SUPPLEMENTS—GOODNESS OF FIT. 
Differences of P. 
0 1: ‘007 873 
. 992 127 172 388 
; ‘819 739 368 321 
; | "451 418 279 486 
‘171 932 *171 932 
We should expect, therefore, in, say, 1000 sets of random 
sampling with 16 classes, about 8 cases of x* between 0 and 3, 
about 172 cases between 5 and 10, 368 between 10 and 15, 
279 between 15 and 20, and 172 over 20. The following table 
shows the results obtained for the more modest number of 100 
sets of trials, and gives very fair agreement with theory, especially 
considering that the assumption of normality can hardly be 
stritly true. The trials were carried out by throwing 200 
beans into a revolving circular tray with sixteen equal radial 
compartments, and counting the number of beans in each com- 
partment. The value of x2 was then computed, taking the 
expected frequency as 200/16 = 125. 
Number of Tables giving a Value of x% 
lying between the Limits on the Left. 
Expected. Observed. 
0-5 0-8 x 
5-10 17-2 20 
10-15 36-8 36 
15-20 279 305 
20 upwards 17-2 135 
If we treat this in its turn as a comparison of observation with 
theory, we find, bracketing the first two groups together, so as 
to reduce the number of classes to four, y*=1-28, whence from 
the tables P is approximately 0°74. That is to say, we should 
expect a worse agreement with theory about three times out 
of four, 
It follows from what was said above that, in any series of trials 
by simple sampling, equal numbers of cases should be found within 
equal intervals of P, e.g. from 10 to 09, from 0'9 to 08, from 
0-8 to 0'7, and so on. The frequency distribution of P, that is to 
377 
P.