2 
Fraternal and Parental Correlation Coefficients 
fraternal correlation coefificient calculated from only two siblings of each family and 
for a jmrental correlation coefficient when only one offspring value from each 
fixmily enters into the calculation. When the material of observation, as is usually 
the case in investigations of inheritance in higher mammals, consists of families of 
varying size, and correlation tables are used in which the same weight is given to 
each observed pair of siblings or pair of parent and oftspring, without regard to the 
size of the family, a rational treatment of the probable error is excluded at the 
outset. With material in hand which makes it possible to examine numerous 
siblings, it is most reasonable to confine the investigation to a constant number of 
offspring from each fixmily. In this case the deduction of formulae for the standard 
deviations of the two correlation coefficients does not present special difficulties, and 
this problem will be solved here. 
We shall suppose that each group of q siblings belongs to the same litter or 
that from other reasons their order of birth is indifferent. Then each pair of 
siblings or each pair of parent and offspring ought to take a like part in the 
calculation, and q siblings give rise to \q{(i — \) pair of brothers and q pair of 
parent and offspring which all of them are entered in the calculation. 
The fraternal correlation can thus be calculated either fi-om a correlation table 
which is made symmetrical so that it contains q{q—\) entries from each fraternity 
or by the formula quoted p. 10 which gives an identical result. 
I. Fraternal Correlation. 
Although this investigation aims especially at fraternal correlation it concerns 
of course other calculations of correlation in which the material consists of classes 
of equal size inside which the individuals are mutually correlated, all of them 
forming like parts. In the following we shall thei'efore name a group of siblings 
a class. 
Suppose we have a material consisting of q individuals from each of n classes 
inside which the individuals are correlated while individuals from different classes 
are uncorrelated. We can then consider such a material as one of many possible 
samples of the same nature and size drawn from a population consisting of classes 
of individuals correlated as mentioned. It is therefore possible to face the problem 
of finding the law of errors for the mean value, the standard deviation of the 
character concerned and further for the correlation coefficient inside a class, supposing 
tliat these are all calculated from a sample like the one now considered. 
Let the sample be 7/1, y.,, y.,^... y,„j with mean value y and standard deviation a. 
No special notation will be introduced for individuals of the same class, but summa- 
tion of products is indicated by S when all factors of the product belong to the 
same class, and by S when factors of the same pioduct belong to two or more 
classes. The summati(ins always extend to all n classes. 
