.■ll'^'l.lc.■lrlo^/ of statisticai. Mr.riions ?7 



It i> til, It wliicll was ()rii;iii.illy sii^^i'sli^d l)\ I.a I'l.uc. It, liow cv ir, 

 wo accept this cxpl.inatioii, wc inu.st accept the fact that the normal 

 law is the exception aiul not the rule. Let us consider why this is 

 true.'- 



I'his method of expl.m.ition rests upon the assimipiiim ili.it the 

 norm.il law is the first api)roximatioii to the freciuencies with which 

 ditTerent values will be assumed liy a \ariahle qiiantit)- whose varia- 

 tions are controlled In- a larj;e inimher of indepfndeiil causes .iciinj; 

 in random fashion. Let us assume that : 



a. The resultant variation is produced b\' n causes. 

 1). The probability p that a single cause will produce an effect A x 

 is the same for all of the causes. 



c. The effect A .v is the same for all of the causes. 



d. The causes operate independently one of the other. 



I'nder these assumptions the frequency distribution of deviations of 

 0. 1, 2 ... n positive increments can be represented by the successive 

 terms of the point binomial \(q + p)" where A' represents the total 

 number of observations. 



L'nder these conditions if p = q and n = «>, the ordinates of the 

 binominal expansion can be closely approximated by a normal curve 

 ha\ing the same standard de\iation. These restrictions are indeed 

 narrow. In practice it is probable thai p is never ecjual to q, and it 

 is certain that n is never infinite. Therefore, the normal (iistril)Uti()ii 

 should be the exception and not the rule. 



There is a more fundamental reason, however, why wc should 

 seldom expect to find an observed distribution which is consistent 

 with the normal law. In what has preceded we have assumed that 

 each cause produced the same etTect A .v, and that the total effect in 

 any instance is proportional to the number of successes. 



Let us assume that the resultant effect is, in general, a function ot 

 the number n of causes producing positive effects, that is, let X =<j>{u). 

 Thus we assume that the frequency distributions of the number of 

 causes and of the occurrence of a magnitude X arc respectively 



y=f{n) 



and 



y.=/i(A-) 



for two values of «, say n and n-\-dn, there will be two \alues of A', 

 say A" and X+dX. The number of observations within this inter\al 

 of w must be the same as that within the corresponding interval of X. 



" Bowley "Elements of Statistics," Part II. 



