224 Criteriun of Goodness of Fit of Psychophysical Curves 
Let X^, 0C.„ ... !)'n, 
be a system of deviations from the means of n variables wliose standard de- 
viations are 
0-3. ■■■ O-n, 
and intercorrclations 
''12 ! ''13 ) '''23 > 
Then the frequency " surface " giving the frequency of occurrence of each possible 
combination of ,7/s is 
z = z.e 
where 
Herein R is the determinant 
R ajfj ' """"^ V R a-i,<TiJ 
•(11), 
.(12). 
R = 
1 '"12 '''J3 
r.2i 1 
''lis 
.(13), 
and Rkk, R^i, are the minors corresponding to 7'kk find rtj. *S'i is a sum over all A.''s, 
and *S'o is a sum over all pairs kl other than k = l. 
When has been calculated, a probability P can be found, from Table XII in 
Peai\son's Tables for Statistician.^. This table is entered by n' = (n + l) and 
and gives values of 
•(14), 
X" 
that is, P is the probability that a raiidovi sample of as bad a fit as the data, or 
woi'se, would be obtained from the theory which is being tested. The kind of data 
for which this criterion was first invented was data in real histogram form, of the 
kind called in earlier sections of this paper a biometric histogram. When the data 
are of this form, Pearson has shown that equation (12) reduces to the very simple 
form 

where in is the theoretical value of -m, and e is m — m, and 8 indicates summation 
over all the cells of the histogram. Psychophysical data of the kind here con- 
sidered, however, as has already been pointed out, are not really in histogram form. 
Although a histogram can be deduced from them, it is only by making certain 
assumptions, and the intercorrclations of the cells of this artificial histogram are 
different from the intercorrclations of a natural dii-ectl}' observed histogram. 
