Karl Pearson and David Heron 
281 
border cases of the table which give Q = + 1 with infinite weight. Also the 
cases adjacent to these where Q, depending upon very small frequency in one 
quadrant, has a very high value, a low probable error, and accordingly a very 
great weight. With the exception of this we took every alternate case in every 
row and so obtained 41 coefficients ; the corresponding tetrachoric r t 's were like- 
wise calculated. The results are exhibited in the accompanying table, the 
notation being that adopted by Mr Yule to mark the divisions. 
There is practical certainty that any extension of the boundaries of this system 
of divisions could only better the position of r t relative to Q. We obtained the 
following results : 
Weighted Mean Q = -834. Weighted Mean r, = '672. 
Weighted s.D. of Q = 0613. Weighted S.D. of r t = -0478. 
We must therefore increase the variability of tetrachoric r, by 28 °/ Q to reach 
the variability of Q. This shows the real degree of difference in the stabilities 
of r t and Q for the correlation-tables of ordinary practice. We note also that the 
weighted mean of r, differs by only "005 from the true product-moment value, 
'677, of the correlation. 
Now let us approach the matter from the standpoint of probable error. We 
have : 
Mean (r t — r,)/probable error = 1'06. 
Mean (Q - Q)/probable error = 172. 
In the case of r, — r, we have 21 below their probable error in value and 20 
above it, just what there should be. Only 5 exceed twice the probable error and 
there should be 4. In other words the distribution of r, in terms of its probable 
error might well have arisen from random sampling of Gaussian material. Now 
turn to Q : in terms of the probable error Q shows an increase of upwards of 
62 °/ 0 on the value for r t . There are only 15 values of Q — Q below this probable 
error, 26 in excess of it. There are 15 values instead of 4 in excess of twice 
the probable error. There are five values in excess of three times their probable 
error, compared with only two occurring in the case of r t . It is, we think, obvious 
that the variations in the case of Q are far greater than those due to random 
sampling*. Our Diagrams XXII and XXIII (pp. 282-3) indicate two points. 
In the upper parts of these diagrams we have plotted r t and Q to the percentage of 
frequency in the quadrant of least frequency. We see at once that if we avoid 
quad ran tal frequencies under 1 °/ o the value of tetrachoric r t is for practical purposes 
equal to the true product-moment r. The reader will recognise the far greater 
scatter of Q. In the lower figures we have plotted r t and Q with relation to their 
probable errors; the full dot denotes the observation, the open dot the end of a line 
* The reader must remember that there is no reason to assert that the errors must be of the order of 
random sampling; there is only one table, not many random samplings from a much larger mass, and 
we take different divisions. But we assert that if the errors be of that order, then the method is as good 
as the data warrant our using. 
Biometrika ix 30 
