126 MR. W. r. SHEPPARD OR’ THE APPLICATION OF THE THEORY OF 
(ii.) Let yQ be another class-index. The lines of division corresponding to these 
two class-indices divide the community into three classes, whose numbers are pro¬ 
portional to quantities Zj, Zj, Z 3 , where Z^ -}- Z, -f- Z 3 = 1 . From § 15 (hi.) it will 
be seen that the mean product of the errors in a and in B is 
AZyZ^jn = {(1 — ajB) — (a ^ 
(iii.) Let the values of Izx^ for the classes C' and C be respectively and Vp, so 
that Vp Vp = Xp. Then it will be found from § 15 (vi.) that the mean product of the 
errors in a and in Lj is — {v-y — v'i)/7i; and that the mean product of the errors in a 
and in is 
— {{^P — ^p) - (^1 - 
The following table shows the general results obtained in this and the last section ; 
for convenience, the divisor n is omitted throughout. 
A, 
a; i 
c 
Ap+i P^p-Ai 
— (''1 — d) 
\ 
+ + pX_y\, - \ = 
— {("p — d) - (‘'1 — d)pAp_i + x\} 
\ 
\+i d^p+Aq-\ + p’Tp-Aq-Ai 
V J 
- 
(Similar expression) 
OL 
1 - 
(1 - a/ 3 ) - / 3 ) 
§ 20 . Mean Squa^'es and Pi'oducts of Errors in Case of Two Attributes .—Let M 
be the measure of a second attribute, its mean value, and the mean qth power 
of the deviation from the mean ; and suppose that each 2 in § 18 denotes the pro¬ 
portion of individuals for which L and M jointly have certain specified values. Let 
Sp _2 denote the mean value of (L — Lj)^ (M — Mj)’, so that S^_o = ^o,a —Ar 
Then it will be found that the error in {i e., tire error produced by taking as 
equal to the average of 7 /, where x and y are the respective deviations of L and M 
from their averages for the n individuals) is of the form SAe, and therefore is dis¬ 
tributed normally ; its mean square being 
[tz{x^if - - q^p,^-yy)- - {tz{x^7f -p^p^y,^x - ?Sp.g_iy)}']/ 
27 1, 9 —1, 2 ~b P 1, 
+ 2p2S^_i_ 1 4" T^p,q~\Pz — Sp,g)/w. 
n 
