XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 305 
accumulate, but only the terms of the third order. There is 
only one second-order term that has been neglected, viz. that due 
to the last bracket in the denominator. Even for much lower 
values of n than that chosen for the illustration—e.g. 10 or 12 
(cf. Qu. 4 at the end of this chapter)—the normal curve still 
gives a very fair approximation. 
TABLE showing (1) Ordinates of the Binomial Series 10,000 (3 + 3)% and 
10,000 - 5 
(2) Corresponding Ordinates of the Normal Curve Y=42r ef 
inomi al Binomial Normal 
Term, Pio ys Tera. Series. Curve. 
32 993 997 24 and 40 196 135 
31 and 33 963 967 23 ,, 41 ge 79 
30 ,, 34 878 880 22, 42 a ad 
29 ,, 3b 753 : 753 21 ,, 43 2:, 23 
28 . 36 606 | cosh Cloo 41008 Ny : 
3715 87 459 457 19° ,, 45 
26 ,, 38 326 324 18 ,, 46 
25 ,, 39 217 | 216 1, 4 
13. But if the normal curve were limited in its application to 
distributions which were certainly of binomial type, its use in 
practice (apart from its theoretical applications to many cases of 
the theory of sampling) would be very restricted. As suggested, 
however, by the illustrations given in Chap. VI, a certain, though 
not a large, number of distributions—more particularly among 
those relating to measurements on man and other animals—are 
approximately of normal form, even although such distributions 
have not obviously originated in the same way as a binomial 
distribution. Take, for example, the distribution of statures in 
the United Kingdom (Chap. VI., Table VI.). The mean stature 
is 67-46 inches, the standard-deviation 2-57 inches (the values are 
worked out in the illustrations of Chaps. VII and VIIL), and the 
number of observations 8585. This gives y,=1333, and all the 
data necessary for plotting a normal curve of the same mean and 
standard-deviation (the process of fitting is dealt with at greater 
length in § 14 below). The two distributions are shown together 
in fig. 49, the continuous curve being the normal curve, and the 
small circles showing the observed frequencies. It is evident that 
they agree very closely. Other body measurements, e.g. skull 
measurements, etc., also follow the normal law ; it also applies to 
certain characters in plants (e.g. number of seeds per capsule in 
20)