Karl Pearson and David Heron 
251 
two tables freed from the effects of ' selecting varying proportions of A 's and B's ' " 
(loc. cit. p. 597). The italics are Mr Yule's. We have rarely come across more 
specious reasoning. A coefficient is selected which for one type of artificial 
selection is not changed, and this peculiarity is termed a special and important 
feature of the coefficient. Another coefficient which is intensely subject to this 
selection is then commended because it can be selected so as to agree with the 
first ; and Mr Yule terms them a " natural pair." Algebraically Mr Yule starts 
with the table ^J - ^ > an d the Q for this is (ad — bc)j(ad + be). This value of Q 
remains unchanged if we write the table \ where the marginal fre- 
V&c | vW 6 
quencies have now been rendered artificially all the same. The original 0 however 
was 
(ad - bc)j\/{a + b) (a + c) (6 + d) (c + d), 
and has changed to 
(vW - »Jbc)/(^ad + \/bc), 
which is Mr Yule's colligation and a function of Q. The original (/> has many 
important properties, it is Pearson's mean square contingency for a fourfold table 
and determines the probability of the two variates being independent ; it is also 
the correlation of means if they be measured from their dividing lines in terms of 
their standard deviations, supposing the material continuous and approximately 
normal. The new $ possesses neither of these really important properties, it is 
the </> of the table after most artificial selection, and that it agrees with Mr Yule's 
coefficient of colligation neither gives validity to that coefficient, nor endows it 
with any new property whatever. 
Mr Yule has not taken the trouble to see what sort of effect his artificial 
selection really has on actual material. We propose to illustrate it on certain 
vaccination data. We give, as Table XXI, the table of Glasgow data published in 
Biometrika, Vol. vil. p. 257. It is not true that the haemorrhagic and confluent 
cases all die, but a very large percentage of them die and it would be not far 
from the fact to represent the data by a fourfold table : 
Deaths 
Recoveries 
Totals 
Vaccinated 
273 
1301 
1574 
TJn vaccinated ... 
65 
50 
115 
Totals 
338 
1351 
1689 
This table is only illustrative, not, of course, a rigorous experience. 
Now we have not assumed Table XXI to be a Gaussian distribution, but we have 
found the means of the arrays by assuming each to be represented by a Gaussian 
curve, a process which, as we have seen, gives a fairly close result even for skew 
material. We have also, in order to get scales of a reasonable character, arranged 
