Apeil 25, 1919] 



SCIENCE 



401 



be based may help to establish it until fuller 

 treatment is possible. 



Professor Phelps has thrown light on the 

 problem by distinguishing between the dis- 

 tribution of B. coli in space and its distribu- 

 tion in time. The former alone is discussed 

 by McCrady"^ in treating of fermentation 

 tubes made from a single sample. The latter 

 furnished the data for suggesting the " geo- 

 metrical mean," which was based on a large 

 number of samples taken at different times 

 from single sources, as, for example, given 

 points on a river. Both methods accomplish 

 the same practical purpose by obtaining a 

 weighted mean which eliminates the undue 

 influence of positive high dilutions and the 

 results differ from each other only by a factor 

 which is nearly constant. Whether we wish 

 to base the method a priori on the theory of 

 probability or upon the actual form of the 

 data, becomes an academic problem, but in 

 practise the simpler is naturally to be pre- 

 ferred. 



The arbitrary application of the conven- 

 tional theory of chance to physical data can 

 always be questioned. Bertrand in his " Cal- 

 cul des Probabilites " calls attention to the 

 fact that if a quantity varies as the law of 

 chance, any observed function of that quantity 

 does not, whereas the choice of the quantity is 

 arbitrary. This distinguishes the mathemat- 

 ical theory of probability from the theory of 

 chance variations of observed quantities. The 

 nimiber and magnitude of the forces acting to 

 change a physical quantity may vary accord- 

 ing to the law of chance, whereas the observed 

 change is some function of those forces. 

 Generally those forces combine as a product 

 instead of a sum and so it is believed more 

 fundamental that proportional variations in- 

 stead of absolute variations follow the con- 

 ventional law. In physics the variations are 

 very small compared to the arithmetic mean 

 value of the observed quantity and the effect 

 may be commonly negligible because the pro- 

 portional and absolute variations approach 

 each other. The average is in such cases a 



iJour. Infect. Dm., 1915, 17, p. 183. 



very good index of the measurement. In 

 biology, and especially bacteriology, the varia- 

 tions, as in the number of bacteria, are many 

 times as great as the mean value and the 

 geometrical effect becomes so pronounced as to 

 require a logarithmic average or a geometrical 

 mean. Francis Galton^ discovered the wide 

 practical application of this law and Mc- 

 Allister^ fully discussed it mathematically. 



In the end, therefore, we are thrown back 

 upon the data themselves to determine the 

 most fitting method of reduction and, as the 

 Pearson School of statistics teaches, the sole 

 purpose of such methods is to obtain some 

 representative value of the data. Fortimately, 

 Allen Hazen has given us in probability paper, 

 a simple and sufficiently accurate graphical 

 method of analyzing such rough data. Pro- 

 fessor Whipple'' has summarized and plotted a 

 large mass of bacteriological results and shows 

 that they follow a logarithmic probability 

 curve closely enough. The results obtained in 

 the Investigation of the Potomac River' show 

 also that the logarithmic summation curves 

 are strikingly symmetrical about the median 

 line. In the results obtained at the Washing- 

 ton Filtration Plant" over a five-year period, 

 the distribution of turbidity readings were 

 found to agree with this form of curve, and 

 the bacteriological results are almost parallel. 

 It is further believed that the practical evolu- 

 tion of the geometrical scale of dilutions in- 

 dicates that where variations are great the 

 arithmetical scale is but an approximation 

 over short portions of the more natural and 

 fundamental geometrical scale. 



2 Galton, Francis, "Geometric Mean in. Vital 

 and Social Statistics," Proc. Boy. Soc, 29, p. 

 365, 1879. 



3 McAllister, Donald, ' ' The Law of the Geo- 

 metric Mean," ibid., p. 367. 



* Whipple, Geo. C, ' ' The Elements of Chance 

 in Sanitation," Jour. FranMin Institute, Phila- 

 delphia, CLXXXII., 37, 205, 1916. 



15 Hygienic Laboratory Bulletin No. 104. Table 

 13, pp. 87-94, and Charts E-H bet. pp. 128-129. 



« Wells, Wm. Firth, ' ' Some Notes on the XTse 

 of Alum in Slow Sand FUteration, " Proc. Am. 

 Water Works Assn., 1913. 



