Miscellanea 
111 
general population, or there is no variahility in the sex-ratio due to individuality. All the 
variation observed is actually due to the random sampling involved in taking small families. 
It appears to iiio therefore that Mr Cobb's note is of \'akie, as it enables Mr Heron to confirm 
his results from another standpoint. For practical purposes the main problem is : Are the 
actual sex-ratios in parent and offspring related 1 The answer is : Not sensibly. Mr Cobb 
suggests that this result is only due to the apparent family not rej^resenting the gametic reality. 
This raised the question of whether any individuality at all exists for the sex-ratio. The 
answer appears to be none, because the standard deviation observed is precisely that which 
would be found were the sex-ratio for any individual that due to a random selection from the 
general population of a group equal in number to his family. 
VI. On the Influence of Double Selection on the Variation 
and Correlation of two Characters. 
By KARL PEARSON, F.R.S. 
In a memoir published in the Phil. Trans. Vol. 200 A, pp. 1— G6, I have dealt with the 
problem of the influence of selecting q characters on the variability and correlation of n-q 
non-selected characters*. My present problem is somewhat different, and as the solution 
provided has been in use, and given in lectures for some years past, it may be desirable to 
publish it. 
There are two correlated variables, 1 and 2, with standard deviations o-i and a-, and correlation 
/•j.). In entering up a record a selection is made of the variable 1, so that its mean is shifted to 
«ij and its standard deviation to .Sj ; let y.^ = SijcT^. In entering for the variable 2, a .selection is 
made so that its mean is shifted to m-i and its standard deviation is changed to s., ; let fio^-sola-i. 
As illustration suppose cards formed of characters in parent and offspring ; we pass through the 
cards, putting a red line through every card not one of the selected parents ; we pass through 
them again and put a blue line through every card not one of the selected offspring; the result 
is that cards may be thrown out because they carry a red line or a blue line or because they 
carry both. The problem is what are the standard deviations 2i and "S,-, and the correlation Ry^ 
of the material left aftei' this double selection. If we assume the material normal the following 
values are readily foUnd by considering the probability of selecting a definite individual pair 
Xi , x-i to be of the form 
const, xe '^-''vrX'^i i^ia-i <^i^J x- 
X 
Thus the correlation surface is 
: const. X e 
2,2 SjSo 2./ 
* It may be of interest to note that the whole of the formulae developed in that memoir are true far 
beyond the range of the Gaussian distributions for which they were proved. They are universally true 
provided we start with a generalised notion of correlation as involving the maximum dependence of one 
variable on an arbitrary linear function of (» - 1) other variables. 
