74 BELL SYSTEM TECHNICAL JOURNAL 



Ciise we find an observed distribution for which ^ = and /32 = 3, it 

 is highly probable that the distribution is approximately normal. It 

 is true, however, that in sampling from a universe in which p = q and 

 ti= x ^ the observed values of k and /So will seldom be exactly equal to 

 and 3 respectively. Then wc must ask what range of values may be 

 expected in these two factors for distributions which are practically 

 normal. For such cases the variations in k and /So are practically 



normal '^ and have standard deviations cr^: = -vl — and "'-3 = \' "v 



where N is the number of observations. Thus, theoretically any 

 series of observations for which the calculated values of k and 02 fall 

 within the ranges OiScrt and 3±3(r^^ niay have arisen from a normal 

 universe. Since, however, the errors vk and ca of sampling are so 

 large, this method does not furnish a very practical test for distribu- 

 tion consisting of only a few observations. This is particularly 

 true since, even for very skew distributions, the values of k and /S^ 

 do not differ much from and 3 respectively (see Table V). If, how- 

 ever, the number of observations is large, the values of k and ^2 in 

 themselves often indicate very definitely that the observed frequencies 

 are not consistent with the normal law. For examjile the calculated 

 values of k and jS^ given for the inspection data in Table II show 

 conclusively that in practically every instance the observed data 

 could not have arisen from a normal universe. So long as we do not 

 use Pearson's system of curves, all that these two factors indicate 

 is that the observed data do or do not conform to the normal law 

 and in this respect their use is limited as is that of the probability 

 paper mentioned above. 



In order to show that the factor ii-> is not in itself a very sensitive 

 measure of the variability from the normal law, I have considered 

 the following special case. Let us assume that the observed dis- 

 tributions can be grouped into two parts depending upon whether 

 or not the observations cluster about the average A'l or A'2 measured 

 from a point which is the arithmetic mean of the entire distribution 

 taken about a common origin. This corresponds to the practical 

 case such as that indicated by Fig. 1 which as alreadv' [lointetl out 

 often occurs in practice. 



" For a critical study of llic coiiilitioiis iiiulcr which the probable errors of 

 these constants have a real siRiiificance, reference should be made to a discus- 

 sion of this problem by Isscrlis in the I'roccedings of the Koyal Society, scries 

 A, \'ol. 92, pp. 23 scq. — 1915. Obviously even for the normal distribution all of 

 the moments will be skew. This follows from a consideration of equation 4. 



