June 19, 1921 wells and wells: counting bacteria 271 



Representing the mode by the geometric mean {Gq), equation (17) 

 becomes 



X = GdGc (19) 



where Gd is the geometric mean of the dilution (D) . Changing varia- 

 bles to the logarithm, this gives 



x=^d + 'c (20) 



as the relation between the arithmetic means of the logarithms, and 



x~\ogX } 



d = \ogGD\ (21) 



c = log Gc 



TABLE 2 

 Constants of Bacterial Frequency 



Number of Bacteria per cc. (s) 10 100 



Arithmetic Mean Bacterial Count (C) 9 .995 100 .001 



Geometric Mean " "(G^) 9 . 54 99 .76 



Standard Deviation {a) 3 .02 9 .55 



Variability (100^) 30 percent 9.6 percent 



The logarithms can be read conveniently from the two-place table. 

 (Tab. 3). 



TABLE 3 

 Two-Place Logarithms 

 1234 5 6789 



To illustrate the computation, an example is worked out. (Ex- 

 ample 3.) 



In any actual experiment the discrepancy between the two means 

 is almost certainly due to experimental errors, and not to theoretical 

 fluctuations of sampling. These errors are largest in the largest dilu- 

 tions. The geometric mean is less affected by large errors in excess 

 than is the arithmetic mean. On the other hand negative plates must 

 be ignored, for a single one would make the geometric mean vanish. 

 The question of what mean to use in this work, however, has little 



