D. E. Lea and C. A. Coulson 



279 



It follows that if equation (37) is used to deduce from an experimentally determined 

 median value r an estimate of m, then this estimate will be subject to a variance 



-^/i*—\ > tne suffix denoting evaluation at the median. Differentiating (36), 

 2N/ \omjo 



dx (x + c) 2 (1 -6 + log m) x + c 



S—L + . 



= (x + c)'' 



d x + c 



(39) 



where d = (l —6 + log m)/a. Hence at the median x = 0, 1^— I =— (cd+l). Thus the 

 variance a 2 m of the estimate of m derived from the median is given by 



\m) N (l+ajc-b + \ogm) 2 N c 2 (cd + l)*' 

 or, inserting the values of a, b, c from (36), 



\mj iV(2-24+loj 



mf 



(40) 



(41) 



make an 



Fig. 35 is a plot of ( — I *JN against m as given by (41). Having used Table 3 to 



estimate of m from the observation of the median value of r, Fig. 3 B is consulted to obtain 

 the standard deviation to be ascribed to the estimate of m. 



m from S [x] = 

 An alternative method of estimating m from experiments in which all or nearly all of 

 the cultures have mutants is the following. Since x is distributed approximately normally 

 about the value x — 0, the mean value of x is zero. An estimate of m from a set of N 

 observations can therefore be made by finding that value of m which makes 



S[x]=S 



rjm — log m + b 



0, 



(42) 



the summation being over the N experimental observations. In using this method a first 

 estimate of m is made by the median method. Inserting this value of m into (36), each 

 experimental value of r is converted into a value of x, and the sum S [x] formed. A series 

 of adjacent values of m are then tried, and the value of m which makes S[x]=0 found 

 (e.g. by plotting S [x] against m). 



The estimate of m obtained in this way is a little more precise than that based on the 

 median. The mean $ [x]/N of a batch of N independent values of x will be distributed 

 (from batch to batch) with variance 1/iV about a mean zero. Suppose that its value for 

 a particular batch is 8, so that 



S[x] = m. 



If m + 8 m is the estimate of m derived from this particular batch (S m being the deviation 

 between the estimated and true values of m), 



h!N=°- 



39 



