IOWA ACADEMY OF SCIENCE 
93 
larger number of variates in this study would have changed the 
secondary modes at 19 and 21-22 to a single mode at 20-21. 
The Rudbeckia hirta heads from Ames do not have the mode at 
19, and the mode at 8 is not so prominent. The heads from Center 
Point show a secondary mode at 7 only, but this irregularity is 
undoubtedly due to the small number of heads counted. The 
curve for the heads from Hedrick very closely resembles that for 
the total number of heads. 
The curve for Rudbeckia triloba is very regular and smooth, 
with a mode at 8. 
The curve for Helianthus grosse-serratus is somewhat irregular. 
The characteristic mode is at 18, with a small secondary mode at 
21. It is probable that the secondary mode would disappear if 
more variates were counted. It does disappear if the classes 
are doubled, as is shown by Table No. 8 and Plate IX. In that 
TABLE No. 3. 
Showing the Count for Helianthus Grosse-serratus, With the Classes 
Doubled. 
Magnitudes 7 8-9 10-11 12-13 14-15 16-17 18-19 20-21 22-23 24-25 
Frequencies... 1 14 100 494 402 144 73 60 7 1 
case, the mode is at 12-13. The curve for the heads from Council 
Bluffs has but one mode, that at 13. So few heads came from 
there, though, that the curve has but little significance. Because 
nearly all of the H. grosse-serratus heads came from Ames, the 
curve for that place is very similar to the curve for the total 
number of heads, having a characteristic mode at 13, and a sec- 
ondary mode at 21. By doubling the classes, this secondary mode 
may be made to disappear, also. 
The formulae for the computation of all the constants were 
taken from “Statistical Methods with Special Reference to Bio- 
logical Variation,” by Davenport, '04. Great care has been taken 
that no mathematical errors enter into the work. To this end, 
all of the calculations were carried out to five decimal places, al- 
though only four are given in this paper. As far as possible, all 
the work was very carefully checked. 
The following formulae have been used: 
A = S(v_ L J)_ 
n 
where A = the mean or average, s = sign of summation, V = the magni- 
tude of any class, f = the frequency, or number of varieties in that class, and 
n = the total number of varieties. 
