KiRSTiNp] Smith 
17 
(8) It is thus, as we aimed at, shown for functions up to the sixth degree that 
by distributing the observations in {n + 1) equally big groups and choosing the places 
of these groups in one special way we can manage to keep the standard deviation of any 
adjusted y within the possible range of observations less than the standard deviation at 
the places of observation. There is every reason to beUeve that the rule holds for 
any degree of function, but as the general proof would be very complicated and as 
almost all practical cases will be covered by functions up to the sixth degree, the 
problem can therefore be left at this stage. 
As we have proved, any other distribution of observations leads to a curve of 
squared standard deviation that has a higher maximum value ivithin the range. This 
special set of {n + 1) groups has therefore a very conspicuous advantage over all 
other distributions of observations. The application of it is however limited in that 
it demands that the degree of the function must be known beforehand and thus the obser- 
vations do not provide any justification for the form of function chosen. If however the 
function has been fully investigated beforehand and there is no doubt about its form, 
(n + I) equally big groujJs of observations placed as indicated are the most desirable 
set of observations jwssible. The approximate values of the places of the groups 
are given in the table below. 
TABLE I. 
Degree of function 
Places of 
observation 
With rougher approximation the intervals between the observations, still 
expressed by the half range as unit, are as follows : 
1st 
2nd 
3rd 
4th 
5th 
6th 
1-0000 
1-0000 
1-0000 
1-0000 
1-0000 
1-0000 
•0000 
•4472 
-6547 
-7651 
•8302 
-0000 
•2852 
•4689 
-0000 
1 
1st degree of function 2 
2nd 
3rd 
4th 
5th 
6th 
1 
1 
2 
12 2 1 
3 3 3 3 
1111 
2 2 2 4 
11111 
3 2 2 3 6 
The six curves of standard deviation are represented in Diagram 1. It will be 
seen that the minima of a curve, if it has more than two, are the lower the 
greater their distances from the middle of the range, so that the variation of the 
standard deviation is greatest in the outermost intervals of the range. 
III. Uniform continuous distribution of observations -with constant standard 
deviation. General formulae. 
(1) As was pointed out in the last section the lumping up of observations in 
groups just necessary to determine the constants of the function in question has 
some drawbacks and cannot be recommended as a universal rule. In many cases 
it is through the observations themselves that we first get to know the form of the 
Biometrika xii 2 
