274 Distribution of numbers of mutants in bacterial populations 



the abscissae are transformed to values of (r/m — log m + 4-5) -1 . Fig. 2 shows that, 

 approximately, 



x= I- ~ — -0-174) /o-086 = (- — ~ 6 — — -2-02) 



\r/m — log m + 4-5 / / \r]m - log m + 4-5 / 



(26) 



is a normal deviate. 



We conclude, in this semi-empirical manner, that when the spontaneous mutation 

 theory is to be compared with experiments falling outside the scope of Table 2 (i.e. experi- 

 ments in which cultures containing more than 64 mutants are frequent), it will be satis- 

 factory for practical purposes to suppose a?= I — ^ -——2-02 J to be normally 



7 



5 - 



4 - 



0-5 



r/m -log m+4-5 



Fig. 2. l/(r/m - log m+4-5) is distributed in an approximately normal distribution. The 

 points are plotted for r = 8, 1(5, 32 and 64, and with m=4, 6, 8, 13 and 15. 



distributed with unit variance about the value 0. r is the number of mutants in an 

 individual culture, m is the mean number of mutations per culture in the batch of parallel 

 cultures. 



The median of the distribution satisfies the relation 

 11-6 



2-02 = 0, i.e. at the median r/m — log m = 1-24. 



(27) 



r/m — log m + 4-5 



This equation provides a means of making a first estimate of m from the count of the 

 number (r) of mutants in the median culture of the batch. 



The quartiles of the distribution (i.e. values of r making P r = 0-25 or 0-75) satisfy the 

 relations : 



at quartiles : r/m - log m = - 0-2 and r/m — log m = 4- 1 , (28) 



34 



