APPUCATIQN OF STATISTICAL METHODS 85 



UK) ohstTvations arc reciuiral. Thus, in Prof. Millikan's *• iletonniiia- 

 tiou of till' t'loctron cluirgo e only ")8 observations were made. Tin- 

 values of a, k, and 183 for this distribution arc .128 units, —.11)13 and 

 2.358. Even though the observe*! distribution is consistent with a 

 iiornial s\ stem of causes, values of k and ^2 may be expected to occur 

 which (litTer from and 3 respecti\eh-, as much as these observed 

 \alues do. In this case e\en if k is real and not a result of random 

 siimpling, the correction to be added to the average in order to obtain 

 the most probable value is insignificantly small. 



Next let us consider the problem of determining the number of 

 observations between any two limits. The physicist is ordinarily 

 concerned with the probable error: that is, the error such that '^ of 

 the observations lie within the range X± probable error. Its mag- 

 nitude for the normal distribution is .6745<r, and the errors are dis- 

 tribute:! symmetrically on either side of the average. It is interesting 

 to note that the magnitude of the probable error is also .674.5(r for 

 the second appro.ximation, but that the errors are not distributed 

 symmctrica!!\- on either side of the average. 



Another important pair of limits is that including the majority 

 of the obser\ations. For the normal law 99.73% of the observations 

 are included within the range ,V±3cr which, therefore, is often called 

 the range. Not a single example has been found, however, of a 

 distribution for which the observed number of observations within 

 this range is less than 95'^c even though the distribution is decidedly 

 skew. In fact it is seldom less than 98V(i- If, however, we have a 

 case such as that represented in Table II where groups of observa- 

 tions have been taken in what is technically known as diflferent 

 universes, and then averaged together, the average result is not the 

 most probable, and the standard deviation of the average is not 

 inversely proportional to the square root of the number of observa- 

 tions. Since this point is of considerable importance, it is perhaps 

 well to state it in a slightly different way. Thus, let us assume that 

 we have a thousand samples of granular carbon which possess inherent 

 microphonic efficiencies differing l)\- comparatively large magnitude. 

 Transmitters assembled from any one of the groups of carbon cover a 

 range of efficiencies. If we choose a sample of 10,000 instruments, 

 0.000 from each of two lots of carbon which do not possess the same 

 inherent efficiency, we cannot expect, for reasons already pointed 

 out, that the observed distribution will be normal. The average of 

 these observations will not in general be the most probable value, 

 and the standard deviation of the average will not be equal to the 



"Millikan, R. A.— The Electron— Universily of Chicago Press. 



