42 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
Table of Results. 
Classificatiom. 
Correlation. 
Mean of sons. 
Mean of fathers. 
A 
•5939 ± -0247 
L 
68"--64 ( - -44 6,32) 
h. 
67"-74 (--087,00) 
B 
•5557 + -0261 
68"-64 (--416,32) 
67"-63 ( --418,86) 
C 
•5529 + -0247 
68"-50 (- -001,16) 
67"-74 (--087,30) 
D 
•5264 + -0264 
68"-53 (-353,71) 
67"-77 (-274,30) 
E 
•5213 + -0294 
68"-60 (-696,57) 
67"-76 (-641,30) 
F 
•5524 + -0307 
68"-53 (-353,71) 
67"-73 (1-023,44) 
Now these results are of quite peculiar interest. They show us :— 
(i.) That the probable error of r, as found by the present method, increases with 
h and h. But the increase is not very rapid, so that the probable errors of the series 
range only between ’025 and ’031. Hence while it is an advantage, it is not a very 
great advantage, to take the divisions of the groups near the medians. It is an 
advantage which may be easily counterbalanced by some practical gain in the method 
of observation when the division is not close to the medians. 
(ii.) While the probable error, as found from the present method of calculation, is 
1-5 to 2 times the probable error as found from the product moment, it is by no 
means so large as to seriously weigh against the new process, if the old is un¬ 
available. It is quite true that the results given by the present process for six 
arbitrary divisions differ very considerably among themselves. But a consideration 
of the probable errors shows that the differences are sensibly larger than the prob¬ 
able error of the differences, even in some case double ; hence it is not the method 
but the assumption of normal correlation for such distributions which is at fault. As 
we shall hardly get a better variable than stature to hypothesise normality for, we 
see the weakness of the position which assumes without qualification the generality 
of the Gaussian law of frequency. 
(iii.) We cannot assert that the smaller the probable error the more nearly will 
the correlation, as given by the present process, agree with its value as found by 
the product moment. If we did we should discard '5213, a very accordant result, 
in favour of '5529, or even '5939. The fact is that the higher the correlation the 
lower, ceteris paribus, the probable error, and this fact may obscure the really best 
result. Judging by the smallness of li and h and of the probable error, we should 
be inclined to select C or the value '5529. This only differs from '5198 by slightly 
more than the probable error of the difference ('033 as compared with '029) ; but 
since both are found from the same statistics, and not from different samplings ot 
the same population, this forms sufficient evidence in itself of want of normality. 
The approximate character of all results based on the theory of normal frequency 
must be carefully borne in mind; and all we ought to conclude from the present 
