XI.—CORRELATION: MISCELLANEOUS THEOREMS. 213 
(wry) = 3(z + 6,)(x + 3,) 
= 2(7), 
and accordingly 
3(ayz,) = 
Shel 7 (5) 
(This formula is part of Spearman’s formula for the correction of 
the correlation-coefficient, ¢f. § 7.) 
6. Influence of Errors of Observation on the Correlation-coefficient. 
—Let x, v, be the observed deviations from the arithmetic means, 
x, y the true deviations, and 8, € the errors of observation. Of 
the four quantities x, 7, 8, ¢ we will suppose # and y alone to 
be correlated. On this assumption 
X(xyyy) = 2(xy) - 
It follows at once that 
Try _ Ta Ty, 
Fay, T Or Oy : 
and consequently the observed correlation is less than the true 
correlation. This difference, it should be noticed, no mere increase 
in the number of observations can in any way lessen. 
7. Spearman’s Theorems.—If, however, the observations of both 
x and 7 be repeated, as assumed in § 5, so that we have two 
measures x, and x,, ¥; and y, of every value of # and y, the true 
value of the correlation can be obtained by the use of equations 
(5) and (6), on assumptions similar to those made above. For 
we have 
= 3 (@1) (075) = 3 (ayy) (yyy) 
(0129) 2(y170) (2129) 2(117) 
= Youn Taw = Tay Ta, g (7) 
Torr Tywe Tozer Tvis 
Or, if we use all the four possible correlations between observed 
values of x and observed values of 7, 
Tenet ogeayet 
r 4 aN ZV WT 8 
= CA ( ) 
Equation (8) is the original form in which Spearman gave his 
correction formula (refs. 6, 7). It will be seen to imply the 
assumption that, of the six quantities , y, 8,, 8, €, €, # and ¥ 
alone are correlated. The correction given by the second part 
of equation (7), also suggested by Spearman, seems, on the 
(6)