GROWTH AND VARIABILITY IN HELIANTHUS 
263 
The number of plants in the several classes is 12, 15, 17, and 14, and indi- 
cates a fairly good random distribution with a tendency to grouping in the 
middle classes. The average quartile position of each class is not far from 
the different mid-class values, a fact which may be taken as further evidence 
of random distribution. 
The facts concerning these plants will be brought out by consideration 
of the means and standard deviations of the positions. Taking the four 
average means at the bottom of table 14 and considering the number of 
plants as frequencies for these values, we find that their mean is 2.552 zb 
.101, and their standard deviation in class units is .8005 =t .0743. Since a 
class unit is .75 of a quartile the standard deviation in quartiles is found by 
multiplying this by .75, viz., .6004 zb .0557. 
The standard deviation in class units deserves further consideration. 
It measures the variation which the class units would have if, without 
reference to grouping, they were arranged in serial order, i. g., as i, 2, 3, 4. 
A simple algebraic calculation will show that the standard deviation of the 
first n numbers is 
In this case, n = 4., therefore, a = 1.12. Now it was found above that the 
actual standard deviation was .8005 =t -0743. This is somewhat below 
the theoretical standard deviation, even with the addition of three times its 
probable error. This relation may be taken to indicate that the variates 
under discussion (the mean quartile positions) are fairly evenly distributed 
in the several classes, but that they scatter less widely from the middle 
classes than would be expected upon the basis of pure chance. 
It now becomes possible to point out a fact of fundamental physiological 
importance, derived from these mathematical relations. Since the means 
of the quartile positions are so nearly distributed by the law of probability, 
it seems logical to conclude that the causes of this condition are also dis- 
tributed by the laws of chance. For example, in table 14, the first average, 
i-333» owes its position to the same cause which determined the position 
of any other average. In other words, the smaller relative size of these 
plants is an expression of the same definite cause as the relatively larger size 
of any of the other groups in this population. 
Perhaps it is well to call attention here to the fact that the discussion 
in the above paragraph relates strictly to the mean quartile distributions 
(average relative size) of the plants and not to the quartile distribution of 
the measurements of a plant through the grand period of growth. The 
latter values are clearly not determined by chance, since under that con- 
dition their standard deviation would be .1988 instead of .8005.^ 
It will be profitable to see if we can discuss the probable nature of the 
^ For the means of arriving at this vaUie, the reader is referred to Pearl and Surface 
(1915). PP- 151-153- Mention should be made that these writers have laid all subsequent 
students of this subject under obligation for their illuminating discussions. 
