92 BINOMIAL AND NORMAL DISTRIBUTIONS Ch. 4 



examination is the same in both instances. In other words, what is 

 of interest is the general form of the frequency distribution of a set 

 of grades and a system for comparing one person's grade with all 

 the other relevant grades. The discussion which follows is intended 

 to show how we can reduce all formulas for particular normal fre- 

 quency distributions to one general — and simpler — formula which 

 preserves all the information which we usually desire from such a 

 formula. 



Multiply through formula 4.21 by o- and then make the following 

 substitutions of variables: let y = o-i/i and let A = {X — /i,)/o-. The 

 result of these substitutions is the following formula for the standard 

 normal frequency distribution: 



(4.23) ^ * -W2 



V2tt 



What has been done by means of these substitutions can be described 

 graphically as follows: (a) Both the vertical and the horizontal axes 

 have been marked off in multiples of the standard deviation, o-; and 

 (6) the peak of the curve (which is above the point where X = jx = 

 nid — MO) has been placed above the point where A — 0. Hence 

 the A^'s which are less than p now correspond to negative values of A, 

 those which are greater than p. now correspond to positive A's. 



The first two columns of Table III give the numbers needed to 

 construct the graph of equation 4.23. Figure 4.24 was constructed 

 by means of this table. Figures 4.23 and 4.24 are essentially the 

 same curve; the only difference lies in the way the vertical and 

 horizontal axes are scaled. In Figure 4.23 A would be under the 

 point where X — 60, would be +1 under the point where X =70 

 because 70 is one times the standard deviation larger than 60, the 

 mean. The A would be —1 under the point where X = 50 because 

 50 is one times the standard deviation smaller than the mean, 60. 

 The other corresponding values of A and X can be determined in 

 the same manner. 



An illustration of the application of standard normal frequency 

 distributions to a generally familiar situation can be obtained from 

 the batting averages of baseball players. The conditions which might 

 affect batting averages may change from season to season or from 

 league to league so that such averages for the different situations are 

 not directly comparable. For example, the ball may be livelier one 

 season than during another; or perhaps the pitching may generally 



