230 Criterion of Goodness of Fit of Fsiivlioplitjsical Curves 
be the number of experiments at each stimulus. Calculate ')^-, the sum of the 
quantities 
{q=\-p'). 
Then in Table XII of Pearson's Tables*, in the column n' = n -\- 1 and the row 
(interpolate if necessary) find the value of P, the probability of obtaining the 
observed p's or a worse set, from the p's by random sampling. 
The probable error of is here approximately 1'35% and the probable error of 
P is approximately 
•6745(P„+,-iV.)%, 
where Pn+i is P itself, and Pn-i is the value found in Table XII by using the 
(/I — l)th instead of the {a + l)th column. 
APPENDIX. 
What value will be obtained for if, in the example used above (normal 
integral hypothesis), we were to proceed by first forming a histogram and then 
treating this histogram as though it were an ordinary directly observed one, 
i.e. using equation (15) above ? The cells of the histogram will be occupied by the 
quantities m = Sp x /j, (observation) or iii' = Sp' x yu. (theory) where Sp is the change 
in p from one stimulus to the next and /x the number of observations at each 
stimulus, here the same throughout. 
p 
5p 
P' 
± - — 5» - 8p' 
e-/m'/j. 
•0233 
•0088 
■0145 
•00021025 
•0239 
•0233 
•0088 
•0034 
•0350 
•0316 
•00099856 
•0285 
•0267 
•0438 
•0900 
•1051 
•0151 
•00022801 
■0022 
•1167 
•1489 
•2400 
•2055 
•0345 
•00119025 
•00.58 
•3567 
•3544 
•2533 
•2611 
•0078 
•00006084 
•0002 
•6100 
•6155 
•2733 
•2164 
•0569 
•00323741 
•0150 
•8833 
•8319 
•0467 
•1164 
•0697 
•00485809 
•0418 
■9300 
•9483 
•0700 
■0517 
•0183 
•00033489 
•0065 
•1239 = /S(e2/»iV) 
whence x' = 300 x •1289 = 37'17, instead of the proper value 19'56. If the calcula- 
tion is performed in this inaccurate way, therefore (by analogy with data which are 
really in histogram form), a very wrong idea of the closeness of fit would be 
obtained. The reason, as stated above, is that the correlations between the cells of 
the histogram derived fi'om an ogive with independently measured |>'s are not such 
as to lead to equation (15). 
* Tables for Statisticians and Biometricians, Cambridge University Press, 1914. 
