68 Clioice in the Distribntion of Observations 
that the negative /x^ gives the lower maximum for a'l. We therefore only have to 
find the conditions for [ct;,] being a minimum when //.j < 0. 
We have 
[(1 + a;U,i)^ + (ju,2 — /x;)] 
0-:,= 
and differentiating with regard to , 
'{u2-fcf+(l -i^iH (86), 
x=i N' iiJ-i-p^l? 
As a < f , we have (f — fx-^) (f + ap,-^ > 0 and 
a (;U2 — i^i) — (f — /xj (f + a/Xj) = (a/X2 — f ) + /Xi (1 — a) < 0, 
from which it follows that 
da-'i 
<0 
x=l 
for any /x^ ^ 0. 
The greatest value fi., can take for our range — 1 to + f is 1, the minimum of 
,1=1 
ol must therefore be found for /Xg = f , for which value (86) passes into 
^9 k 4^(1 
N \ ' 1 
which, since /xj 5 0, is a minimum and equals ^ . 2 (1 + a^) when /x^ = 0. 
The </) {x) distribution ought accordingly to consist of two equally big groups 
at the ends of the range and the actual distribution to be chosen for a function of the 
first degree, the standard deviation of which is a linear function of the variable, should 
be two groups at the ends of the ivorhing range with numbers projjortional to the squared 
standard deviations at these flaces. 
(8) For a continuous distribution from — 1 to 1 with frequencies proportional 
to the squared standard deviations we have 
= 0 and ^2 = \i 
and the maximum o-f, ^ ( ^ + 3 ) 
the actual distribution is tfj {x) = ^ "^^^ . ^ . 
Table V contains besides the maxima of cr^ from these two distributions those 
obtained from a distribution for which ^ (x) is constant with two additional clusters 
N 
at — 1 and 1 each consisting of ~ of the observations. 
° 4 
The actual distribution is, since 
ti. Q 1 
= i + i = 
