BINOMINAL SUMMATION 101 



Numerical Results 



Since it is easy to compute the binomial summation directly when 

 either c or n — c is small, the practical value of an approximate 

 formula depends on its efficiency for large values of these quantities. 



The analysis given below under the heading "Derivation of the 

 Approximate Formula" indicates that the successive RaS in the 

 series for 5 decrease when Vc — 1 and -sjn — c increase. Therefore, 

 when these two quantities are large, a few terms of the approximate 

 formula (3) may be expected to give satisfactory results. As a 

 matter of fact, the formula gives good results when Vc — 1 and 

 ■yjn — c are not large. To confirm this statement the Tables ^ given 

 at the end of this paper are submitted. In the 4th column of each 

 table are given 10^ times the true ^^alues of 



p = Z{1) p'i^ - py- 



In the columns headed Ai, A2 and A3 are given 10^ times the differences 

 between the true values and those obtained by applying formula (3) 

 with the first, second and third approximations to 5 respectively. 

 Table I in Czuber's " Wahrscheinlichkeitsrechnung " was used for 

 evaluating the probability integral in equation (3). 



The range of values of P covered by the tables is such that at the 

 lower end of each section P > .0005 while at the upper end P < .9995, 

 except where this latter condition would call for a value of c < 2. 

 Of course, a larger or smaller range might have been given. The 

 decision as to this question was based on the fact that several writers 

 on the theory of statistics, when dealing with the normal law of 

 errors, speak of an error exceeding 3 or 4 times the standard deviation 

 as being a very improbable event. In order to keep the number of 

 pages required for the tables within reasonable bounds computations 

 were made only for even values of c. 



The values of a = up used are such that each of the values p = 1/2, 

 p = 1/10 and p = 1/20 occurs twice; likewise each of the values 

 w = 100, n = 50 and n = 30 occurs twice. 



A greater degree of accuracy than that indicated by the tables can, 

 of course, be obtained by working out and using R4, i?5 • • • ; for this 

 purpose, recourse should be had to equation (12) below and the 

 details immediately following it. The only practical limitation to 

 the use of formula (3) would appear to be the number of places given 



3 I am greatly indebted to Miss Nelliemae Z. Pearson of the Department of 

 Development and Research both for supervising the work of my computers and 

 contributing personally several sections of the tables. 



