278 Distribution of numbers of mutants in bacterial populations 

 is normally distributed about median 0, it follows that 



^logm=- -6 = 1-24. (37) 



This equation enables an estimate of m to be made from an experimentally determined 

 value of r . With its aid Table 3 has been constructed, which enables m to be obtained for 

 any value of r up to 4400. While the derivation of m from the median is not the most 

 efficient way of utilizing the experimental data from a statistical standpoint, it is the 

 quickest satisfactory method, and is useful for making a preliminary estimate even if 

 a more elaborate method is to be employed in making the final estimate. 



Table 3. Preliminary estimation of mfrom median value of r 



Thus if the middle culture of the series has 50 mutants, interpolation in the table between r =49-2 and 

 r =55-8 gives /•„/>« =3-81, so that m = 50/3-81 =13-1. This is the mean number of mutations per culture. 



The precision of an estimate of m made in this way from counts of N cultures may be 

 determined by calculating a m jm. We shall make use of the approximate result that x is 

 distributed in a normal distribution with unit variance. The probability of .z lying between 



1 2 1 C x 



x and x + dx is -..- - <H r2 dx. The probability of its lying between and x is , e~ ix2 dx, 



vw' T J yl^Jo 



and for observations in the neighbourhood of the median (^ 2 <^1) we may write this as 



j~= -. Thus |-+ llct . 1 is the probability of getting an observation ^x, and I-— t^—A 



V(2tt) \2 V(27r)/ j h \2 v '(2tt>; 



is the probability of getting an observation ^ x. Thus the probability that, of N = 2s + 1 

 observations, s shall be ^.r, s shall be ^x, and one shall be between x and x + dx is 

 (providing x is in the neighbourhood of the median) 



(2x4-1)1/1 



(2s + l)! 



(U x —\ 8 ( l - r V s ,h ' = (2 * +1)! /i %*\' < lr 



\2 s l(2n)f [■> j(2n)J 4(2n) (2°s\f\ ir ) ~Jfrr) 



for s^> 1. Thus the median value of a set of A' = 2,s + 1 values of x is distributed about x = 



.,, 77 77 



with variance — = — . 



4s ' 2A 



3» 



