174 
STxCW Variation, a Rejoinder 
in excess of the theoretical distribution 100 (^-h^V, but within the limits of a 
random sample. It is quite conceivable that if the returns for each individual 
coin were analysed it would be found that those of one exceeded in proportion of 
heads the limits of random sampling, and that the coin proved to be loaded when 
delicately tested. Thus as I have shown in my memoir on errors of observation, 
we have not only to test for general goodness of fit, but also to consider the 
probable errors of the fundamental constants of the distribution. Because the 
general distribution of frequency is given within the limits of random sampling 
by a normal curve it does not follow that the system will be mesokurtic. 
Consider for example the two curves : 
and 
2(r'{m,-^)] ' ■ v'27ro- \/(mo-|)r(m2 + ^) 
They are both symmetrical, they both for any value of or which is moderately 
large are indistinguishable in appearance from the Gaussian curve. If they 
represented actual observations, we should try to fit them (i) by finding the area, 
(ii) by finding the standard deviation. The former for both curves is N and the 
latter for both curves is a. Hence we should fit them with 
N 
y = 7^ e (0- 
But this in both cases would be incorrect. Both cases would only pass into the 
Gaussian curve when nii and are theoretically infinite, practically large. No 
Gaussian fitting could distinguish one of these curves from the other. Why? — 
Because it does not proceed further than the standard deviation. To measure the 
dift'erence between either of the above distributions and the Gaussian curve we 
must proceed to higher moments. Let iY/t„ be the nth moment about the mean, 
i.e. if X be the mean value of x, 
where the limits of the integral are those of the range. Then if I3., = fijfi-i-, we 
easily find : 
'^1 = ni and mo = ' 
2(^,-3) "-• 3-^, 
2mi-3' 2wt2+3' 
Thus we reach one of the conditions for the Gaussian curve, i.e. ySo = 3, in either 
case when m, and ii\ are considerable, but if /S^, be > 3, wi, will be positive and if 
/3jj< 3, will be positive. Now since ^* \ is always less than | , it is easy 
