ns 
:^IR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
of the different values of « (a' — a) + {/3' — yS) + c (y' — y) + • • • for different 
random selections of n individuals, o, b, c, . . . being any constants. 
(1.) If we only require the mean and the mean square, we can most conveniently 
use the formulae of § f 2. Suppose an indefinitely great number of random selections 
to be made. Then the proportion of cases in which p come from A and the remaining 
n — p> from the other classes is 
Hence 
\n 
'p'^n — p 
a‘ 
(I - a)" 
-p 
(i.) the mean value of a' is 
p—a rpp p—n u, ^ 1 
S - ^ (1 - = rx S - = ot ; 
p=o |Pp^ V p=i \P ^ V 
so that the mean value of a.' — a is zero ; and 
(ii.) the mean square of a' is 
p=>! „ 
% _Lr 
p = 0 -p 
a^(l - aY-^ • ^ = ir- Y 
/r 
p='' 
•’ V _ L 
p=o 'P\'p~ p 
a”(I O + Pi 
= n~~ {n (/i - 1) a- 4“ 7(a] = a- -j- a (l — j 
SO that the mean square of a' — a is a (1 - a)/n. 
(iii.) Similarly the mean value of a is 
j) = n q=n 
\n 
5=0 'P^qp — P — q 
a'’yS'^(l — a - fd) 
n-p-q . 
p 
n n 
'll — 2 
= 1 5=1 Yp_ — I \ q — 1 |/i — Ji — q 
rH'-' (1 — a — 
= a /3 — a. ^jn 
and therefore the mean product of a' — a and (3' — /S \h — a /Bpn. From these three 
results it follows that 
(iv.) the mean value of rx [a —«) + /> (/3' — /3) + c (y' — y) + ... is zero ; 
(v.) the mean square is 
a- a (1 — a)/n + 1)-{3 (1 — + c“y (1 - y)/n + . . . 
— 2aba.[3jn — ‘lacc/.yin — 2hc^y n — . . . 
= [(«.'« -p -f c-y -f- . . . ) — + -‘/3 + cy + • • • )■}/« ') 
