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200 Skew Variation, a Rejoinder 
existence at all, its limits lie between — oc and + oo , i.e. the whole range of the 
normal curve. But in order to get a range limited at one end, not the whole series 
of values of X corresponding to x are taken. A value is selected for X, such that 
X becomes impossible after a certain value of x. In other words, x is a character 
which although following the normal curve is abruptly terminated as far as X is 
concerned at a value with a finite frequency ! Kapteyn takes* 
x = {X + K)'i-M, 
where k, q and M are to be determined from the data. 
For example, in Professor Weldon's data for the measurement of foreheads of 
Garciruis inoenas, Kapteyn (p. 39) finds that with our notation 
a = -002,204, 
1/- -002,561, 
K=- 0-57S1, 
2-21. 
Thus when A' = — k, we have x = — M = — a roughly, or the Gaussian frequency 
curve for x is to be abruptly cut olf, and about 15 per cent, of its tail discarded. 
If it be said that this could be achieved by natural selection of foreheads, the reply 
is the simple question : Please show what physical character in a crab is given by 
an abruptly truncated normal curve ! The fact is no such character has ever been 
met with, and it must be recognised that x represents a wholly fictitious variable 
having no physiological relation to the character A' at all, but introduced solely to 
reduce the frequency by hook or by crook to that fetish distribution the Gaussian 
curve. 
We can now sum up the objections to Kapteyn's method. 
Theoretical : 
(i) There is no justification whatever for assuming that some character x 
actually exists which obeys the normal law of distribution, and that the observed 
character is a function of this. Some characters are found as a rule in any organism 
which obey the normal law, but no two characters in an organism have ever 
been found to be the one a mathematical function of a second, they are always 
imperfectly correlated. 
(ii) Kapteyn's hypothesis involves if his normal character were a physiological 
entity, that distributions of organic chaiacters should occur which would be 
represented by fragments of Gaussian curves, or such curves abruptly curtailed. 
We have no experience of such distributions in actual vital statistics. If they 
did exist they would contradigt the first two axioms on which the Gaussian law 
itself is based, and would thus deprive that law of the sole justification for its 
application. As it cannot be supposed that all skewly distributed characters X in 
an organism are functions of one and the same x, for in this case they would be 
* This becomes the Galton-McAlister curve for the limit q — 0. 
