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 permuta- 
tions of sign. We should therefore only expect to get an equal 
or greater total value of x2 and tables all showing positive associa- 
tion, 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 compart- 
ments 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-distri- 
butions 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. 
184