74 Choice in the Distribution of Observations 
we find, according to (89), 
< = ^ (1 + 2a/x, + aVJ { 1 + .. ) (90), 
^2 J 
and a'^ = ( 1 + 2a;it2 + a^/x^) (91 ) . 
iV ^2 ~ Ml 
As for any skew distribution of observations we can find a corresponding 
symmetrical distribiition with the same p.^ and fx^, both these expressions are a 
minimum for /Zj = 0. 
We have already shown in (2) of Section VII that any possible values of jx^ and 
/L(,4 can be produced by three symmetrical groups of observations, so that by intro- 
ducing the variables v and y determined by 
and = ?'*y, 
and limited by 1, 
we do not leave out any possibilities. 
From (90) we then get 
(72 
af,„ = ^ (1 + 2ayv^ + a?yv'^), 
which for a > 0 is a minimum when y = = 0, and for a = 0 is for any y 
and v^. 
For a < 0 we find, since 
^==^2ay(l + a?;2) and < _ _ , 
that for a constant y, a;,^^ has the least value when is as great as possible, that 
is for = 1 . 
The minimum of al^ is then 
<, = ^{l + (2 + a)ay}. 
which, since a (2 + a) < 0, is a minimum when y takes its greatest possible value 1. 
The minimum is thus 
< = ^'(l + a)l 
Hence we conclude that : 
0-2 
when a > 0, ct;,^^ is a minimum and equal to for N observations at a; = 0, 
when a = 0, a'i^^ is a minimum and equal to ^ for any distribution for which /j.^ = 0, 
and 
n a < 
of observations at ± 1. 
when a < 0, o-f,,, is a minimum and equal to "^^^ ^'^^''^ equally big groups 
