ee WA ER 
SEED-EAR CHARACTERS AND PRODUCTIVENESS IN CORN 7 
The coefficients of partial correlation of the first order were com- 
puted according to the well-known formula— 
) 1 Sy = SG Pans Fach ce 
) aed V1 —riovl —Tg¢ 
The values of 1 —r? were taken from the tables prepared by Miner (4). 
The partial .coefficients of higher orders and the coefficients of 
multiple correlation were computed by the methods described by 
Tolley and Ezekiel (8). 
- CORRECTION FOR HETEROGENEITY OF DATA 
Let X be the mean of a variable for a series of years in which 
X,, X,, ... X, are the annual means. Similarly, let oxq be the 
“total”? standard deviation, as determined from the mingled records 
of all seasons, and cx be the square root of the weighted mean of the 
squared annual standard deviations. Then, 
7 Y2 7. ye 72 
o x(t) ed = 5s (1) 
in which N,, N,, ... N, are the numbers of records in the individual 
seasons and JN is the total population. The quantity within the 
brackets will be 0 only when X,, X,, ... X, are identical, in which 
case o%y)=c%. If the annual means are not identical the quantity 
within the brackets will be positive, and o} q) will be larger than «2? by 
this amount. 
Using a similar notation, let p,,(,) be the mean product moment 
of the deviations of X with those of an iSuecinted! variable, Y, as 
determined from the mingled records for all seasons, and let p,, be 
the weighted mean of the annual mean product moments. Then, 
[Mx EN)XY, bi: N,Xa¥ 
= Avy Dore? aon 
Pry (t) — “<8 = Pry (2) 
The quantity within the brackets will be 0 if either X,, X,, ... X, 
or Y,, Y,, ... Y, are identical, or both, or if the fluctuations of the 
annual means are such as to bring about equality accidentally. In 
the absence of these conditions the quantity within the brackets 
will have either a positive or negative value, and p,,() will differ 
from p,, in this amount. : 
The coefficient of correlation between two variables as computed 
from the mingled records for a series of years is affected by one or 
both of the above propositions, depending upon whether the means of 
one or both variables remain constant. In the correlations between 
yield and the different ear characters the mean annual yield remained 
constant at 100 per cent, whereas the means of the ear characters 
changed from season to season. Consequently, the standard deyia- 
tion of yield and the product moments for yield with the ear char- 
acters were correct as determined from the mingled records, but the 
_ standard deviations of the ear characters were unduly large and the 
coefficients of correlation correspondingly too small. 
