278 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
first approximation. Turning to Table V., we see that variation in height is greater 
for males than females ; but while very sensible for husbands and wives, and sons 
and daughters, it is insignificant for fathers and mothers. This superiority of male 
to female variation, as measured by the coefficient of variation, is in accordance with 
the usual belief that the male is more variable than the female, but it is entirely out 
of accordance with the great bulk of the statistics I have so far reduced. The belief 
seems to have arisen from a very loose notion of how variation is to be estimated- 
These stature statistics of the English middle classes seem to some extent anomalous. 
For example, I find from statistics of stature in the German working classes :— 
Male coefficient of variation = 4*0245, 
Female ., „ ... — 4*2582. 
Hatio of female to male coefficient = 1*058, thus more than reversing the highest 
English ratio, that of husbands and wives. It is noteworthy that, while the varia¬ 
tion is thus reversed, the ratio of the mean heights equals 1*078, and remains practi¬ 
cally the same. These remarks are introduced in order to prevent any too hasty 
generalisation as to the nature of male and female correlation based on a current 
belief in the greater intensity of male variation. 
(ii.) Coefficient of Correlation and Coefficients of Variation .—Let x and y be two 
correlated organs, and let f and y be corresponding deviations from the mean values 
m 1 and m 2 . We shall suppose that £ and 17 are so small that the squares of the ratios 
and 17 /m 2 are negligible as compared with the first powers. Let r be the 
coefficient of correlation of x and y , oq, cr 2 their standard deviations, v lf v. 2 their 
coefficients of variation, and let 2 be any function f(x, y) of x and y with a deviation £, 
corresponding to f and 77, and a standard deviation, mean and coefficient of variation 
respectively 2, M, and Y. 
Differentiating 2 = f (x , y) and remembering our hypothesis as to the smallness 
of the variations, we have : 
£ — + f y y- 
Squaring : 
Summing for every possible value of f and 17 , and dividing by n the total number 
of correlated pairs : 
S(P) 
n 
S(F) , 
—f‘~ „ + fi ~vr + 2 A/; 
n 
n 
y 
n 
or, 
—ffo v + ff oy + 2 f x f y (x x x 2 X 
s m 
n<T x <T : , 
Now, if there were no correlation, we should have: 2 s = ff erf + ff cr/; hence 
