THEORY OF STATISTICS. 
For this comparison n’ is 8, x2 is 26:96, or practically 27, and P 
is about *000001—a value much more nearly in accordance with 
that suggested by the mean. 
Such a regrouping of the frequency distribution by the runs of 
classes that are in excess and in defect of expectation would appear 
often to afford a useful and severe test of the real extent of agree- 
ment between observation and theory. In the second example 
the signs are fairly well scattered, and the regrouping has a com- 
paratively small effect ; the mean being in almost precise agreement 
with expectation. The regrouped distribution is :— 
SITCOREeE, a Expected 
requency. Frequency. 
0 447 459 
1 1145 1108 
2-3 1977 2022 
4 380 364 
5-6 139 143 
7-8 8 5 
Total. . : E 
Here nn’ is 6, x*is 5°52, and P 0°36, so that the deviations from 
expectation are still well within the range of fluctuations of 
sampling. 
The value of P is the probability that a set of observations 
will occur giving a group of deviations from theory, s.e. a value 
of x, which is more improbable than that observed. If, to take 
the second illustration above, we were to repeat 4096 throws of 
twelve dice a large number of times, noting the throws of sixes, 
we should expect to get a worse fit to theory, z.e. a value of x? 
greater than 5 81, roughly speaking 56 times in every hundred 
trials. 
, The value of P corresponding to ¥2=0 ig necessarily unity, 
for it is certain that all values of x2 must exceed zero. If the 
value of P corresponding to x2=1 is P,, then 1-2, is ithe 
frequency of values of y2 between 0 and 1. Similarly, if the 
value of P corresponding to x2=2 is P, then the frequency of 
values of x? between 1 and 2 is P,— P,, and so on. Thus, for 
16 classes (n”=16), we find in the tables :(— 
376 
4096 4096