176 
Slew Variation, a Rejoinder 
As a rule the data available in craniological investigations ai'e too sparse to 
give any real test of mesokurtosis, and this is the true reason why we must content 
ourselves with the Gaussian curve. 
Again Mr Powys found out of twelve frequency distributions for the stature of 
men and women that 11 were leptokurtic and the twelfth essentially mesokurtic. 
This tendency to leptokurtic distributions — which can hardly be due to chance — 
is actually given by Ranke and Gi'einer as a case in favour of the Gaussian curve ! 
{Anmevkiuig S. 32G). They further cite Fawcett and Lee in the following 
manner : 
In der an letzter Stelle zitierten Arbeit ist der Nachweis einer bestimuit gerichteten 
Awynimetrie fiir die Mehrzahl der ilasse und zwar in der nach Fechner zu erwartenden Richtung 
besonders beachtenswerth. 
They do not say that of the 24 curves given by Fawcett and Lee 14 are 
leptokurtic and that Fechner's curve can only represent platykurtic distributions. 
They do not draw attention to the fact that the Fechner curve would be impossible 
for the whole of Powys' stature data, and for 12 out of Macdonell's 26 curves for 
the English skull ! In other words, if the Ahweichinigen of Fawcett and Macdonell 
and Powys' data are to be used as an argument at all, 38 out of these 62 
distributions diverge from the normal curve in a manner which cannot possibly 
be represented by Fechner's theory ! : 
If we turn from the condition for mesokurtosis to those for differentiation of 
mode and mean and for skewness we meet other considerations. So far we are 
not dependent for anything we have said on any theory of frequency other than 
the Gaussian. On that theory yS2 = 3 and if the difference — be significant 
the distribution cannot be Gaussian. If we want to distinguish between the mode 
and the mean, we cannot start from the Gaussian theory, because that theory 
supposes the two values absolutely the same. On the other hand if we consider 
asymmetry, we ought to have, within the limits of random sampling, all the odd 
moments zero, i.e. 
Now it is of very little practical value testing the high moments because their 
probable errors are excessive. The probable error of /i, for the normal curve 
= •67449 a/ -i^cy'' and of = "67449 a / -j^a^, or in terms of a as our unit is 
thirteen times as large. These are the gross errors ; the percentage probable 
errors are of course infinite. As a rule it is hardly worth testing these conditions 
beyond /tj. We determine whether the third moment is zero within the limits of 
random sampling. If we wish a relative magnitude we can take /3i = /ia'/M/, a 
quantity which occurs over and over again in frequency discussions. The probable 
error of /3i is obviously zero for the normal curve, becaiise /3i is of the square 
/is = ^5 = = • ■ • = M2il-1 
-0. 
of the order of small quantities. The probable 
