J. F. Tocher 
315 
applying Sheppard's correction to the moment the square of each standard 
deviation involved in the calculation) both absolute, and relative to 
Although the absolute differences appear small they are relatively large in three 
cases, LB J', BH J', and LS ^. In all the other cases the relative differences 
are small and are not significant — the regression is tndy linear. It would thus 
appear that, in the case of the males, there is a probable significant departure 
from linearity in the regression curves of these three pairs of characters among 
the "general insane" population. On plotting linear regression curves, it is 
seen that this departure from linearity is mainly but not altogether caused by 
numerically small groups at the ends of the regression lines. (See Diagram IX.) 
The non-linearity however is not so very great and it seems scarcely worth while 
undertaking the statistical labour of fitting skew regression curves to the results. 
For comparative purposes, the correlation coefficients in the case of Z & -B was 
determined by three different methods with the following result: 
Ordinary method. 6 method ■■ . Contiugency method t- 
r -5026 -5010 -5019. 
(3) Means and their Differences. Problem (c). 
In this section the following problem is considered : Do different parts of 
Scotland differ sensibly fi'om each other, assuming the "general insane " population 
to be an anthropometric sample of each local population ? This problem can be 
answered by discussing the individual asylum means and the extent of the difference 
of each from the i-emainder of the "general insane" population. Asylum means, 
no matter what character be selected, show diffei'ences as we pass from asylum 
to asylum. Tables XV. to XXI. of Supplement^ give the values of the means of 
all the characters measured for each asylum, the general mean being the last line 
on every table. The probable errors are given in every case, but in order to note 
whether or not individual asylum means differ significantly from the means of the 
remainder of the "general insane" population, the differences have to be studied 
with respect to the standard deviation of sampling of these differences. If ?>i = the 
mean of any character at any one of the asylums and M' = the mean of the same 
character for the remainder of the asylum population, m — M' is the difference 
between the local mean and that of the rest of the same population. Then, if 
<T = standard deviation of any character at any one of the asylums and 1,' = standard 
deviation of the same character for the remainder of the population, 
n 
* Using fourfold table. See Pearson, Phil. Trans. Vol. 195 A, pp. 1 — 47, 79 — 150. 
+ See Drapers' Company Research Memoirs, Biometric Series i, on "The Theory of Contingency, <Sc. 
K. Pearson. 
J Biometrika, Vol. v. Supplement, pp. 92 — 96. 
