Karl Pearson and David Heron 
269 
In forming these weighted means we have left out the value of Q = 1 + "0000 
and of r t = ? ± ?*. In the first place had we included this value of Q, its weight 
would have been infinite, and whatever the value of r t and its probable error may 
be, it follows that Q must be more unstable than r t . We have got rid of this 
auomalous Q, although Mr Yule's theory gives it infinite weight, and dealt only 
with the remaining 15 cases; but to do this is like considering the assets of a 
bankrupt, after we have disregarded the claims of his principal creditor. The 
above table, however, shows that Q is, notwithstanding this disregard, much worse 
than r t ; in fact 43 , 4°/ 0 must be added to the variability of r t to reach that of Q. 
If we measure the deviations from the weighted means in terms of the probable 
errors as before we find for tetrachoric r t : the mean =2*71 and for association 
Q : = 3'31. Thus again, although since Q = 1 is excluded, we find to a lesser 
extent tetrachoric r t more stable than Q. 
(D) Lengths of Ivy Leaves. 
Mr Yule directly selected the lengths of growing ivy leaves as an especially 
skew distribution upon which to test tetrachoric r t , and this in a case where the 
table had been deduced for homotyposis, i.e. with all the local lumpiness which 
arises from that method of treatment. If we take the 4 x 4-fold of our Table XVII 
on p. 222, nine values of Q and tetrachoric r t are available. They are : 
1—2 
2—3 
3—4 
1— 2 
r t = -6998+ -0046 
§=•8531 ±-0027 
r t = -6406+ -0039 
§=•8655 ±-0040 
r t = -5572 ± -0058 
§-=•9167 ±"0085 
2—3 
^=•6406+ -0039 
(^=■8655+ -0040 
»-,= -5731 + -0033 
§=•6768 ±-0033 
r t = -5218+ -0046 
§=•7570 ±-0048 
3-4 
r t = -5572+ -0058 
§=•9167 ±"0085 
r t = -5218+ -0046 
§=•7570 ±-0048 
^=•5548+ -0052 
§=•7508+ -0045 
We have the following results : 
Weighted Mean 
Weighted 
Standard Deviation 
Tetrachoric r t 
Association § 
•5920 
•8024 
•0570 
■0757 
* On the difficulty, in fact idleness, of attempting to determine r t from tables with zero frequency in 
one quadrant : see Appendix I. That quadrant as we there show might have -5 in it, or, indeed, 
the material from which it is drawn unity. In the former case 20 terms in the ?' r equation will not 
suffice to determine the value of r t . In the latter case r t might swing over from - 1 to a small positive 
quantity. In fact the process of finding tetrachoric r t in such cases warns its user that it is indeter- 
minable. Users of Q will assert that the relationship is perfect with a zero probable error ! 
