500 S. E. LURIA AND M. DELBRUCK 



Substituting here the previously found value of (t — to) and neglecting the 

 second term in the brackets, we obtain: 



(n) var, = Ca 2 N t 2 . 



Comparing this value of the likely variance with the value of the likely 

 average, from equation (8), we see that the ratio of the standard deviation 

 to the average is: 



(12) Vvar~r/r = VC/ln (N t Ca). 



It is seen that this ratio depends on the logarithm of the mutation rate and 

 will consequently be only a little smaller for mutation rates many thousand 

 times greater than those considered in the experiments reported in this paper. 

 In the beginning of this theoretical discussion we pointed out that the 

 hypothesis of acquired immunity leads to the prediction of a distribution of 

 the number of resistant bacteria according to Poisson's law, and therefore to 

 the prediction of a variance equal to the average. On the other hand, if we com- 

 pare the average, equation (8), with the variance, equation (n), (not, as above, 

 with the square root of the variance), we obtain 



(12a) var r = rN t Ca/ln (N t Ca). 



Equation (12a) shows that the likely ratio between variance and average is 

 much greater than unity on the hypothesis of mutation, if (N t Ca), the total 

 number of mutations which occurred in our cultures, is large compared to 

 unity. 5 



It is possible to carry the analysis still further and to evaluate the higher 

 moments of the distribution function of the number of resistant bacteria, or 

 even the distribution function itself. The moments are comparatively easy to 

 obtain, while the calculation of the distribution function involves considerable 



5 In some of the experiments reported in the present paper we did not determine the tota' 

 number of resistant bacteria in each culture, but the number contained in a small sample from 

 each culture. In these cases the variance of the distribution of the number of resistant bacteria 

 will be slightly increased by the sampling error. The proper procedure is here first to find the 

 average number of resistant bacteria per culture by multiplying the average per sample by the 

 ratio 



volume of culture 

 volume of sample* 



second, to evaluate the mutation rate with the help of equation (8); third, to figure the likely 

 variance for the cultures by equation (n); fourth, to divide this variance by the square of the 

 ratio (13) to obtain that part of the variance in the samples which is due to the chance distribution 

 of the mutations. The experimental variance should be greater than this value, on account of 

 the sampling variance. The sampling variance is in all our cases only a small correction to the 

 total variance, and it is sufficient to use its upper limit, that of the Poisson distribution, in our 

 calculations. Consequently, when comparing the experimental with the calculated values, we 

 first subtract from the experimental value the sampling variance, which we take to be equal to 

 the average number of resistant bacteria. 



