16 THE MATHEMATICAL TREATMENT OF DATA 



If the mutants arise spontaneously, we will expect the same average 

 number of mutants, since the same total number of cells is being tested 

 in each case. But in the second experiment the deviations should be 

 greater, since in some tubes a mutant may arise early during growth 

 and give rise to a large percentage of mutant cells; alternatively, in some 

 tubes mutants may chance to arise very late, so that the percentage of 

 mutant cells in these tubes will be very small. Therefore there is a clear 

 difference in prediction by the two models. The induction model predicts 

 the same variance in both experiments; the spontaneous arisal model 

 predicts a variance in the second experiment which is much greater than 

 the variance in the first experiment. 



In several control experiments, in which the number of resistant 

 bacteria was determined in different samples from the same culture flask, 

 it was found that the mean number of mutants equaled the variance with 

 reasonable fidelity. For example, where the mean was 3.3, the variance 

 was 3.8; where the mean was 51.4, the variance was 27. Now, in the 

 second part of the experiment, the number of resistant bacteria was 

 determined in the set of similar samples from different culture tubes. 

 Here the means were very different from the variances. For example, 

 where the mean was 3.8, the variance was 40.8; where the mean was 48.2, 

 the variance was 1171. 



Using tables of the variance ratio (with the proper number of inde- 

 pendent measurements for each variance), it was shown that the experi- 

 mentally found ratio was far outside any chance fluctuation of the ratio 

 away from unity. Thus the mechanism of spontaneous, noninduced mu- 

 tation was first demonstrated. 



3. The chi-squared test 



Chi squared (x 2 ) is defined as 



,2 T,i 0* — a) 2 



X" 



s 2 



'th 



Now, for a largo number, n, of independent measurements, this ratio 

 should be approximately n, since the numerator is n times the experi- 

 mental value of the variance and the denominator is the theoretical value 

 of the variance obtained by using an essentially infinite number of meas- 

 urements. So the tables of chi squared show the number of independent 

 measurements as a parameter. Up to this point, chi squared is just an- 

 other variance ratio test, with the denominator being a theoretical vari- 

 ance (approximated by taking an inordinately large number of measure- 

 ments or estimated by theoretical methods beyond the scope of our 

 treatment). 



