BINOMIAL AND NORMAL DISTRIBUTIONS 



Ch. 4 



tunately, it can be shown mathematically that if n is fairly large and 

 p is not far from 1/2, the numbers obtained from C n> r -p T 0- — p) n ~ r 

 by setting r successively equal to 0, 1, 2, • • • , and n are much the same 

 as those obtained from (1/ V 2-ir-a) -e _(X ~ m) /2<r , wherein X replaces 

 r, np = n, a = vnpq , and e = the base for natural logarithms. In 

 particular, if p = 1/2 so that n = n/2 and a = Vw/2, it is found 

 that the approximation is very close for n = 20 or more. Table 4.21 

 shows the approximation when n = 20. 



TABLE 4.21 



Illustration of the Goodness with Which the Normal Frequency 

 Curve Fits the Binomial Frequency Distribution When p = q = 1/2 



and n = 20 



r or X Binomial Normal Error 



r or X Binomial Normal Error 



If the relative frequencies calculated the two ways shown in Table 

 4.21 are plotted on a common set of axes, Figure 4.22A is obtained. 

 Graphically, the normal frequency distribution fits this binomial dis- 

 tribution almost perfectly at the points where the binomial distribu- 

 tion exists. 



The sum of all the relative frequencies (ordinates) for the binomial 

 frequency distribution is 1 because it is the sum of the probabilities 

 for all of the (n + 1) mutually exclusive events which are possible 

 under the specified conditions. Likewise the sum of all the ordinates 

 of the normal curve at the points where X = 0, 1, 2, . . . , 19, and 20 

 will add to approximately 1. If rectangles of width 1 and heights 



1 



Vi = 



ViOtt 



,-(Xi- io) 2 /io 



where i = to 20, inclusive, are constructed as in Figure 4.225, their 

 total area also is approximately 1. Moreover, the total area of the 



