228 Crifei'ion of Goodness of Fit of Psychophysical Curves 
We have 
(8) The Probable Error of and P. 
P9 
(from eqn. 18). 
If the accurate values of p' were known, the variation of x^ would be due entirely 
to the variations in the observed values p. In point of fact, of course, the p"s 
which are available are themselves functions of the p's : but like Pearson in his 
1914 article on the probable error of a coefficient of contingency*, and for the same 
reason and with I think the same justification, we shall assume that the p"s do not 
vary. Then the mean square deviation 
.,^>.s\.,(^i^.>^^..s^.,^ C^V 
Therefore the probable error of x" calculated in the way suitable for the Constant 
Process and other processes for fitting psychometric functions is 
•6745(7^., = 1-349 (20). 
In the case above where ;^-=:19%56, its probable error is therefore about 5 9, so 
that we have 
;^2= 19-6 + 5-9. 
We must next find x" ^^^^^ its probable error for the arctan. hypothesis. The 
calculations are partly carried out above in finding »S'(.'r^). Completing them we 
obtain the following table : 
p' 
ci' 
q'p' 
X- 
x'^jp'q' 
■0795 
•9205 
•07318 
•00315844 
•0431 
•1086 
•8914 
■09680 
•00670761 
•0693 
■1682 
•8318 
■1.3991 
•00265225 
•0189 
■3259 
•6741 
•21968 
•00094864 
•0043 
■6464 
•3530 
•22856 
•00132496 
•0058 
•8222 
•1778 
•14718 
•00373321 
•0254 
•8872 
•1128 
•10006 
•00183184 
■0183 
■\mi = S{x'^lp'q') 
For arctan.. 
For (7), 
where 
Probable error of x^ = 1"349 = 10^0. 
-^2 = 55.53 + lo^O. 
^2=19-6 + 5^9. 
Difference = 35^9 ± 11-6, 
ll-6 = \/lO-0^ + 5^9l 
* Biomctrika, Vol. x. 
