal figure selected as being close to the true value. Cases 

 Nos. 5 and 6 illustrate how the mode of the WiQi) function which 

 always lies at A = cjla may be shifted either down or up and the 

 extent of modification in the resulting errors which may be expected. 



