Choice in the JJistribution of Observations 
TABLE II. 
Degree of 
function 
Ratio of number of 
observations at each 
end of the range to 
the total number 
Maximum of 
<T 
1 ■ 
•2500 
1^581 
1-414 
2 
2 
■2203 
b862 
1732 
3 
3 
■1708 
2^161 
2-000 
4 
4 
- -1411 
2^467 
2-236 
5 
5 
•0978 
2-878 
2-449 
6 
6 
■0850 
3- 149 
2-646 
7 
A comparison between our maximum and Vn + 1 shows the price we have to 
pay for information about the degree of the function. For lower degrees the 
maximum only differs quite insignificantly from Vn -\- I, but with increasing 
degree the difference grows relatively greater for the sixth degree, being about 
one-fifth of Vn +1. — 
The curves of standard deviation for the three sets of distributions are given 
in Diagrams 3 — 8, while Diagram 9 represents the six curves just reached. 
It seems likely from the form of the Oy curves that two clusters of observations 
placed at the outermost of the maxima besides the two clusters at the ends of the 
range would produce a Oy curve with a lower maximum than the one we have 
succeeded in getting for the functions from the fourth to the sixth degree. But 
then again the position of these new clusters would depend on the degree of the 
function and thus make the proceedings more complicated ; and what is more at 
the same time as the maximum of the curve approached + 1 the distribution 
of observations would incur the disadvantages of the grouping in {n + 1) clusters. 
On the whole the distribution arrived at seems to be satisfactory and certainly 
marks a great progress from the uniform distribution. 
VII. Observations ivith varying standard deviation. 
(1) In Section I we have already given the formula for the standard deviation Oy 
of an adjusted ij when the standard deviation s,j of an observation is ct V f {x). 
It is 
1 
X 
X^ 
1 
"h 
... 
.. m„ 
X 
ni; . . . 
... W„+i 
. . . 
... ?H„+2 
a;" 
ni„+2 ••• 
... Hign 
