90 



SCIENCE 



[N. S. Vol. LI. No. 1308 



sion, the variation between the counts being 

 attributed to errors in sampling. He then 

 raises the question as to the most probable 

 number of bacteria present, and, after point- 

 ing out that, according to custom, the arith- 

 metic mean of the counts (18.3) would be 

 regarded as the most probable number, proves 

 this to be untrue by showing the frequency 

 distribution to be highly asymmetrical, as 

 follows : 



Counts Frequency 



6-10 6 



11-15 3 



16-20 4 



21-25 3 



26-30 2 



31-35 



36-40 1 



41-45 1 



Although Dr. Johnstone discusses this dis- 

 tribution, and, by employing Galton's graph- 

 ical method, determines certain constants, he 

 fails to answer the question he raises. 



In cases of this kind it seems as though 

 the simplest procedure is to find some func- 

 tion of the measurements whose frequency 

 distribution is Gaussian, and apply the prob- 

 able error to that function. The reason is 

 that an asymmetrical distribution implies 

 that some influence other than " chance " is 

 operative, and substitution of a function 

 whose distribution is Gaussian enables their 

 separation. In the particular case at hand, 

 and it is typical of many within the province 

 of biology, this function is the logarithm. 

 This is easily demonstrated by grouping the 

 logarithms of the counts with respect to a 

 deviation of =t0.1 from their mean (=1.2046) 

 as follows : 



Logarithm Frequency 



0.505-0.704 



0.705-0.904 2 



0.905-1.104 5 



1.105-1.304 6 



1.305-1.504 5 



1.505-1.704 2 



1.705-1.904 



The arithmetic mean of the logarithms 

 (1.2046) is the logarithm of the geometric 



mean of the counts (=16.02), the geometric 

 mean, by definition, being the twentieth root 

 of the product of the twenty counts. Accord- 

 ingly, the Gaussian distribution of the 

 logarithms shows that the counts cluster in 

 approximately constant ratio about their geo- 

 metric mean, or, to express it otherwise, that 

 variations in the count are compensatory in 

 the geometric mean. This signifies that 

 variation in the count is not primarily 

 attributable to errors in sampling and that 

 each count is not an estimate of the number 

 of bacteria present per c.e. in a homogeneous 

 emulsion, but rather that conditions favor- 

 ing the propagation of bacteria fluctuated in 

 an " accidental " way either during the period 

 in which the twenty samples were removed 

 from the emulsion, or from place to place 

 within the emulsion, or both. "Whether or 

 not this interpretation be correct, the log- 

 arithmic frequency distribution demonstrates 

 that something of like nature occurred. In 

 any case the most probable number of 

 bacteria per c.c. corresponding to the most 

 typical condition of the emulsion is the geo- 

 metric mean of the coimts (16.02) ; and, in 

 the same sense, 250X16.02 = 4,005 is, of 

 course, the most probable nmnber of bacteria 

 in the whole emidsion. 



The reliability of this estimate may be ap- 

 proximated by applying the probable error to 

 the logarithms. The standard deviation of 

 the logarithms, a, is 0.224, the probable error, 

 or, better, the " probable departure " from the 

 logarithm of a single count is 0.6745 o- = 

 ± 0.1511 and the probable departure from the 

 logarithmic mean is 0.1511/V20 = ±0.0337. 

 It follows from tabulated values of the prob- 

 ability integral that, had the entire 250 c.c 

 been examined, it is as likely that the 

 logarithmic mean would have been within 

 1.2046 rt 0.0337 as that it would have been 

 outside these limits, while the odds are about 

 4.6 to 1 that it would have been within 

 1.2046 ±2(0.0337), about 22 to 1 that it 

 would have been within 1.2046 ± 3(0.0337), 

 and nearly 142 to 1 that it would have been 

 within 1.2046 ± 4(0.0337). The numbers cor- 

 responding to these logarithms are the limit- 



