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 agreement
 between observation and theory. In the second example
the signs are fairly well scattered, and the regrouping has a comparatively
 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