128 MR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
in k is (— 6/2\., + i — xp/'Z/j..,) k. For the mean squares and mean products of 
0, (p, xp, we have the table— 
1 
S,I 
r-i 
^2 
M-x| 
S3, 1 1 
^2,2 
S 3 . 2 -S?,, 
S], 3 1 
Mi — m\ 
from which it will be found that the mean square of the error in k is 
§ 21 .—Test of Inde'pendence of Tivo Distributions.—Yov an illustration of the 
application of the theory of error to testing statistical hypotheses, let us take the 
case of two independent distributions. The criterion of independence of the distri¬ 
butions of two measures L and M is that, if a denotes the proportion of individuals, 
in the complete community, for which L lies between any two values L' and L", and 
if denotes the proportion for which M lies between any two values M' and M", 
then the proportion for which both these conditions are satisfied is aj^. Hence, in 
order to test the hypothesis of independence w^hen n individuals have been obtained 
by random selection, we must arrange them in a table of double entry, thus:— 
Values of L. 
Values of M. 
Total. 
XT to M”. 
W to M"'. 
&c. 
L' to L”. 
&c. 
Pi 
L" to L”' .... 
i 
'>h2 
Pi 
Total . . . 
'll 
. 
n 
then form a new table by dividing each number in this table by n, so as to show the 
proportions in the different classes ; and then consider whether the discrepancies 
between these proportions and the corresponding proportions in a table showing 
independent distribution are such as might be accounted for by random selection. 
