PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 277 
deviations in man and woman, we must alter the deviations in the same ratio. 
Freeing ourselves from this particular ratio, we may take as a measure of variation 
the ratio of standard deviation to mean, or what is more convenient, this quantity 
multiplied by 100. We shall, accordingly, define Y, the coefficient of variation, as 
the percentage variation in the mean, the standard deviation being treated as the 
total variation in the mean ; since the p.e. = ’674,506 X S.D., Y multiplied by 
’674,506 may be called the “probable percentage variation.” Of course, it does not 
follow because we have defined in this manner our “ coefficient of variation,” that this 
coefficient is really a significant quantity in the comparison of various races ; it may 
be only a convenient mathematical expression, but I believe there is evidence to show 
that it is a more reliable test of “ efficiency” in a race'"' than absolute variation. At 
present, however, we will merely adopt it as a convenient expression for a certain 
function, and proceed to examine its relation to correlation. 
Let m ]} m 2 be the means of two correlated organs ; oq, c r, their standard deviations ; 
r their coefficient of correlation; Y l5 Y 2 their coefficients of variation; and Rj, R 2 
the respective regressions for deviations d. 2 and d x of the two organs. 
Now 
R, = »• — <4 = r It X ^ 
1 tr, Z Vo TO, 
or 
and similarly 
Ifi Y 1 d, 
— = r — X —, 
TO, Vo TO, 
K 2 Vo d, 
— = r W X — 
in ^ V j rn l 
But we see that the amounts d 2 /m 2 and d 1 /m 1 are equally significant deviations in 
the case of the second and first organ, while the amounts R 1 /m 1 and R 2 / m 2 are 
equally significant regressions in the case of the first and second organ.t 
It follows, therefore, that the significances of the mutual regressions of the two 
organs are as the squares of their coefficients of variation. 
Hence inequality of coefficients of variation marks inequality of mutual regressions. 
Now coefficients of variation are rarely, if ever, equal for the same organ in corres¬ 
ponding classes of men and women. In dealing with male and female skull measure¬ 
ments for a great variety of races, this inequality is often very marked, and, therefore, 
differences of variation tell, especially in mutual regression in the case of sexual 
selection and inheritance from the opposite sex. They are sufficient, I think, to pre¬ 
clude Mr. Galton’s theory of the mid-parent from being considered as more than a 
* By “ race efficiency,” I would denote stability, combined with capacity to play a part in the history 
of civilization. I hope later to publish details of variation, especially in skull measurements of different 
races of man, the data of which I have been for some years reducing. 
t For example, 1" and -pf" I term equally significant deviations or regressions in the stature of man 
and woman, and 1” and pf'' in the stature of woman and man. 
