Te THEORY OF STATISTICS. 
Stirling (1730). If n be large, we have, to a high degree of 
approximation, 
|n= 20m se 
Applying Stirling’s theorem to the factorials in equation (1) we 
have 
Hy 5 
heh ime 5) 
The complete expression for the normal curve is therefore 
FE 
7 Nor. : (6) 
The exponent may be written 22/c2 where c= v2.0, and this is 
the origin of the use of 2 xo (the “modulus ”) as a measure 
of dispersion, of 1/ 2.0 as a measure of “precision,” and of 20? 
as “the fluctuation” (¢f. Chap. VIIL § 13). The use of the factor 
2 or a/2 becomes meaningless if the distribution be not normal. 
Another rule cited in Chap. VIIL, viz. that the mean deviation 
is approximately 4/5 of the standard-deviation, is strictly true 
for the normal curve only. For this distribution the mean 
deviation =o N/2/r=0-79788 . . .. 0: the proof cannot be given 
within the limitations of the present work. The rule that a 
range of 6 times the standard-deviation includes the great 
majority of the observations and that the quartile deviation is 
about 2/3 of the standard-deviation were also suggested by the 
properties of this curve (see below §§ 16, 17). 
12. In the proof of § 9 the assumption was made that % (the 
half of the exponent of the binomial) was very large compared 
with # (any deviation that had to be considered). In point 
of fact, however, the normal curve gives the terms of the 
symmetrical binomial surprisingly closely even for moderate 
values of n. Thus if »=064, k=32, and the standard-deviation 
is 4. Deviations # have therefore to be considered up to +12 
or more, which is over 1/3 of k As will be seen, however, from 
the annexed table, the ordinates of the normal curve agree with 
those of the binomial to the nearest unit (in 10,000 observations) 
up to z= +15. The closeness of approximation is partly due 
to the fact that, in applying the logarithmic series to the 
fraction on the right of equation (3), the terms of the second 
order in expansions of corresponding brackets in numerator and 
denominator cancel each other: these terms, therefore, do not 
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