84 



BELL SYSTEM TECHNICAL JOURNAL 



lion the most probable value is at a distance — — from the arithmetic 



mean. Many distributions have been found for which k lies between 

 .5 and unity, and, therefore, this difference is from }i to ]/2 of the 

 standard deviation. Thus, the efificiencies of certain standard types 

 of transmitters are found to conform to such a law, and the difference 

 between the modal and a\'erage values is of the order of magnitude of 

 0.4 mile. 



Obviously the geometric mean of the sound intensities (Fig. G) 

 and not their arithmetic mean is the most probable. The difference 

 between the two is quite large. The difference between the arithmetic 

 mean and the modal value for groups of data such as gi\en in Fig. 1, 

 Tables II and VI are quite large. To use again the illustration ot 

 the alpha particles the observed most probable number is 4 ; whereas, 

 the observed average ''* is 3.87. Judging from the best theoretical 

 distribution the most probable number of alpha particles is 3. Choos- 

 ing the number 3 it is seen that either of the other two numbers differ 

 from this by approximately J 2 the standard deviation. Such results 

 are, however, not confined to the work of the present investigation 

 nor to the examples previously cited as is evitlenccd by the data given 

 in the last column of Table VIII. 



T.ABLE VI II 



* Elderton "Frequency Cur\'es and Correlation," published by C. & E. Layton, 

 London, 1906. 



\Vc should not leave this phase of the discussion, however, without 

 pointing out that in a large number of purely physical experiments a 

 sufficient number of observations has not been taken to make it pos- 

 sible to choose the best theoretical distriliution. In general more than 



"Of course, such an average has no significance, except for a continuous 

 distribution. 



