\\.\l' ", LI" '] li.lruil.irtin,, clxv.ii 



This " most reasonable value" is very different from tin- " in<Ht likely value"; it i.s 

 the value of p most probable on the basis of our past ex|>en<-n<-.- *f the fre- 

 quency distribution of similar p's. 



Statistical workers cannot bo too often reminded that there is no validity in a 

 mathematical theory pun- and simple. The application of l5a\.V Th.-.,r.-iu must be 

 based on the: experience that, where we are a priori in ignoranei -, all values are 

 equally likely to occur. This is not, the e.-use in the present illustration, and we 

 must use our past experience in the same way as we should use our past experi* n<> 

 of equal frequency; this appeal to experience h;us equal validity with the appeal 

 to the experience that where we are ignorant all values are equally likely. We see 

 that our new experience scarcely modifies the old, and this is what we should 

 naturally conjecture would be the case. Thus if we increase the size of our sample, 



the T'- r-v term becomes very small, and we approach nearer and nearer the 

 m(n 1) 



value '59182, obtained by distributing our ignorance equally. But past experi < -n< 

 will bias the value obtained from the new material for a long time, and we see 

 that according to the weight of the past experience p may vary from '40225 to 

 59182. It will thus be evident that in problems like the present the indiscriminate 

 use of the " most likely value " p is to be deprecated. 



(2) If we take the case where we have some past experience, but it is not as in 

 (1) a dominating factor, we must start, in order to be successful in our approxi- 

 mation, with something nearer to p'than to p. Let us write pr = p^ and /5r = p *. 

 then if we have made a guess at p, e.g. p\, po 2 = Pi>' will be our first approximation, 

 and our second will be p 2 + e', where e' is given by 



" 



......... (H), 



and (1 -po 4 ) # n+ i = (2 



Now if we can make a good guess at p so that e' is small, then it will be sufficient 

 to put E n+ i = E n = E', and solve for E', substituting as before in (li) to find /. 



Illustration. The correlation between length and breadth of the skull is at 

 present not very definitely known. Its mean value is about '30, but the values so 

 far determined can range from about to *60. Such a range (see Table XXII) would 

 mark a standard deviation of about 10. What is the most reasonable value, p, when 

 a sample of 25 skulls shows a correlation of '50 *. 



Using equation (xxxvii) on p. clxxiii and Table LP we find 



p = -49217. 

 Now p = '30000. 



13. II. * 



