J. F. Tocher 
145 
satisfactorily represented by the normal curve. The tables of the probability 
integral are therefore not applicable and do not give the probabilities. They can 
be found however when the type and the constants of the curve which fits the 
hypergeometrical distribution have been determined. Tables* for these extreme 
cases are in the course of production, but they involve laborious calculation and it 
may be some time before they are ready. Accordingly special stress must not be 
laid on the differences found where the value of p is such as to give a significantly 
asymmetrical distribution of samples from which the probabilities of greater or 
lesser values in future samples are found. 
The form in which each difference has been expressed and studied requires 
notice. It is obvious that, in considering differences and their standard deviations, 
one may take the observed absolute numbers and expected absolute values — that 
is, in the notation herein used, y" and ?//. Again one could take the observed 
and theoretical percentages — that is the difference 100 {(i/s"/n) —p} ', or reckoning 
T/g' in each case as 100, one could take the difference as 100 {(y/'/ys ) ~ 
Now it is easy to see that '^npq{N — n)j{N —1), reckoned as a percentage, is 
100 *Jpq (N — n)/7i{N —1), the standard deviation with which 100 {(ys'/n)—])} has 
to be compared. Expressed as a coefficient of variation, it is also easily seen to be 
100 '^q {N — n)/7ip {N — 1), the variability constant (decreasing as n increases) with 
which 100 [{ys'lVs) - 11 has to be compared. Thus there are for selection, according 
to convenience, in the statistical analysis, the three ratios 
(1) iy^' - ys')Hnpq {N - n)/(N - 1). 
(2) 100 [{ys"/>i) - 2)}HlOO'pq {N - n)/n {N - 1). 
(3) 100 {(ys"/ys') - l)/100Vr7(7V - n)lnp {N- 1). 
It is perfectly obvious that the above ratios, applied to the data, will give 
identical results. These ratios will, throughout this memoir, be called relative 
local differences (RLD), this term being the one introduced by the writer in a 
previous investigation to denote the local differences in the physical characters of 
the Scottish insanef. In determining relative local differences, the first expression, 
which deals with the absolute figures, has been the one used, the calculations 
having been performed in duplicate. Since the percentages in district groups have 
been calculated, it was found convenient to use the second form in cases where 
it was necessary to compare certain of these districts with the general population. 
The following table (Table V.) constructed to illustrate, by means of maps, the 
relative local differences in the physical characters of the Scottish insane f will be 
used throughout the memoir both in the text and in the maps, and defines the 
terms used to indicate the significance or non-significance of the observed results. 
From what has already been said, these relative local differences when n is fairly 
* Biometrika, Vol. v. p. 175. 
t Tocher : Biometrika, Vol. v. Part ni. pp. 317—318 ; also Table VIII. of that memoir. 
Biometrika vi 19 
