52 
Variation and Correlation in Brain-Weight 
approach to linearity of the regressions is that given by Pearson*. The two 
constants which are of the greatest significance here are, (a) the mean square 
deviation of the means of the arrays from the regi'ession line, S^/", and (6) a con- 
stant 7], called the correlation ratio, giving the mean redaction in variability of 
an array as compared with the whole population. Evidently 
Si,/' = (Tm' - (iX 
and v = ~ 
cr 
whence by simple substitution S,/- = (?;■* — ?■-) o-- (iii), 
where in these equations cr is, as usual, the standard deviation of the variates 
about the mean for the whole population, r is the coefficient of correlation between 
the two variables concerned, and a^j is the standard deviation of the means of 
the arrays about the mean of these means. The deviation of 1,m from zero, and 
of r] from r, measure the deviation of the system from linearityf. 
I propose to discuss the relation of brain-weight first to age alone, then to 
stature alone, and finally to both age and stature together. 
In Table XIV are exhibited the values of 2ji/ and t) for the correlations 
between brain-weight and age discussed in this paper. In calculating ajyj from 
which to obtain r) according to the relation given above, in this and all other cases 
the means of the arrays were weighted with the number of cases on which they 
were based. This procedure of course gave the mean of the means of the arrays 
the same value as the general population mean calculated from the elemental 
frequency distribution. 
TABLE XIV. 
Analytical Constants for Linearity of Regression. Brain-weight and Age. 
Eace and Series 
r 
1 
2^ 
? 
s 
? 
s 
? 
Swedisli (Total) ... 
„ (Young)... 
Hessian (Total) . . . 
(Young)... 
Bavarian (Total) ... 
„ (Young)... 
Bohemian (Total) ... 
- -2493+ -0310 
- -1705 -f- -0405 
- -1673 ± -0300 
- -0750 + -0393 
- -1225+ -0290 
- -0100+ -0353 
- -2045 -1- -0335 
-•2336-1- -0418 
-•1512+ -0585 
- -3598 ± -0350 
-•1650+ -0499 
- -2405 + ^0354 
•0114+ -0412 
- ^2558+ ^0449 
•2876 
•2251 
•2002 
•1411 
•1962 
•0676 
■2441 
•2770 
•2143 
•3864 
•1961 
•3481 
•1958 
•3033 
•1434 0- 
•1469 0- 
•10996 0- 
•1195 a 
■1533 a 
•0669 a 
•1333 0- 
■1489 o- 
•1519 0- 
•1409 0- 
•1060 0- 
•2517 0- 
-1955 0- 
•1628 0- 
Boy. Soc. Proc. Vol. lxxi. pp. 303 — 313, especially the footnote, pp. 303, 304. Since the above 
was written a very full treatment of the whole subject of skew correlation and non -linear regression by 
Prof. Pearson has appeared as "Mathematical Contributions to the Theory of Evolution, XIV." Drapers' 
Company Research Memoirs, Biometric Series II. 1905. 
t Of course the deviation of -q from r must be considered numerically simply, because rj is necessarily 
a positive quantity from (ii) above, since neither ctm nor a can be negative. If the difference between ?} 
and r were taken with reference to sign in cases where r is minus, an altogether false notion of the 
degree of departure of the regression from strict linearity would be obtained. The degree of this 
departure will always, of course, be immediately given by whatever the sign of r. 
