THE NORMAL CURVE 95 



being included which lie between 550 : 449 and 449 : 550. 

 The points thus obtained are so close together that the 

 eye can scarcely distinguish whether they are joined by 

 straight or curved lines. We have, in fact, arrived at 

 a close approximation to the normal curve. 



The curve thus approximately indicated may be 

 seen to be closely similar to the one shown in Fig. 5 ; 

 in fact, the two curves are of such a kind that by 

 altering the vertical and horizontal scales in one of 

 the figures in a suitable ratio their form could be made 

 practically identical. 



The figure arrived at in this way approximates to a 

 mathematical curve which is intelligible to the mathe- 

 matician from the formula y = e~**. The theoretical 

 curve is really arrived at by supposing n in the ex- 

 pression (i +i)* to become indefinitely great. Prac- 

 tically, by making n very large we can get as near an 

 approximation as we may wish to the normal curve of 

 theory. Even in the case of relatively small values of 

 n the approximation to the normal curve is fairly close, 

 as may be seen by comparing together Figs. 6 and 7. 



The example of tossing up coins was only taken 

 as a means of illustrating the more general assumption 

 of an event or a magnitude depending upon a number 

 of causes of equal strength, which in the long-run act 

 with equal frequency in two opposite directions. We 

 can understand that human stature may afford a 

 comparable case, when we consider the large number 

 of bones and cartilages the lengths of which must be 

 added together in order to make up the total stature 

 of any individual, and that the separate length of each 



