Editorial 
129 
differences depending on the classification chosen. The essential feature of the 
sample is that the frequency f(iund between these dichotomic ordinates is not exactly 
the pi'oportion to be expected on the average, i.e. between Xg and x^^^ it is not 
— Hj but differs from this by - ^n^, where hii^, Sjig+i are the deviations of 
random sampling*. 
Let h,., X,., a be quantities of the sampled population and A,., x,., a refer to the 
sample. Further let (j,.s be the a of the sample as found from the ?'th and sth 
dichotomic interval, Thus if x,. > .r, or r < s, measuring x positive to the right- 
hand, we have 
(/(,. - A,) cr,.s ^= X, - X, = X, -x, = a (/<,. - h,). 
Accordingly to a first approximation : 
So-,-s hhr — hJis 
Thus 
1 [ 2 n,. f , ^ 2 ns f-, w,A 4 n,. 
(45). 
Now it is convenient to put this in a somewhat different form. Let two 
dichotomic ordinates be taken and suppose the range between them to represent 
100 units of the character in question. It is convenient to call this unit in general 
a notace, while mentace, sanitace, munditace repi'esent special units of intelligence, 
health, cleanliness, etc. Let these dichotomic ordinates be and q, p <q. 
Then 100 = a (hp — hq) and accordingly : 
100 1 (2 n, _ «,\ ^ J_ 1« A _ 
' ''■'l-'m- (46). 
In actually calculating this value of the a,^^ as found from the ?-th and sth 
dichotomic ordinates we must of course substitute for hp, h,j, h,-, hg, Hp, H,,, Ti,., h,, 
the values provided by the sample in the usual way. 
Now it will be clear that it is not easy to determine which pair of dichotomic 
ordinates will give the best, i.e. lowest, value of a^^^. For while it is disad- 
vantageous to select s near to r owing to the presence of h,. — in the denominator, 
the increasing correlation of A,, and A,, as s approaches r makes the radical smaller 
and smaller. 
Our previous investigations, however, indicate to us the neighbourhood where 
Cc^s is a minimum and we shall endeavour to choose our pair of grades out of the 
* We suppose tbe subscript s to refer to the «th dichotomic ordinate numbered from the right-liand 
end of the distribution. 
