IX.— CORRELATION. 177 
Hence the sons of fathers of deviation « from the mean of all fathers 
have an average deviation of only 0-522 from the mean of all sons ; 
ve. they step back or “regress” towards the general mean, and 0-52 
may be termed the “ratio of regression.” In general, however, 
the idea of a “stepping back” or “regression” towards a more 
or less stationary mean is quite inapplicable—obviously so where 
the variables are different in kind, as in Tables V. and VI.— 
and the term “ coeflicient of regression” should be regarded simply 
as a convenient name for the coefficients 4, and 6,, RR and CC 
are generally termed the “lines of regression,” and equations (6) 
the “regression equations.” The expressions “ characteristic lines,” 
““ characteristic equations” (Yule, ref. 8) would perhaps be better. 
Where the actual means of arrays appear to be given, to a satis- 
factory degree of approximation, by straight lines, we may say 
that the regression is linear. It is not safe, however, to assume 
that such linearity extends beyond the limits of observation. 
14. The two standard deviations 
8,=0, n/1-12 8,=0, 1-12 
are of considerable importance. It follows from (7) that s, is the 
standard deviation of (z-6,.7), and similarly s, is the standard 
deviation of (y — b,x). Hence we may regard s, and s, as the 
standard errors (root mean square errors) made in estimating « 
from y and y from « by the respective characteristic relations 
x=05.y y =bya. 
s, may also be regarded as a kind of average standard deviation of 
a row about RE, and s, as an average standard deviation of a 
column about CC. In an ideal case, where the regression is 
truly linear and the standard deviations of all parallel arrays are 
equal, a case to which the distribution of Table III. is a rough 
approximation, s, is the standard deviation of the z-array and s, 
the standard deviation of the y-array (cf. Chap. X. § 19 (3)). 
Hence s, and s, are sometimes termed the “standard deviations 
of arrays.” 
15. Proceeding now to the arithmetical work, the only new 
expression that has to be calculated in order to determine 7 5,, 0, 
$x» and s, is the product sum 3(zy) or the mean product p. Asin 
the cases of means and standard deviations, the form of the 
arithmetic is slightly different according as the observations are 
few and ungrouped, or sufficient to justify the formation of a 
correlation-table. In the first case, as in Example i. below, the 
work is quite straightforward. 
Ezample i., Table VII.—The variables are (1) X—the estimated 
12