272 JOXJRNAL OF THE WASHINGTON ACADEMY OE SCIENCES VOL. 11, NO. 12 



theoretical importance. It is in space and time distributions that large 

 variations and skewness may arise, and then the means dififer signifi- 

 cantly. In many such cases the geometric mean is more probable 

 than the arithmetic mean, as will be shown in a later paper. 



Example 3 

 Explanatory Form 



Dilution 

 D 

 1 cc. 



10 cc. 

 100 cc. 



1000 cc. 



10,000 cc. 



Bacterial 

 count 



C 



Too many to 

 count. 

 (200) 

 130 

 190 

 82 

 53 

 None 



DC Deviation 



d 

 logZ? 



13,000 

 19,000 

 82,000 

 53,000 



29.000 

 23 

 40 

 11 



c 

 log C 



2.11 

 2.28 

 1.91 

 1.72 



Sums 167,000 103,000 



Arith. means 42,000 ± 13,000 



Arith. mean of logs = x = 4.51 



Geometric mean = antilog x = 32,000 ± 14,000 



f Geom. mean^Gx^Gi) Gc 



5 

 2.5 



Number of bacteria 

 per cc. 



I Arith. mean := X z^ DC = 



Actual Routine Form 



DC d d+c 



13,000 2 4.11 



19 .28 



82 3 .91 



53 .72 



4 1 167 



4 1202 



d + c Deviation 



8.02 

 2.01 



4.11 

 4.28 

 4.91 

 4.72 



18.02 



4.51^ 



0.40 

 .23 

 .40 

 .21 



1.24 

 .16 



32,000 ± 14,000 

 (3.2 ±1.4) X 10^ 

 42,000 ± 13,000 

 (4.2 ± 1.3) X 10^ 



Dev. 

 40 

 23 

 40 

 21 



124 



DC = 42,000 

 Gx = 32,000 



a4,000. X = 4.51 ± 0.16 



It should be emphasized that the logarithm itself furnishes a most 

 convenient scale for the expression of results in bacteriology. Thus, 

 instead of bothering to find the antilogarithm of x, this value itself can 

 be used; x = S.67 means quite as much as X = 4.7X10^ = 4700, when 

 equally familiar, and it is much more convenient for very large num- 

 bers, as well as for graphical work. Moreover, the significant changes 



