8 BULLETIN 1821, U. S. DEPARTMENT OF AGRICULTURE 
In the correlations between the different ear characters the product 
moments and the standard deviations of both variables were affected 
by the fluctuation from season to season. 
The following corrections were applied to eliminate these effects 
of heterogeneity: - 
The standard deviations of the ear characters as determined from — 
the mingled records were adjusted according to equation 1 to obtain 
the weighted means of the squared annual standard deviations. The 
square roots of these values were considered to be the true average 
standard deviations for the period with the effect of the fluctuation 
in the seasonal means eliminated. They are the only standard 
deviations reported for the ear characters and were used in computing 
all of the reported coefficients of variation and correlation. The 
standard deviations as obtained directly from the correlation tables 
were used in computing the probable errors of the means, however, 
as in this connection variation from season to season must be con- 
sidered. 
The product moments for yield with the other variables were not 
affected by heterogeneity and were used as determined from the 
tabies. The product moments for the pairs of associated ear charac- 
ters were adjusted according to equation 2 to obtain the weighted 
means of the annual product moments, and these were used in com- 
puting all coefficients of correlation between ear characters. This 
method of correcting for heterogeneity is illustrated in connection 
with Table 4. 
The coefficients of correlation reported here express, then, the ratio 
of the average of the annual product moments of the deviations of 
two associated variables over a series of years to the product of their 
average standard deviations for the same period, both averages being 
properly weighted. Such a coefficient measures the true average 
correlation of the variables under all of the varying conditions that 
obtained during the different seasons. It is purely a function of the 
variation of the two variables within the itciont seasons and is 
unaffected in any way by the fluctuation of the annual means. 
The regression equations based upon these average correlations give 
the best measure for the deviations of one variable in terms of the 
deviations of the other if written in terms of deviation. They may 
be converted to absolute values, however, only by using the indi- 
vidual annual means in the conversion, giving rise to a different 
regression equation in absolute values for each season. 
In discussing this method, Dr. Sewall Wright kindly called the 
writers’ attention to his paper (10), in which the same coefficient 
was derived from a somewhat different point of view. In Doctor 
Wright’s proof the average of the individual coefficients was treated 
as the correct measure of correlation, and the coefficient as used here 
was shown to approximate that average when variation in the indi- 
vidual coefficients was no greater than may be expected in random 
sampling. Here the coefficients reported are considered as the correct 
measure of correlation, which the average of the annual coefficients - 
will apprommate when they fluctuate no more than may be expected 
in random sampling. 
