GROWTH AND VARIABILITY IN HELIANTHUS 
quartile would be dispersed only in one direction, viz., upward, while those 
which started in the second quartile might be dispersed either upward or 
downward on the scale. Be that as it may, however, the plants which 
started in the second quartile showed a greater tendency to deviate toward 
a higher quartile than toward the lower one, and those starting in the third 
quartile showed a greater tendency to deviate toward a lower than toward a 
higher quartile. 
Tables 5 and 6 show the quartile positions at successive observations- 
of plants which started in the third and fourth quartiles respectively. It 
will economize time and space, however, to bring the observations on all 
the distributions together in one table and there to discuss the characteristics 
of the several quartiles. 
Table 7 presents a summary of the observations on the quartile distri- 
butions of all the plants starting in the various quartiles. It shows, for 
the plants which started in the various quartiles, the number and percentage 
of observations falling in the several quartiles at successive dates. The 
first line shows the number and percentage of the observations falling in 
the different quartiles for the plants which started in the first quartile, and 
so down the table. The table brings out still more plainly the tendency 
for a group of plants to remain in or near that quartile in which they started. 
By use of a method of comparison which has been given by Pearl and 
Surface (1915), it is possible to obtain a concrete, definite, quantitative 
expression of this tendency for plants to retain the quartile position with 
which they started. 
The measure of this tendency calls for the consideration of certain 
aspects of the theory of probability. If nothing but pure chance were 
operating to determine the quartile positions of these plants, it is evident 
that they would fall in one quartile as often as in another. The results 
would, consequently, be comparable to those obtained by making an equal 
number of throws of four-sided dice. After the allocation of the plants to 
quartiles, they were measured ten times, consequently there were 150 
observations on the quartile positions of plants starting in the first and third 
quartiles and 140 observations on those starting in the second and fourth 
quartiles. It will be necessary, first, to consider what the results would 
have been if nothing but pure chance were operating to determine the 
distribution of the plants. It is well known that the chances of success 
of an event where p = the chance of success and q = the chance of failure, 
are p-q = I. In this case, the chance of success would be very close to 34 
and that of failure to 
The theoretical standard deviation of the percentage of successes in n 
events would be o- = -ylp-q/n which will give the standard deviation from 
the theoretical mean percentage, provided we are dealing with a purely 
chance distribution, such as throwing four-sided dice. 
The standard deviation of the theoretical mean for quartile I is, there- 
fore, 
