4 BULLETIN 1468, U. S. DEPARTMENT OF AGRICULTURE 
lems remain unsolved the efforts have resulted in the accumulation of 
a large mass of data which can be utilized for an analysis of linkage. 
The data available as to the linkage relations of these factors 
show that stability of crossover rates is no more to be expected than 
stability in other biologic processes. Almost any possible grouping 
of the data brings to light significant differences in the rate of cross- 
ing over, if the numbers involved are sufficiently large, and it is the 
purpose of the writers to consider some of these differences and their 
causes. 
The rate of crossing over in the formation of gametes must be in- 
ferred from zygotic ratios, and observed changes may be due (1) 
to changes in the proportion of crossover gametes formed, (2) to a 
differential death rate of crossover gametes, (3) to selective 
fertilization, or (4) to a differential death rate of zygotes. 
Strictly speaking, changes other than those specified in 1 are not 
changes in the rate of crossing over. Both 1 and 2 are gametic and 
there is little hope of being able to distinguish between them except 
through refinements of cytological methods. 
It is not a simple matter to make even the general distinction be- 
tween gametic and zygotic variation in the rate of crossing over. 
Selective fertilization as a source of variation is .eliminated when the 
crossing over is measured by back crosses of which one parent has 
the gametes all of one kind in respect to the characters under 
consideration. 
A differential death rate of zygotic classes may be detected by a 
comparison of linkage and repulsion values. But to do this the 
comparison must be made between distinct progenies, and the experi- 
ments show that sister progenies often differ significantly in crossing 
over. 
Another method of detecting differential death rates of zygotic 
classes is to cross two individuals having the gametic compositions 
A a b b and a a B b. This will yield a population with all four 
classes and no opportunity for linkage. Here again individual 
diversity precludes the possibility of generalization. With prog- 
enies involving the same two characters there are some that show 
significant departures from equality of the four classes, while in 
closely related progenies no difference appears. 
In view of the variability of the crossover ratio it was thought to 
be desirable to consider the factors that were associated with this 
instability before considering the conditions that bring about changes 
in the rate. Accordingly, in the present bulletin the data are first 
scrutinized from the standpoint of variability and then considered 
from the standpoint of the factors associated with changes in the 
rate. 
STATISTICAL TREATMENT 4 
In dealing with variations and differences in the percentages of 
characters or of crossing over, the significance assigned to the results 
* Explanation of Statistical Symbols.— N= number of subgroups, rc=number of seeds, if=mean, 
M p =mean percentage, PE= probable error, D/E= difference divided by the probable error, i?d=probable 
error of a difference, p=percentage, g=100 minus p, Q=Yule's coefficient of association, r=oroduct mo- 
ment coefficient of correlation, S=summation, o-=standard deviation, o-j=standard deviation of x array, 
<r 2 =standard deviation of y array, o- : _ 2 =standard deviation of x—y array, <r 50 =standard deviation corrected 
to 50 per cent, a 60P = standard deviation corrected to 50 per cent and for size of population, H= harmonic 
mean, Xl =0.67449/ ^ 
