juiyi, I920 Universality of Field Heterogeneity 281 



p 's of the same combination plot, C^, will furnish a measure (on the scale 

 of o to ± I ) of the heterogeneity of the field as expressed in capacity 

 for crop production. If this correlation be sensibly o (under conditions 

 such that spurious correlation is not introduced), the irregularities of 

 the field are not so great as to influence in the same direction the yields 

 of neighboring small plots. As heterogeneity becomes greater the cor- 

 relation will also increase. The value of the coefficient obtained will 

 depend somewhat upon the nature of the characters measured, some- 

 what upon the species grown, somewhat upon the size of the ultimate 

 and combination plots, and to some degree upon the form of the combina- 

 tion plots. 



Knowledge of the values of the correlations to be expected must be 

 obtained empirically. 



Let 5 indicate summation for all the ultimate or combination plots of 

 the field under consideration, as may be indicated by C„ or p. Let 

 p be the average yield of the ultimate plots and o-p their variability, and 

 let n be constant throughout the m combination plots. Using the for- 

 mulae of an earlier memoir (j) in a notation which is as much sim- 

 plified as possible for the special purposes of this discussion, 



{[S{C^')-S{p')] lm[nin-i)]] -p' 



\''2 



PiP, 



This formula assumes the combination plots to be of uniform size — 

 that is, to contain each the same number, n, of ultimate plots. It may 

 be desirable or necessary to have some of the combination plots smaller 

 than the others. 



Such cases are frequently met in practical work. For example, the 

 wheat field of Mercer and Hall is laid out in a 20 by 25 fold manner. 

 This permits only 2 by 5, 4 by 5, or 5 by 5 combinations of the same size 

 throughout. One of Montgomery's experiments with wheat covered 

 an area of 16 by 14 plots which may be combined in only 2 by 2 or 4 by 2 

 fold groupings to obtain equal areas suitable for calculation. In each of 

 these cases other groupings are desirable. 



The formulae are quite applicable to such cases; the arithmetical 

 routine is merely a little longer. The formula is as above, but p and 

 o-p are obtained by a (w-i)-fold weighting of the plots,^ where n is the 

 variable number of ultimate plots in the combination plot to which any 

 p may be assigned — that is, 



p = S[(n-i)p]/S[n(n-i)], 



S[{n-i)p^] /S[in-i)p]Y 



2_ 



P S[n{n-i)] 



/ S[in-i)p] V 

 \S[n{n-i)]J 



1 That is, each ultimate plot is multiplied by the number less one of the plots in the combination plot to 

 which it is assigned. 



