: THEORY OF STATISTICS.
Nelumbiwm, Pearl, American Naturalist, Nov. 1906). The question
arises,” therefore, why, in such cases, the distribution should be
approximately normal, a form of distribution which we have only
shown to arise if the variable is the sum of a large number of
elements, each of which can take the values 0 and 1 (or other two
constant values), these values occurring independently, and with
equal frequency.
In the first place, it should be stated that the conditions of the
deduction given in § 9 were made a little unnecessarily restricted,
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Stature tn inches,
Fig. 49.—The Distribution of Stature for Adult Males in the British Isles
(fig. 6, p. 89), fitted with a Normal Curve: to avoid confusing the
figure, the frequency-polygon has not been drawn in, the tops of the
ordinates being shown by small circles.
with a view to securing simplicity of algebra. The deduction
may be generalised, whilst retaining the same type of proof, by
assuming that p and ¢ are unequal (provided p—g¢ be small
compared with Jpg, of. § 3), that p and ¢ are not quite the
same for all the events, that all the events are not quite independent,
 or that » is not large, but that some sort of continuous
variation is possible in the values of the elementary variables,
these being no longer restricted to O and 1, or two other discrete
values. (Cf. the deduction given by Pearson in ref. 13.) Proceeding
 further from this last idea, the deduction may be rendered

306