VARIATION— GENERAL RESULTS. 



23 



The chief constants, both physical and algebraical, of the distribu- 

 tions are given in table 4. The unit is 1 leaf. 



Table 4. —Analytical constants for variation in leaf-number. 



Constant. 



Total frequency 



/*2 



Ms 



Mi 



i3i 



VPi 



^2 



/3,-3 



Series I, 

 II, and 

 III com- 

 bined 

 (Distribu- 

 tiou No. 

 177). 



2328 

 1.5888 

 -1.0503 

 6.5040 

 .2793 

 .5285 

 2.6028 

 - .3972 

 -1.6323 



Plant 2 of 

 Series IV 

 (Distribu- 

 tion No.71), 



922 

 1.1747 



- .8331 

 4.4627 



.4281 



.6543 



3.2341 



.2341 



— .8163 



Constant. 



Skewness 



Modal divergence 

 Standard deviation 



Mean 



Mode 



Modal frequency. . 



Range 



Lower end of range 

 Upper end of range 



Series I, 

 II, and 

 III com- 

 bined 

 (Distribu- 

 tion No. 

 177.) 



—0.6332 



— .7961 



1.2573 



8.6465 



9.4425 



704.58 



6.5028 



4.2547 



10.7575 



Plant 2 of 

 Series IV. 

 (Distribu- 

 tion No. 74) 



—0.4432 



— .4804 



1.0838 



8.7581 



9.2385 



344.90 



9.8186 



1.1408 



10.9594 



The fact that the criterion «, {= 2/3, — 3A — 6) is in both cases nega- 

 tive indicates that curves of Type I are demanded. The equations to 

 the curves for the two distributions are as follows: 



Series I, II, and III: 



2y-704.5833 

 Series IV, plant 2: 



V^^ 5.1878/ V^ 1.3150/ 



.5794 



1.9760 



.7209/ 



The histograms and their fitted curves are shown in figs. 4 and 5. 



From both the constants of table 4 and the curves themselves it is 

 clear that the distributions deviate very far from normality. There is 

 no doubt that both the kurtosis (cf . Pearson, :05) and the asymmetry of 

 the curves are significant. In both cases the skewness is negative, the 

 mode lying above the mean. The amount of the modal divergence is 

 considerably greater in the case of the first curve (Series I, II, and III 

 combined) amounting there to more than three-fourths of a leaf. Since 

 frequency curves of Type I are of limited range, we have given for these 

 two curves the theoretical range of variation. It is of interest to com- 

 pare this with the observed range. In the Carp Lake race the observed 

 range is between 5 and 12 leaves, inclusive. The theoretical range is 

 between 4.25 and 10.76 leaves, a total of 6.51, as against the observed 7. 

 Thus the total theoretical range is only about a half leaf different from 



