220 Criterion of Goodness of Fit of Psijcho^jhysiml Curves 
In biometrie language, the problem is to fit a normal curve to data for which the 
" tails " are undefined as to range, although their areas are known. This problem 
was solved by Mviller {op. cit.) as folloAvs : 
(3) TJie Constant Process. 
Call the stimuli Sj, s.,, s-^, ... s,i, 
and the proportions jj , j).,, j>j , ... pn, 
then we have n equations 
p-l-^ e-'^'dx = 0 (1), 
TT ■' 0 
to find the mean and the precision h. We retain for the present this form of the 
integral as being more familiar to psychol(»gists. The more modern form would 
have the standard deviation instead of the precision as the second unknown. 
These equations are slightly inconsistent with one another. No pair of values 
S and h will exactly satisfy all ii equations ; instead of giving zero they leave small 
residuals v^. 
Mviller assumed tacitly that these n equations if based on the same number of 
exjDeriments each, are of equal importance or weight*. We shall allow this 
assumption to pass for the present but shall return to it later. 
If we now make the usual assumptions of the Method of Least Squares, we can 
take as the best values of S and h those which make 
S (i^i") a minimum, 
where the summation is over the n stinnili or n equations. The conditions that 
this should be so are 
^ S (i'l") = 0 for constant 
2 {Vi-) = 0 for constant It 
an 
Unfortunately, the n equations however are very far from being simple and linear 
as in usual apj^lications of Least Squares. To avoid this difficulty we look up in 
tables of the Probability Integral (which psychologists call Fechner's Fundamental 
Table) those n values of 
y = h{s-S) (3) 
which correspond exactly to our n values of p. These equations are not yet linear 
in S and h, but if we write 
c = Sh (4) 
they become j — hs + c = 0 (5), 
* There is unfoitunately a possibility of ambiguity of language here as the word weight also occurs 
in the particular example we are using as illustration, where weights of 84 grams etc, are employed. 
.(2). 
