ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 127 
Let X and Y be the values of L and M corresponding to class-indices a and ^ ; 
and let ■|(l — )() be the proportion of individuals for which L exceeds X and M 
exceeds Y : thus ^ is necessarily greater than either a or (B. Let the constituent 
parts of corresponding to (1 — x) ^(1 + x) be ^ and respectively, 
so that, if a representative selection of N individuals is made, the value of 
^ (L — Li)^ (M — Ml)'' for the (1 — x ) individuals for which L exceeds X and M 
exceeds Y is N oy g, while for the remaining N (1 -b x) if ^ s- Then it can be 
shown by the methods of §§ 18 and 19 that the following tables give the mean 
products of the errors in the quantities concerned, the divisor n being omitted :— 
Li 
^p, ? 
Ml 
Si.i 
‘8(, 1 1 
®P, 9+1 ,8,^ 1 (J'hp g_i/l3 
/<». 
S|, M 1 
m lSi,m 
^p. 9 +m ^^^p—m 5 /* 7 n+lq —i 
iSp, j + i + lSp_i^ I 
+ mg^(„,_lSp_,_i/l2 — 
S;+r,s ^X;_iS,i.i g vX;i,iSr_l, 5 
sS;_iSr, ,_1 + Zr/\;_iS,._i Ai 
+ Z.sX(_iSr, j_lSi^ 1 X(Sg j 
®p+r, g+« P^p— 1, g^r+1, fi 2 ^p, 9—l^r, s+1 
’"Sp+l.gSr — i,, ,_,_,Sr_g_i 
+ jP?'Sp_i ,Sr_i A2 "t 3 ^Sp. 9—I®r— 
“f" 2 ^sSp — 1 ^ 9^r, s—1^1,1F 9—«—i/n 
-Sp. 9 S.,, 
X 
(°'l, 0 ^i,o) 
— A.o) —A,o) + X^d 
{ ? ^P, 9^ P^P ” 1 , Q (*^ 1 . 0 ^ 1 , 0 ) 
('^O, 1 ^ 0 , 1 ) g} 
; 
V 
(similar expressions) 
X 
a 
/3 
X 
1—' 
1 
(1 - x) (1 + «) 
(1 - x) (1 + /3) 
a 
1 - a' 
I -H a + /i — a/3 — 2x 
I-/F 
Suppose, for instance, that we are considering the error in Si_ i/\/X,,p ..2 = L 
Let the errors in X.,, in S,_ i, and in fxo be 6 , (f), and if/ respectively ; then the error 
