90 BINOMIAL AND NORMAL DISTRIBUTIONS Ch. 4 



rectangles is approximately the same as the area under the normal 

 curve, as the reader can verify visually. 



With the preceding remarks in mind, consider the following two 

 facts: (a) To obtain the exact probability that r will have one of 

 the values from a to b, inclusive, we need to sum the ordinates, 

 C n< r (l/2") for n = 20 and r = a, a + 1, a +2, . . . b. (b) The opera- 

 tion described in a is approximately the equivalent of finding the 

 area under the normal curve between the points X = a — 1/2 and 

 X = b -f 1/2- The operation of a is very laborious; hence if b can 

 be accomplished with much less work and is satisfactorily accurate, 

 it should be the better method. As a matter of fact, this is the case, 

 as will be shown by some of the subsequent discussion of this chapter. 



If the relative frequency of occurrence of a normally distributed 

 measurement, X, is denoted by y\, we have the following general 

 formula for y x : 



(4.21) yi = -^-e-' x -^ W . 



27TO- 



Hence, if the /*, and o- are known and the measurement X is known 

 to have a normal distribution, we can graph the frequency distribu- 

 tion by the usual methods of algebra. For example, if /a = 60 and 

 o- = 10, formula 4.21 becomes 



(4.22) y x = —7= e 



V2tt(10) 



-(X - 60) 2 /200 



Table 4.22 was prepared from this formula, and Figure 4.23 then 

 was constructed from the pairs of values (A", y x ) in that table. It 

 will be left as an exercise for the student to verify the values given 

 for i/i by using Table VI (end of book) to obtain e~ w , where w = 

 (X — 60) 2 /200. Thereafter, division by 10 gives the numbers in 

 Table 4.22, under the heading y x . 



The following information can be obtained easily from Figure 

 4.23: (a) The normal distribution curve is symmetrical about a 

 vertical line through the point where A" = /a = 60; (b) the median 

 X, the modal X, and the arithmetic mean of the X's are equal and 

 each is equal to 60; and (c) after (X — 60) becomes at least twice 

 the size of the standard deviation, either positive or negative, the 

 corresponding ordinates, y lf are very small. In fact, when (X — 60) 

 becomes three times the size of the standard deviation, the corre- 

 sponding y x is practically zero. Hence, it follows that for a truly 



