Co THEORY OF STATISTICS. 
Given only the above data, investigate whether there is probably 
a partial association between child and grandparent. 
If there were no partial association we would have— 
(4B)(BO) , (4B)(BO) 
AC) == 
HET Rar 
_ 2524 x 2231 1060 x 947 
Ag ST 
=18450+ 513-2 
= 2358-2. 
Actually (AC)=2480; there must, then, be partial association 
either in the B-universe, the S-universe, or both. In the absence 
of any reason to the contrary, it would be natural to suppose there 
is a partial association in both; 7. that there is a partial 
association with the grandparent whether the line of descent 
passes through “light-eyed ” or “not-light-eyed ” parents, but this 
could not be proved without a knowledge of the class-frequency 
(480). 
10. The total possible number of associations to be derived from 
n attributes grows so rapidly with the value of » that the evalua- 
tion of them all for any case in which =» is greater than four 
becomes almost unmanageable. For three attributes there are 9 
possible associations—three totals, three partials in positive 
universes, and three partials in negative universes. For four 
attributes, the number of possible associations rises to 64, 
for there are 6 pairs to be formed from four attributes, and 
we can find 9 associations for each pair (1 total, 4 partials 
with the universe specified by one attribute, and 4 partials 
with the universe specified by two). For five attributes the 
student will find that there are no less than 270, and for six 
attributes 1215 associations. 
As suggested by Examples i. and ii. above, however, it is not 
necessary in any actual case to investigate all the associations 
that are theoretically possible ; the nature of the problem indicates 
those that are required. 
In Example i., for instance, the total and partial associations 
between 4 and D were alone investigated ; the associations between 
A and B, B and D were not essential for answering the question 
that was asked. In Example ii., again, the three total associations 
and the partial association between 4 and C' were worked out, 
but the partial associations between 4 and B, B and C' were 
omitted as unnecessary. Practical considerations of this kind will 
always lessen the amount of necessary labour. 
nA