182 
Miscellanea 
Thus if n variates are equally correlated (e) among themselves, and equally correlated (p) with 
another variable, we shall not indefinitely increase the accuracy with which the last variable will 
be predicted from the others by increasing indefinitely the number of the variates n. 
Illustration. The c(3efiicient of multiple correlation is I'equired as we increase the number of 
brothers from whom a prediction of a character in a given brother is made. The fraternal 
correlation =-5. 
I- of Brothers 
1 
•5000 
2 
■5774 
3 
•6124 
4 
•6325 
5 
•6455 
6 
•6547 
10 
•6742 
00 
•7071 
Comijare against these results two parents only in a population where there is no assortative 
mating and the parental correlation = -5. Here e = 0, p=-b and ?i = 2, .•. ^=i^2 = -7071, or two 
parents will give more information than 10 brothers and sisters, and as much in fact as an 
indefinite number. Suppose the parents tend to select their like, i.e. suppose there is assor- 
tative mating in the population, .say, f=^15, then with the same inten.sity of parental correlation 
/;=^6594, 
or, two parents will give us more information than six brothers and sisters. 
Now this illustration brings out the real nature of the effect of increasing the number of 
variables from whicli we predict. Such increase has very little value, if those variables are 
fairly highly correlated with each other. To be effective they must be highly correlated with 
the variate we wish to predict and correlated very slightly with each other. 
Even in this case there is a limit to the degree of correlation reached when the number of 
variates is indefinitely increased, namely p/\/f, and it is clear that if p be small and t fairly large, 
no very great increase of correlation is obtained if we use an indefinitely great number of variates. 
For example if p=^05 and e = ^5, we find ^^ = -0707 only. Even if p were '10, we should only 
raise R to ^1414, could we predict from an indefinitely large number of such correlated variates*. 
Indeed as long as e is not less than p we gain singularly little by combining large numbers of 
variates. For example if p were '4, and f = ^4 ten such variates would only raise the correlation to 
•5898, and an indefinitely large number to •6325, which is less than double the single correlation. 
Yet there are apparently many persons who believe that by taking a number of low correlations, 
a high relationship can be reached ! 
Actually there is a limit to what relations can possibly exist between a variate .r,, and a series 
of equally correlated variables ... .^'„ . Since R must be less than unity, we have 
np^ — 1 
<1, 
1 
Thus if »=10and p = ^5, t must be >^1667. Or, it would be impossible for 10 variates to 
have a correlation '5 with another variable, and a zero correlation with each othei-. 
* Even if p were •lO and e as low as "10 we should not raise R for endless variates of this order of 
correlation above •31&S, while from compounding ten such variates we should only obtain a correlation 
about double that of a single variate, i.e. J? = ■2294. 
