Godfrey H. Thomson 
229 
The difference is therefore three times its probable error and is just significant. 
The final conclusion is therefore that in this particular case the arctan. hypothesis 
is just significantly worse than the normal integral or <f) (7) hypothesis, but that the 
latter itself is very improbable. The P of the normal integral hypothesis it will be 
remembered was "007. The P of the arctan. hypothesis can be found from Table XII 
of Pearson's Tables. The entry has to be made with ?i' = 7 + 1 = 8, and = 55"5, 
and we find given P = "000000, i.e. it is less than '0000005, showing how very im- 
probable the arctan. hypothesis is. 
The probable error of P is discussed by Professor Pearson in the Pliil. Mag. for 
April 1916 and he shows that the standard deviation 
<r,> = K^IP,,-P„_,} (21), 
and using e(]uation (19) wc get 
It must be borne in mind that n = /; + 1, where n — number of variates. In our 
case therefore, n is one more than the number of stimuli. P„'_2 is similarly 
obtained from Table XII of Pearson's Tables by entering with the column with 
heading one less than the number of stimuli. For the above <^ (7) hypothesis 
we have 
%^=19-6, 
Number of stimuli = 7, 
P or Pg = '007 approximately, 
P«--002 
ap = (P3 - P„) X = -005 VliF6 = -022, 
Probable error of P = -67450-^ = -015. 
Therefore for the <p (7) hypothesis the criterion of goodness of fit is in this case 
P = -007 ±'015. 
It is most improbable, therefore, that P is at all large, and the fit is significantly 
a bad one. The probable error of P for the arctan. hypothesis is too minute to be 
found from the table. 
The calculations we have performed have been for Urban 's Subject IV {heavier 
answers). For his other data similar calculations can be carried out. The arctan. 
hypothesis is usually worse than the normal integral, but not always significantly 
worse, and the normal integral itself is an atrociously bad fit to the data in 
most cases. 
. (9) Sumvianj of Rules for Testing and Comparing Goodness of Fit 
of Psijchometric Curves. 
Let there be n stimuli, and let 
Pi, V2, Ih ■ ■ ■ Pn 
be the theoretical frequencies at these stimuli, and 
Pu P-i, Ih ■■■ Ihi 
the observed values. Let fx^, ^i.,, fx-^ ... iji,,^ 
