62 



nr.LI. SYSTEM TF.CllXICAL JOLKX.IL 



were fitted to such a distribution, would it be possible to detect easily 

 any great difference i)etween theor\- and obser\ation? 



Let us c()ni|)ari' the two distributions. The data are given in 

 Table III. I-"irsi, the a\'crage \-alue must be the most probable in 

 order to be consistent with the normal law. It is, because the observed 

 most probable \-alue corresponds to 10 successes, and the average of 



Fig. 7 



tlie h\p()ihelic;ilK- obserxed distribution is 9.998. This under ordi- 

 nary circumslaiucs would be considered a close check between theory 

 and practice. 



The normal dislribulion is sii\en in the third column of the table. 

 Even though there is a difference between the frequencies given in 

 the second and third columns, would the average observer be apt to 

 conclude ih.ii the h\ imihelically obser\-ed distribution is other than 

 normal? IK- would probably base his answer upon a graphical 

 comparison sucli as gi\'en in Fig. 7. The solid line represents the 

 normal curve; whereas the frequencies given in the second colunm 

 of Table III arc represented by circles. It is obvious that the normal 

 law appears to be a \'ery close approximation to the terms of the 

 binomial expansion. 



Thus we see that for even a small number of causes the difference 

 between p and q may be ([uite large, and \et the difference between 

 the distributions given by the binomial expansion and that given b\- 

 the normal law is apparently small and not easily to be detected li\ 

 ordinary methods. As ;; increases the closeness of fit does likewise. 



