Miscellanea 
109 
IV. The Effect of Errors of Observation upon the 
Correlation Coefficient. 
By J. A. COBB. 
The correlation coefficient r= ^^'^^^ , where the vakies of x and y are unaffected by errors of 
observation. If we assume all the .^'s and y's to be subject to errors a, /3, with standard 
deviations a^, o-y, the errors of and 1/ in every pair being indeijeudent, ^^''"''^^ will become 
iV O"^ (Ty 
So ^ - 
r 
In the special case where <Tx=<t,i and o-j^^ — (r„j, ^ becomes — — ^. The correctness of 
determinations of correlation coefficients has often been considerably affected by the neglect of 
this point. The question of the inheritance of sex-ratio is an interesting one in this connexion. 
The sex-ratio of an individual is the value which the ratio of male to total births in his 
fraternity would tend to assume if his fraternity were infinitely large. The standard deviation 
of error of sex-ratio due to taking small families is approximately \/~~ i where wi is the 
number of the family and p, q the ratios of male and female respectively to the whole in all the 
families. The total standard deviation of the sex-ratio in families of the same size is due to the 
deviation ^^^^ *° ""x; the variability of the sex-ratio of families. Therefore 
where ctq is the total observed standard deviation. I have in this way calculated o-^ from various 
data, and it turned out to be between '03 and '04. Now in the case of families of nine children 
— = •028. So if the observed sex-proportion is used in forming a correlation table between the 
sex-ratios of parents and ofifspring in families of nine, the resulting correlation coefficient must 
2 , M 
be multiplied by r— ■ = ^--^^L^T^r—=24 to give the true correlation between the sex-ratios 
a-/ (•035)'' 
of parents and ofifspring. 
V. On Heredity in Sex. Remarks on Mr Cobb's Note. 
By KARL PEARSON, F.R.S. 
Mr Cobb's point is an important one, though as a statistician I am never convinced by 
unpublished statistics which are said to give a certain result. But Mr Cobb's criticism would 
I think apply to all correlation treatment of inheritance. If we take the male offspring of a 
father of definite arm-length, their mean based upon four or five cases may differ very widely 
from the mean of all the offspring the father would have, if we supposed him capable of an 
indefinite number of ofl'spring. In the same way he himself is only a single random sample 
from an indefinitely great array which might theoretically be attributed to his parents. I do 
