.ifri.ic.iriox or si.nisric.ii. }fr:rffoi>s 55 



witli ilic .isMiinption of tin- imrm.il law. I ikUt tlu'so coiulition^ this 

 law is (MTliaps as ^ixxl .m appnixiination as any. 



Thi" fiiiKiaiiR-nlal assumptions undiTlyiiin iho original cxpl.ination 

 wore later i)n)Uj;ht into (iiii'stion. What a priori reason is there for 

 assuming that the arithmetic mean is the most probahle \'alue? Why 

 not chcH)se some other mean.'' " Thus if wc assume that the median ' 

 value is the most prohalile. we obtain as a special case the law of 

 error represented t>\- the following equation : 



y^Ae-''''^ (2) 



where v represents the frecjuency of occurrence of the (le\iation .v Irom 

 the median value and e is the Xapcrian base of logarithms. Both 

 .1 and h are constants. If. however, we assume that the geometric 

 mean is the most probable, we have as a special case the law of error 

 represented by the follow ing equation : 



y = jle-h'{\ogX-\oga)' (3) 



where in this case ,v is the frequenc\' of occurrence of an observation 

 of magnitude X, "a" is the true value, and A and /; arc constants.' 



Knough has been said to indicate the significance of the assumption 

 that the arithmetic mean is the most probable value, but. why choose 

 this instead of some other mean? No satisfactory answer is available. 

 So far as the author has been able to discover, no distribution represent- 

 ing physical data has even been found which approaches the median 

 law. Several examples have been found in the study of carbon 

 which conform to the law of error deri\cd upon the assumption that 

 the get^metric mean is the most probable. If the arithmetic mean 

 were observed to be the most probable in a majority of cases, we 

 might consider this an a posteriori reason for accepting the normal 

 law. We find the contrary to be the case. 



Furthermore, we find in general that the tlistribution of errors is 

 non-symmetrical about the mean value. In fact, most of the distri- 

 butions which are given in textbooks dealing with the theory of 

 errors and the method of least squares to illustrate the universalit\- 



' .\n average or mean value may lie defined as a quantity derived f fdm a 

 given set of observations tiy a process such that if the observations l]ccame all 

 equal, the average will coincide with the oltservations, and if the oliscrvations 

 arc not all equal, the average is greater than the least and less than the greatest. 



'If a series of n observations arc arranged in ascending order of magnitude. 

 the median value is that corresponding to the observation occurring midway 

 between the two ends of the scries. 



* A ver>- interesting discussion of the various laws that may be obtained by 

 assuming different mean values is given in J. M. Keynes' ".\ Treatise on the 

 Theory of Probability." 



