62 
Choice in the Distrihntion of Observations 
Table III contains for a series of values of a the values of v, (1 + av^) and u 
of the two distributions above and the maximum Oy for the three distributions. 
TABLE III. 
a 
V 
1 + av^ 
Maximum of 
a,i — from 
best distri- 
bution 
Maximum of 
cr,, ^ - from 
distribution 
for which 
u 
JN 
Maximum of cr,, — — 
(7 
frnm nistl-l nn^'.ion 
N 
for which </> (.r) = — 
4 
and clusters of 
N 
— at ± 1* 
4 
0 
1-0000 
1-000 
1-414 
2-000 
roooo 
1-581 
1 
G 
1-0000 
1-167 
1-650 
2113 
1 0000 
1-760 
1 
3 
1-0000 
1-333 
1-886 
2-231 
1-0000 
1-944 
1 
■iy 
•8836 
1-390 
2-100 
2-352 
1-0000 
2-131 
2 
3 
•8071 
1-434 
2-284 
2-477 
•9289 
2-316 
?T 
•7510 
1-470 
2-448 
2-603 
-8502 
2-483 
1 
•7071 
1-500 
2-598 
2-733 
•7797 
2-637 
2 
■5559 
1-618 
3-330 
3-540 
-5762 
3-438 
3 
•4782 
1-686 
3-908 
4-382 
-4925 
4-173 
4 
•4278 
1-732 
4-404 
5^241 
-i612 
4-899 
The difference between the maxima from the two first distributions taken as a 
proportion of the maximum of the first decreases from 41 per cent, at a = 0 to the 
minimum 5 per cent, at a = 1, and then again increases to 19 per cent, at a = 4. For 
small a, that is in practice a = 0, and again for a > 3, for which the difference is 
greater than 12 per cent., the third distribution may therefore be useful as giving 
a much smaller maximum value than the purely continuous distribution and at 
the same time offering some justification for the form of the function. 
(4) We shall next, still assuming that /(x) = (1 + ax^Y, consider the choice 
of observations for &, function of the second degree. 
According to (66) and (67) we find 
1 ' . 
i IJ'2H-i - f^t + 2/X1/X2/X3 - /XI>4 - j 
(77), 
and ^ = 1 + 2a/X2 + a^i^i , 
where the (m's are the moment coefficients about x = 0 of the distribution ^ (x) 
which is connected with the actual distribution ifi (x) by the relation 
'P{x)==kcf> {x) .fix). 
From any distribution cf) (x) which has /Xj and iu.3 ? 0 we can form a symmetrical 
I {(f> (x) + </>(— x)} which has the same /xg and 11x4 as 0 (x). We shall prove that 
the maximum a; obtained from the symmetrical distribution is always lower than 
that obtained from the skew. 
