98 BINOMIAL AND NORMAL DISTRIBUTIONS Ch. 4 



It is learned from Figure 4.31 that, for a normal population, 



.07 of the X's correspond to A = —1.48; 

 .27 of the X's correspond to A — —0.65; 

 .73 of the X's correspond to A ^+0.41; and 

 .93 of the X's correspond to A — +1.50. 



In terms of the X's, we have the following facts obtained from the 

 relation: A = (X — /jl)/o-: 



.07 of the X's are ^65-; 



.27 of the X's are ^ 70+ ; 



.73 of the X's are ^78-; and 



.93 of the X's are ^86-; 



therefore, the required numerical limits on the letter grades are as 

 follows: 



A = 86 on; B = 78 to 85; C = 71 to 77; D = 65 to 70; 

 F = below 65. 



The preceding applications of Figure 4.31 have given approximate 

 answers to the questions asked, and these answers are as accurate 

 as the graph used and our ability to read values from it will allow. 

 It seems rather obvious that a more accurate and, if possible, more 

 convenient method is desirable. A method of this sort is available 

 through the use of statistical tables. They perform essentially the 

 same service as Figure 4.31. Although their derivation is not appro- 

 priate to this book, the reader can simply keep in mind the fact that 

 the information obtained from Table III is the same as that which 

 can be derived directly from Figure 4.31, but is in a more accurate 

 and convenient form. 



It will be left as an exercise to rework problems 4.31 to 4.33, inclu- 

 sive, using Table III in place of Figure 4.31 as was done above. 



It is worth while to investigate a set of data from Chapter 2 to 

 see if it seems to be following a normal frequency distribution, at 

 least approximately. Actually it is not feasible at this level of statis- 

 tics to decide this matter rigorously; but some useful information 

 can be obtained nonetheless. 



Consider first the ACE scores of Table 2.01, their frequency dis- 

 tribution in Table 2.42, and the graph of Figure 2.41. Obviously, 

 some approximation is introduced by using such a summary — espe- 

 cially one with only 12 class intervals — but the approximate distri- 

 bution will serve the purpose here. The graphs of Figure 2.41 would 



