XIL.—CORRELATION : MISCELLANEOUS THEOREMS. 217 
The standard-deviations o, a, are given by 
0 2=025 - &=(4)(a)/N? 
02 =025 — 72 =(B)(B)/N* 
Finally, 
(zy) =1{(4B) + (a) - (4B) - (aB)} - N 5. 
Writing 
(4B) - (4)(B)/N=3 
(as in Chap. III. §§ 11-12) and replacing & 7 by their values, 
this reduces to 
(xy) =. 
Whence 
N.§ 
r= —pmm———, 12 
JADED vi 
This value of » can be used as a coefficient of association, but, 
unlike the association-coefficient of Chap. III. § 13, which is 
unity if either (4B)=(4) or (48)=(B), r only becomes unity if 
AB)=(4)=(B). This is the only case in which both frequencies 
iz and (4) can vanish so that (4B) and (af) correspond to 
the frequencies of two points X; ¥;, X, ¥, on a line. Obviously 
this alone renders the numerical values of the two coefficients 
quite incomparable with each other. But further, while the 
association coefficient is the same for all tables derived from one 
another by multiplying rows or columns by arbitrary coefficients, 
the correlation coefficient (12) is greatest when (4)=(a) and 
(B)=(B), i.e. when the table is symmetrical, and its value is 
lowered when the symmetrical table is rendered asymmetrical by 
increasing or reducing the number of 4’s or B’s. For moderate 
degrees of association, the association coefficient gives much the 
larger values. The two coefficients possess, in fact, essentially 
different properties, and are different measures of association in 
the same sense that the geometric and arithmetic means are 
different forms of average, or the interquartile range and the 
standard-deviation different measures of dispersion. 
The student is again referred to ref. 3 of Chap. IIL for a 
general discussion of various measures of association, including 
these and others, that have been proposed. 
11. The Correlation-coeflicient for all possible pairs of N values 
of a Variable.—In certain cases a correlation-table is formed by 
combining N observations in pairs in all possible ways. If, for 
example, a table is being formed to illustrate, say, the correlation 
between brothers for stature, and there are three brothers in