66 
Choice in the Distribution of Obt<ervations 
.(84). 
Hence = i (1 + 1 
and /^4= 3(1 + «^')J 
For a ^ 3 we have found (••^ = 1 which according to (84) involves jm^ = f^i, so 
that only the distribution above consisting of three groups can realise the requisite 
conditions. 
When a > 3 we have v < 1 and therefore < , so that it must be possible 
to satisfy the equation (84) by a continuous distribution of observations. However 
is decreasing so slowly for increasing a that practically the distribution deter- 
mined by (84) cannot differ much from three groups of observations. 
Our results are accordingly that /or a function of the second degree, of which the 
standard deviation of the observations is a (l + ax^). we get the best function for a'^ 
when a 5 3 6^/ taking three groups of observations at the middle and the ends of the 
range, each group proportional to the sqtiared standard deviation at the 2}lace, and when 
S + 2 V'2 > a > 3 by taking three groups of observations determined by (82) and (83). 
(6) From (78) we find 
o-?' = A7 (1 + 2a/x.^ + a^fJ-i) - , 
J.V ^4 fj,,, 
which, when fi.^ and jj.^ are found in accordance with (82) and (84), determines the 
maximum ay arrived at from our special three groups of observations. Besides the " 
numerical evaluation of this standard deviation, we give in Table IV below the 
maximum of o-y obtained from a distribution for which 0 (x) is constant from — 1 to 
1, that is, since, according to (67), 
the distribution i/j (x) 
k 
1 + ax')'^ N 
1 I 2 I 
1 -h fa + - 
9 
That maximum is determined by 
. 9 being the maximum ol obtained from a rectangular distribution of observations 
with the standard deviation a. 
The last column of the same table gives the maximum arrived at when </) (x) 
is the rectangular distribution with clusters at — 1 and 1 for which a'l = al . For 
this distribution consisting of -22026 iV observations at + 1 and at — 1 and 
•5595 N = 2cN uniformly distributed from — ] to 1, we have found as given in 
