FOUNDATIONS OF THEORETICAL STATISTICS. 
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In a similar manner the exact form of Sheppard’s correction may be found for other 
curves ; for the normal curve we may say that the periodic terms are exceedingly minute 
so long as a is less than <r, though they increase very rapidly if a is increased beyond 
this point. They are of increasing importance as higher moments are used, not only 
absolutely, but relatively to the increasing probable errors of the higher moments. 
The principle upon which the correction is based is merely to find the error when the 
moments are calculated from an infinite grouped sample ; the corrected moment therefore 
fulfils the criterion of consistency, and so long as the correction is small no greater 
refinement is required. 
Perhaps the most extended use of the criterion of consistency has been developed by 
Pearson in the “ Method of Moments.” In this method, which is without question of 
great practical utility, different forms of frequency curves are fitted by calculating as 
many moments of the sample as there are parameters to be evaluated. The parameters 
chosen are those of an infinite population of the specified type having the same moments 
as those calculated from the sample. 
The system of curves developed by Pearson has four variable parameters, and may 
be fitted by means of the first four moments. For this purpose it is necessary to confine 
attention to curves of which the first four moments are finite ; further, if the accuracy 
of the fourth moment should increase with the size of the sample, that is, if its probable 
error should not be infinitely great, the first eight moments must be finite. This 
restriction requires that the class of distribution in which this condition is not fulfilled 
should be set aside as “ heterotypic,” and that the fourth moment should become 
practically valueless as this class is approached. It should be made clear, however, 
that there is nothing anomalous about these so-called “ heterotypic ” distributions 
except the fact that the method of moments cannot be applied to them. More¬ 
over, for that class of distribution to which the method can be applied, it has not 
been shown, except in the case of the normal curve, that the best values will be 
obtained by the method of moments. The method will, in these cases, certainly be 
serviceable in yielding an approximation, but to discover whether this approximation 
is a good or a bad one, and to improve it, if necessary, a more adequate criterion is 
required. 
A single example will be sufficient to illustrate the practical difficulty alluded to 
above. If a point P lie at known (unit) distance from a straight line AB, and lines be 
drawn at random through P, then the distribution of the points of intersection with 
AB will be distributed so that the frequency in any range dx is 
1 dx 
J l+(x-mY* 
in which x is the distance of the infinitesimal range dx from a fixed point 0 on the line, 
and m is the distance, from this point, of the foot of the perpendicular PM. The distri- 
