64 BELL SYSTEM TECHNICAL JOURNAL 



Binomial Expansion (p+q)". We ha\e already seen that the 

 distribution is approximately normal when p = q and « = » . Following 

 Edgeworth", Bowley -" shows that if p^q but Wiethe frequency 

 y of the occurrence of a deviation of magnitude .v is given by the 

 following expression where k represents the skewness -' of the 

 distribution : 



-'=^(''"'"£)C'~^(v-iSO]- 



(6) 



This will be referred to as the second approximation. 



If p is very small, but pn=\ is finite, we ha\e the so-called law of 

 small numbers -- which was first derived by Poisson. The successive 



terms of the series e^^i 1 +X + 1 !-■••) represent the chances of 



^ 2 3 / 



0, 1, 2 ... H successes. Theoretically, if we are dealing with a dis- 

 tribution of attributes,'" it is always possible to calculate the values of 



" Camhridgf Thilosoi.hical Transactions, Vol. XX, 1904, pp. 36-65 and 11.M41. 



^ Loc. cit. 



" In statistical work the practice is followed of using the moments ot the 

 distribution for determining the parameters of the fre(iucncy curve. The i th 

 moment *<; of a frequency distribution about the arithmetic mean is by delinition 



2yx' 



In calculating such moments it is necessary to consider the observations as 

 grouped about the mid-point of the class interval and unless this interval is 

 very small certain errors arc introduced wliich can be partially eliminated liy 

 applying Shejipard's corrections as given bv liim in Hiometrika, \\<\. Ill, pages 

 308 seq. If A.v be taken as unity, we ha\e 



pc=«= , 



Hi = pqn=a- S pqn 



^, = pqn{q-p) --ii^-^t? 



lj., = 3(pqny- + t>nq(l-(if'(j) ' pqn 



and if /• is approximately ecpial to cy and « is large we luue crj.. = '^ -and 



'' It is of interest to note that several investigators have derived this law 

 independently. Thus H. Batenian derives this expression in an appendix to 

 the article of Prof. Rutherl'ord and H. Geiger previously referred to. This 

 is, in a way, an illustration of the apparent need of a broader dissemination of 

 information relating to the application of statistical methods of analysis to 

 engineering and physical data. It is also of interest to note that this law has 

 been used to advantage in the discussion of telephone trunking problems. 



"'If the classification is based upon the presence or absence of a single 

 characteristic, this characteristic is often referred to as an attribute. 



