Godfrey H. Thomson 
225 
It is not correct therefore to use equation (15) above. It is more accurate and 
withal exceedingly simple to apply equation (12) direct to the p's. Since the 
latter are independent, all the intercorrelations r are zero. Therefore R is unity, 
Rkk is unity, and Rki is zero. Equation (12) therefore becomes 
and as the distributions of each p will be binomial in form provided the experimental 
conditions remain constant enough we have 
<r' = p'q'/H' (17)*, 
where fu, = the number of experiments on which p is based, and jt)' = 1 — q', so that 
^-ng) <'«'t. 
Herein the x's are the differences between observed p'n and theoretical jt»"s. The 
pi'obability P is then found as before. 
* If we look upou the judgments heavier, as suggested in an earlier paragraph, as being comparable 
with drawing black balls out of a bag containing black balls and white balls in the proportion p' and 
1 -p', then the probable error of p is -67449 v/^ (- — ^ , («■ being the number of judgments of which 
V fx 
PfjL are of the category heavier. 
For the chances of obtaining 0, 1, 2, .../«,- 1, or black balls in a drawing of are given by the 
terms of 
q' being 1 -^j': that is, the chances of obtaining 
0 12 M-1 fi 
p = - , - , -, ■■ — , or - . 
fj. fx. fl fx. fj. 
The s. d. of the above binomial is >J fx.p' q' and the s. d. of p therefore -'Jfxp'q'= . 
fX. \ fJ. 
t Compare Professor K. Pearson on " Goodness of Fit in Statistics and Physics,'' Biometrika, 1916, 
XI, pp. 239 — 261, especially p. 257. 
We can check our equation (18) by treating the matter from first principles, and not as a special ease 
included in Pearson's formulae. We have, from this point of view, n quantities^ which are independently 
measured, and n quantities p' which are theoretically given. The variations from p' are binomial in 
form, that is, they are appro.rivuttely Gaussian. The probability of an error 
_ IXX- 
is therefore it>^. = - Jl^ ^ e (a). 
The probability of the whole set of observed values Pi, pi, Pi, ■■■Pn occurring is the product 
2 = ^1^2 •■• (b). 
Write this z = Z(te^^-^' . 
Then ^^^s 
from equation (a). 
P'<l') 
