254 
Miscellanea 
Hence x 2 = 705 x 590 x -000,0273 = 1 1 -36. 
This gives us P=-02G5, or the odds are about 37 to 1 against the boys and girls being 
random samples of the same material. These are nothing like the odds obtained in the scarlet 
fever and measles differentiation of pigmentation ; but they are sufficiently considerable to 
render it desirable in future investigations to keep the sexes separate. 
The above illustrations were simply selected because I happen to have Dr Macdonald's 
results before me, and it seemed desirable to ascertain whether there was a selective disease 
incidence by a method which would not appeal to statistics of the general population made for 
another purpose. The main point brought out is that the excess incidence of measles over 
scarlet fever in the persons recorded as fair-haired is much beyond the limits of any random 
sampling differences. 
Many other problems to which the method can be applied will occur to the biometrician, who 
is "in active practice." 
IX. On a Correction to be made to the Correlation Ratio »/• 
By KARL PEARSON, F.R.S. 
It is well known that as the square of the correlation ratio always involves the sum of the 
squares of differences of means, it must always take a positive value. Hence its mean value 
even, when its value for an indefinitely large population would be zero, must for finite samples be 
positive. Thus such values as -05 to -10 of ij may not denote small but significant values of the 
correlation ; they may denote solely the measure of rj's mean value for actual zero correlation. 
The observed value of rj ought to be compared with 
where rj is the mean value of rj for zero correlation and E n the probable error of j;. 2?, has been 
determined by me for any value of rj and = -67449/ViV, when there is no correlation*. 
Let n x be the total frequency of the population falling in the subrange, centred at x 9 , and 
let y Xfl De the mean value of all the ^-characters associated with this subgroup of %'s. Let y be 
the mean, and o- y the standard-deviation of all the y-characters. Then by definition : 
a ^Kpfe p -y) 2 } _o-,,; 
6(r >' ) - a,/ + da/ 
or, neglecting products of small quantities, 
(J,J 2 O-y 2 (Ty 2 
8 (o-.l/ 2 ) o <W 2 
- 5 T 5- . 
°v «v 
Now the variation of a-y 2 can either be positive or negative ; hence if we take the mean values 
of both sides of the above equation 
Mean S (o\„ 2 ) 
Mean S (j/ 2 ) = ~-^- L . 
Now if a M 2 be sufficiently large, tj fairly large, S (o\ v 2 ) may be either positive or negative with 
random sampling, but if »? be either zero or unity, the variation of a- M 2 must always be wholly 
* "On the General Theory of Skew Correlation and Non-linear Regression." Drapers' Company 
Research Memoirs, 1905, Dulau and Co. 
