EKROR TO CASES OF NORMAL DISTRIBUTION AND COREELATION. I3l 
deteraiined when the mean value Lj and the semi-parameter a are determined. 
When the values of L for n individuals obtained by random selection are given, the 
values of Lj and of a can be found in either of two different ways. 
(1.) We can find the average and the standard deviation (square root of average 
square of deviation from the average^) of the n individuals. The average will differ 
from Li by an error whose mean square (§18 (i.)) is a'jn, so that the probable error 
of Li as found in this way is Qaj\/n ; and (§ 18 (ii.)) the square of the standard 
deviation wdll differ from a- by an error whose mean square is (X4 — \l)/n = 2 a^fn (§ 5 ) ; 
so that the probable error in a will be Qa/\/ ‘In. These are familiar results. 
(2.) The other method is that which has been mainly used by Mr. GALTON.t 
Let a and be ai^ two class-indices, and let X and Y be the corresponding values 
of L in the complete community. Then, if x and y are the abscissae corresponding 
to class-indices a and /3 in the standard normal figure {i.e., if ordinates at distances 
X and y from the central ordinate divide the figure into areas whose ratios are 
1 + a : 1 — a and 1 + /8: 1 — /S respectively), we have 
X = Lj + ax 
Y = L, -f- ay 
Whence 
L, {xY - yX)j{x - y)l 
a =: (X - Y)/(a; - ^) J. 
Now let ^ and rj he the errors in the observed values of X and of Y; i.e., let a and /3 
he the class-indices of X -f ^ and Y -f- ly in the collection of n individuals. Then, if 
we deduce the values of Lj and of a from (ii.), the resulting erroi’s are — {y^ — x'r))/ix — y) 
and (^ — r))/{x — y) respectively. Now the eirors ^ and y are due to errors — 2 i^/a 
and — 2 z'r}Ja in the class-indices of X and Y’^, where 2 and z are the ordinates of the 
standard normal figure corresponding to abscissae x and y ; and therefore (§ 19) the 
mean squares and mean product of ^ and y are a" (1 — a^)/ 4 n 2 ", cr(l — ^')linz'\ and 
cr{(l — a^) — (a ^)]/4:nzz. Hence the probable errors in Lj and in a, as found 
from (ii.), are respectively Q.'E/x/n and where 
-■'7/ (1 — «/3) — jS) 
(x - yf 42.' ■ + 
H“ = 
^ 412 + \ 
{ 
I 
J 
. (iii.). 
* It seems couveuient to use tbe term “staiidaicl deviation” in this sense, as denoting a quantity 
which has a definite value for the particular data, 
t GALTcr, ‘ Natural Inheritance,' p. 61.'. 
s 2 
