THEORY OF STATISTICS. 
if d denote, as in Chap. IX., the deviation of the mean of an 
array of X’s from the line of regression, we have by the relation 
of Chap. IX,, § 11, p. 172 
a. (1 LL, 72) =, + 02. (6) 
Substituting for o,, from (2), that is, 
of =o =v . (7) 
But ¢, is necessarily positive, and therefore 7,, is not less than 7. 
The magnitude of o, and therefore of %2—72 measures the 
divergence of the actual line through the means of arrays from 
the line of regression. 
It should be noted that, owing to the fluctuations of sampling, 
r and 7 are almost certain to differ slightly, even though the 
regression may be truly linear. The observed value of #%- 7? 
must be compared with the values that may arise owing to 
fluctuations of sampling alone, before a definite significance can 
be ascribed to it (cf. Pearson, ref. 19, Blakeman, ref. 22, and the 
formule cited therefrom on p. 352 below). 
22. The following table illustrates the form of the arithmetic 
for the calculation of the correlation-ratio of son’s stature on 
father’s stature (Table III. of Chap. IX. p. 160). In the first 
column is given the type of the array (stature of father); in the 
second, the mean stature of sons for that array; in the third, the 
difference of the mean of the array from the mean stature of all 
sons. In the fourth column these differences are squared, and in 
the sixth they are multiplied by the frequency of the array, two 
decimal places only having been retained as sufficient for the 
present purpose. The sum-total of the last column divided by 
the number of observations (1078) gives o,,,2 = 2058, or ,,, = 143. 
As the standard-deviation of the sons’ stature is 2:75 in. (cf. 
Chap. IX., question 3), 7,,=052. Before taking the differences 
for the third column of such a table, it is as well to check the 
means of the arrays by recalculating from them the mean of the 
whole distribution, ¢.e. multiplying each array-mean by its fre- 
quency, summing, and dividing by the number of observations. 
The form of the arithmetic may be varied, if desired, by working 
from zero as origin, instead of taking differences from the true 
mean. The square of the mean must then be subtracted from 
2(f-m2)/N to give o,,,2% 
If the second correlation-ratio for this table be worked outrin 
the same way, the value will be found to be the same to the 
second place of decimals: the two correlation-ratios for this table 
are, therefore, very nearly identical, and only slightly greater 
than the correlation-coefficient (0:51). Both regressions, it 
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