258 H. S. REED 
(7 
and for quartile II is 
(7 
and similarly for the other quartiles as shown in the next to the last column 
in table 7. 
A range of ± 30- from the theoretical mean ought to include all, or 
nearly all, of the observations; therefore there should be few if any obser- 
vations above 35 percent or below 15 percent if the distributions were 
governed by pure chance. An examination of table 7 shows that, as a 
matter of fact, the percentages of observations falling within a given quartile 
do deviate more widely than the limits to be expected under the laws of 
pure chance. As in the case of the maize plants studied by Pearl and 
Surface (19 15), the greatest positive deviation© were in that quartile in 
which the sunflower plants started, and the greatest negative deviations 
were in the quartile farthest from that in which the plants started. Further- 
more, it will appear that plants starting in quartiles I and IV deviated more 
widely from their respective theoretical means than plants starting in 
quartiles II and III deviated from their means. From this it appears that 
plants which start as small, medium, or large individuals in a population, 
though they vary more or less, tend to stay in or near the class in which they 
started. 
We may next attempt to get a concrete expression of the observed 
deviations which have been discussed above in a general way. We shall 
find that the root-mean-square deviation of the observed percentage values 
(or their actual standard deviation) from the theoretical mean is an excellent 
measure of their tendency to deviate from the theoretical results of simple 
sampling, and that it is a measure of those influences, other than chance, 
which determine the locus of the given group of plants in the population, 
because it shows how widely the plants starting in any given quartile 
depart from the position which would have been most probable according 
to the laws of pure chance. 
We may proceed' to compare the theoretical standard deviation of the 
mean with the observed deviations of the percentage from these theoretical 
mean percentages. This is conveniently done by obtaining the actual 
standard deviation from the theoretical mean for the plants, starting in the 
various quartiles. The deviations of the several percentages from their 
theoretical mean percentages are obtained and each is squared, then the 
actual standard deviation equals ^lUd^jn. The values are shown in the 
last column of table 7. 
Inspection of the results shows at once that the actual standard devia- 
tions are considerably greater than the theoretical. This can only indicate 
X 75>o6 
50 
= 3.533, 
= /^4.56 X 7544 = ^^^38, 
\ 140 
