TO THE THEORY OF EVOLUTION. 
17 
No. 
r. 
h. 
Jc. 
q 2 . 
Q3- 
Q:> 
1 
•5939 ± -0217 
•0873 
- -4163 
•7067 
•6054 
•6168 
•6100 
2 
•5557 + -0261 
- 
•4189 
- -4163 
•6688 
•5657 
•5405 
•5570 
3 
•5529 ± '0247 
— 
•0873 
- -0012 
•6828 
•5809 
•5699 
•5813 
4 
•5264 + -0264 
+ 
•2743 
+ -3537 
•6345 
•5331 
•5200 
•5283 
5 
•5213 ± -0294 
+ 
•6413 
+ -6966 
•6530 
•5511 
•4878 
•5160 
6 
•5524 ± -0307 
+ U0234 
+ -3537 
•7130 
■6118 
•6169 
■6138 
7 
•5422 ± -0288 
+ 
•6463 
+ -5828 
•6693 
•5673 
•5136 
•5452 
8 
•2222 + -0162 
+ 
•3190 
+ -3190 
•2840 
•2268 
•2164 
•2251 
9 
•3180 + -0361 
+ 
T381 
+ -0696 
•3959 
•3185 
•3176 
•3183 
10 
■5954 ± -0272 
+ 1-5114 
+ -7114 
•7860 
•7100 
•6099 
•6803 
11 
•4708 + '0292 
+ 
•0865 
- -0054 
•5692 
•4712 
•4720 
•4715 
12 
■2335 + -0335 
+ 
•0405 
+ *0054 
•2996 
•2385 
•2385 
•2385 
13 
•2451 ± -0205 
+ 
•2707 
+ -0873 
•3103 
•2473 
■2456 
•2470 
14 
•1002 + -0394 
+ 
•4557 
+ T758 
T311 
•1032 
•0993 
•1029 
15 
•6928 ± -0164 
+ 
•5814 
+ -5814 
•8032 
•7108 
•6699 
•6897 
Now an examination of this table shows that notwithstanding the extreme ele¬ 
gance and simplicity of Mr. Yule’s coefficient of association Q 2 , the coefficients Q 3 , 
Q,, and Q 5 , which satisfy also his requirements, are much nearer to the values 
assumed by the correlation. I take this to be such great gain that it more than 
counterbalances the somewhat greater labour of calculation. If we except cases (6) 
and (10), in which h or k take a large value exceeding unity, we find that Q 3 , Q 4 , and 
Q 5 in the fifteen cases hardly differ by as much as the probable error from the value 
of the correlation. If we take the mean percentage error of the difference between 
the correlation and these coefficients, we find 
Mean difference 
?? ?5 
5? ? J 
of Q 3 = 24'38 per cent. 
Q 3 =: 3 95 ,, 
Q, = 2-94 
5) 
Q 5= 272 
Thus although there is not much to choose between Q t and Q 5 , we can take Q 5 as 
a good measure of the degree of independent variation. 
The reader may ask : Why is it needful to seek for such a measure ? Why cannot 
we always use the correlation as determined by the method of this paper ? The 
answer is twofold. We want first to save the labour of calculating r for cases where 
the data are comparatively poor, and so reaching a fairly approximate result rapidly. 
But labour-saving is never a wholly satisfactory excuse for adopting an inferior 
method. The second and chief reason for seeking such a coefficient as Q lies in the 
fact that all our reasoning in this paper is based upon the normality of the frequency. 
We require to free ourselves from this assumption if possible, for the difficulty, as 
is exemplified in Illustration V. below, is to find material which actually obeys 
within the probable errors any such law. Now, by considering the coefficient of 
regression, roq/oq = S(.r?/)/(No- 1 o- 3 ), as the slope of the line which best fits the series 
YOL. CXCV.-A. 
D 
