J. Davidson 143 



Since only five infections were made on each of the varieties tested 

 it is clear that only the roughest estimate of the standard deviation can 

 be based upon the data for a single variety. For a standard deviation 

 estimated from a sample of five is not only subject to very large errors 

 of sampling, but is distributed in a markedly skew manner ; on the other 

 hand, the causes of variation, whatever they may be, must be closely- 

 analogous in all the varieties tested; and this fact should enable us to 

 make use of the information supplied by the whole of the material, to 

 estimate with some accuracy the probable error to be ascribed to each 

 value. 



The process of obtaining a single probable error from the deviations 

 of a number of distinct groups has been applied successfully in cases 

 Avhere it may be assumed that the groups are equally variable ; as is the 

 case when a correlation ratio is determined on the assumption of the 

 equal variability of the arrays. In the case of the infestation numbers 

 no such assumption is a priori plausible, for setting aside the lowest 

 mean which is evidently exceptional, the 18 means range from 286 to 

 1037. Moreover, an inspection of the figures for the individual plants 

 shows that the higher numbers are, as is to be expected, actually the 

 more variable. There exists, however, a class of distribution, of which 

 the Poisson Series is the classical example, in which the variance is 

 proportional to the mean. 



To test whether this is the case with the infestation numbers, the 

 quantities 



1 W/X2 



X 



were calculated for each variety, where x is the mean and ^2 the second 

 moment of each sample; n = 5 in every case but two, for which only 

 four counts Avere available. In these cases the quantity was increased 

 by one-third, and used with the others to obtain the aggregate. Adding 

 the 18 quantities obtained from the several varieties, and dividing by 

 18 {n — 1), we find that on the average the variance is nearly 28-5 times 

 the mean. If now the variance is proportional to the mean we have for 

 each variety „ ^o r- 



and the probable error is thence calculated with 18 times the weight 



with which it would be calculated from a single variety. 



A precise test of the accuracy of the above assumption is afforded by 



the distribution of 



nji2 



a" 



. 10—2 



