2: THEORY OF STATISTICS. 
one family with statures 5 ft. 9, 5 ft. 10, and 5 ft. 11, these are 
regarded as giving the six pairs 
5 ft. 9 with 5 ft. 10 5 ft. 10 with 5 ft. 9 
o sy, ite 1] Bub Ll : 
3 16.10, n ™ » oD 16.10 
which may be entered into the table. The entire table will be 
formed from the aggregate of such subsidiary tables, each due to 
one family. Let it be required to find the correlation-coefficient, 
however, for a single subsidiary table, due to a family with & 
members, the numbers of pairs being therefore NV (V —1). 
As each observed value of the variable occurs &/ —1 times, 
t.e. once in combination with every other value, the means and 
standard-deviations of the totals of the correlation-table are the 
same as for the original IV observations, say # and o. If 2; 2, 
Zz. .. .be the observed deviations, the product sum may be 
written 
TZo + XX +2124 F oo a 
+ Xo%; + Xg +22, + 
+ Xg%y + XT + Xgl + 0. 
+ 
=2,{3(2) — @} +2 {3(@) = 2} +2 {3(w) ag} +. LLL 
= a l-pl-mlo lL = Vg, 
whence, there being N(& — 1) pairs, 
No? 1 
pita ag rn dns nl 2 
For ¥=2, 3,4... . this gives the successive values of r= -1, 
-1, —1.... Itis clear that the first value is right, for two 
values x, x, only determine the two points (#;, x,) and (x, ,), 
and the slope of the line joining them is negative. 
The student should notice that a corresponding negative 
association will arise between the first and second member of the 
pair if all possible pairs are formed in a mixture of 4’s and a’s. 
Looking at the association, in fact, from the standpoint of § 10, 
the equation (13) still holds, even if the variables can only assume 
two values, e.g. 0 and 1. This result is utilised in § 14 of Chapter 
X1Y. 
12. Correlation due to Heterogeneity of Material.—The following 
theorem offers some analogy with the theorem of Chap. IV. 
§ 6 for attributes.—If X and Y are uncorrelated vn each of two 
records, they will nevertheless exhibit some correlation when the 
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