APl'lJCATION OF ST.IT[.<;TIC.IL METHODS 65 



/>, (/ >iii(l /( from tlio momoius of ilir ilistrilmtion.-^ ICvcn when p, 

 q ;uul « are known, the arilhinetic involved in calculating the terms of 

 the binomial is often prohibitive, and, therefore, it is necessary to 

 obtain certain approximations corresponding to the three laws of 

 error; that is, normal, second approximation, and the law of small 

 nimilK^rs. Tables for the normal law and for the law of small numbers 

 are reailily available in many places, while those for the second ap- 

 proximation are i;i\en by Bowley." 



ICven under conditions where the binomial expansion does not hold, 

 Kilgcworth has shown that it is possible to obtain the following general 

 approximation : 



This holds providing the observations are influenced by a large number 

 of causes, each of which varies according to some law of error but 

 not necessarily to the normal law. 



Gram-Charlier Series. Gram, according to Fisher,''' was the first 

 to show that the normal law is a special case of a more generalized 

 system of skew frequency curves. He showed that the arbitrary 

 frequency function F{X) can be represented by a series of terms in 

 which the normal law is the generating function <^ {X). Thus 



F{X) =co<t>{X)+c,4>'{X)+C2<t>"{X)+ ... (8) 



where Co, C\. Cj, etc., are constants which may be determined from the 

 moments of the observed data. This series is similar to that already 

 mentioned in the above equation (7) which Edgeworth has obtained 

 in several different ways. This law is of interest from the viewpoint 

 of either a physicist or an engineer in so far as it gives him a picture 

 of the casual conditions consistent with an accepted theoretical 

 curve. Thus, if either the causes of variation are within a certain 

 degree not entirely independent, or the errors are not linearly ag- 

 gregated, the observed frequency distributions may be expected to 

 conform to an equation such as 8. This equation has been found to 

 fit a much larger group of observed distributions than the normal law 



" See footnote 26. 



^ Sec for example Pearson, K. — Tables for Biomctricians and Statisticians — ■ 

 Cambridge University Press. 

 " Fisher, Arne — Theory of Probabilities — page 182, 



