218 Criterion of Goodness of Fit of Psycliophysical Curves 
Any suitable curve which happened to occur to one might of course be employed. 
For example, a parabola of higher order can be used, and the curve tan""' 6 has also 
been used. But clearly the whole experiment suggests that an error function of, 
some sort is wanted, and as early as 1860 G. T. Fechner {op. cit.) suggested that 
such numbers formed the integral of a normal or Gaussian curve. One usual 
argument is somewhat as follows, using for clearness terms applying directly to the 
above example. 
The existence of a hypothetical point is postulated, called the limen or threshold 
for the judgment heavier, such that above this point the subject always returns the 
answer heavier, and below it he always returns some other answer, not heavier. 
But this limen is supposed to be fluctuating from moment to moment, either really 
or apparently, owing to changes in the physical, physitjlogical and psychological 
conditions of the experiment. If at one moment the answer heavier is returned, 
for the variable 96 grams, then at that moment the limen is below 96 grams. 
Later the answer lighter, or the answer equal may be returned for 96 grams, and at 
that moment the limen is above 96 grams. The values p in the above table will 
then be integrals of the distribution curve of this limen. 
(2) Pecaliarities of Psychophysical Data from the Point of View of 
Curve Fitting. 
The problem of fitting a distribution curve integral to such data, say in the 
first place the probability integral, has certain peculiarities which differentiate 
it from many biometric curve-fitting problems. 
Usually, when we are required to fit a normal curve, we are given the data in 
histogram form : 
mj 
m. 
Ms 
rti: 
That is, a number M of direct measurements is made, and are found to fall into 
a certain short range, in.2 into another adjacent range, and so on. To fit a Gauss 
curve requires the mean and the standard deviation, and these quantities can 
be directly found from such a histogram, Sheppard's adjustments being used if 
necessary. 
Quantities analogous to our proportions p can be formed from such a biometric 
histogram, viz. : 
Pi = m^/M, 
P2 = + m^jM, 
Pz = (^1 + mo + ms)/M, 
