Karl Pearson and David Heron 
175 
The two series of results just considered show us that with <f> constant we can 
make Q rise to unity and with Q constant we can make <f> fall to zero. It is there- 
fore always possible to determine or to find in actual practice series of tables in 
which an ascending order of Q is accompanied by a descending order of <£. That 
is to say the two coefficients will flatly contradict each other. There is no basis 
whatever for Mr Yule's assertion that in Sheffield the correlation of vaccination 
and recovery is highest, while in Homerton and Fulham it is lowest. With the 
same three values of Q that he gives, we can make any order of the $'s we please. 
In the actual tables the relative order is not that given by Mr Yule ; the order of 
tetrachoric r agrees with that of </> ; so does the coefficient of mean square con- 
tingency and further the probability of independence, i.e. the probability that 
death is independent of vaccination. These are given as r t , C 2 and P in the 
last three columns of the table on p. 173, and they entirely reverse Mr Yule's 
judgment. P is of course in a different class to any other of the coefficients, 
but we return to this point later. 
Mr Yule states that any table may be reduced without modification of the 
association to its equalised form ; for example the tables 
998,667 
666 
666 
1 
999,333 and 
667 
300,000 
200,000 
200,000 
300,000 
500,000 
500,000 
999,333 
667 
1,000,000 
500,000 
500,000 
1,000,000 
are "equivalent," both have Q = '385, but the inferences to be drawn from the two 
tables had they originated independently are quite different. In the first case the 
probable error of Q = '288, and the result is not definitely significant. In the 
second case the probable error is '001 and there is no doubt of the significance. 
If Q 0 be the Q of an original table and Q E of the equalised table, then we have 
Mr Yule's vaccination data : 
p. is. of Q„ 
P. E. Of Q E 
Sheffield 
■007 
•003 
Leicester 
•065 
•022 
Homerton and Fulham 
•007 
•005 
Mr Yule has not entered into this question of the probable errors of the series 
of his modified tables. The statistician, however, whose long experience enables 
him closely to associate given types of tables with given degrees of reliability, 
is largely deceived when an " equivalent table " is presented to him of which the 
probable error as it stands may be A- to yi^ or less of the true probable error of 
the coefficient given. 
We have shown in this section of our discussion that the two coefficients Q and 
<j> cannot both be valid. Mr Yule nowhere adequately assigns the type of cases to 
which the one or the other should be applied. He tells us that he hopes the 
