148 
MR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
cation of the observed individuals with regard to the true means of the complete 
community be 
Below Lj. 
Above L,. 
(D) 
(C) 
Below M] .... 
V — 0 — (p — yjr 
R + 
(B) 
(A) 
Above Mj .... 
R + 0 
V + e 
Then the erroneous values L'l and M'l are obtained by shifting the medians so that 
this table may present the appearance of the former table. Thus Li is shifted so as 
to transfer 0 \fj individuals, and Mj is shifted so as to transfer d -\- cf). In the first 
case the particular individuals are in the class for which L = Lj (to a first approxi¬ 
mation) ; and the median of M for this class is at M,, so that half of the 0 xjj are 
put from class (C) into class (D), and half from class (A) into class (B). Similarly 
half of the 6 (j) are put from (A) into (C), and half from (B) into (D). Hence 
P'=P — — 
and the error in D is 
^ TT (^ + V-') / (P + P) = (^ + 
This error is distributed with mean square D (tt — D) / n ; and therefore the probable 
error in D as obtained by the second method is Q\/D (tt — a)/v/ n. 
This probable error is of course greater than the probable error due to using the 
method of (1.). since v/D( TT - D) > sin D. 
(3.) Suppose that, instead of taking the medians, we fix on any two class-indices 
a and yS, and divide the total community into four classes (A), (B), (C), and (D) by a 
double classification with regard to the corresponding values X and Y of L and M 
respectively, thus :— 
Below X. 
Above X. 
Total. 
(B) 
(C) 
Below T . . . . 
+ d) + V = v'” 
I (1 + d) 
(B) 
(A) 
Above T . . . . 
i (1 - /3) - V = V' 
V 
I (1 - d) 
Total .... 
I (1 -«) 
1 
