LE THEORY OF STATISTICS.
The association between inoculation and protection from attack
is positive for each estate, but for only one of the tables is the
value of P so small that we can say the result is wery unlikely to
have arisen as a fluctuation of sampling. Adding up the values
of x2, the total is 28-40, and entering the column for n'=7 (one
more than the number of tables considered), we find
P 0
23 000094
29 000061
whence by interpolation the value of P is ‘000081, i.e. we should
only expect to get a total of x%s as great as or greater than this, on
random sampling, 81 times in 1,000,000 trials. We can therefore
regard the results as significant with a high degree of confidence.
We may, I think, go further: for all the observed associations
are positive, and in six cases there are 2% or 64 possible permutations
 of sign. We should therefore only expect to get an equal
or greater total value of x2 and tables all showing positive association,
 not 81 times in 1,000,000 trials but 81/64 or, roundly, 1-3
times. P for the observed event (3(x?)=28'4 and all associations
positive) is therefore only ‘0000013.
Experimental Illustrations of the General Case.—The formule
for the general case, as for the special case in which the frequencies
with which comparison is made are given a prior, can be checked
by experiment.
The numbers of beans counted in each of the sixteen compartments
 of the revolving circular tray mentioned on p. 374 above
were entered as the frequencies of a table (1) with 4 rows and
4 columns, (2) with 2 rows and 8 columns, and the value of x?
computed for each table for divergence from independence. For
the two cases we have
w=(3x3)+1=10
and n'=(1x7)+1=38
respectively. Differencing the columns for P corresponding to
these two values of n’, we obtain the theoretical frequency-distributions
 given in the columns headed “Expectation” in Table A,
The observed distributions of the values of x? in 100 experimental
tables are given in the columns headed “ Observation.” It will be
seen that the agreement between expectation and observation is
excellent for so small a number of observations. If the goodness
of fit be tested by the x2 method, grouping together the frequencies
from x2=15 upwards, so that n’ is 4, x* is found to be 2-27 for
the 4 x 4 tables and 4:36 for the 2 x 8 tables, giving P=052 in
the first case and 0:22 in the second.

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