354 



bers in the series and treat these departures according to the following 

 formula : 



Index of variability of the plats equals 



I ,-) op , / Sum of all the (Departures)^ 

 \ Number of departures less 1, 

 which formula may be interpreted as meaning that from the squares 

 of the departures added together and divided by the number of plats 

 less 1 we derive an index called the '' probable uncertainty of 1 meas- 

 ure," or " the probable variability of 1 plat as compared with all the 

 plats of the series." Again, knowing this uncertainty of any one 

 measure, we find the " probable uncertaint}^ of the average of n meas- 

 ures " by the following formula : 



Index 

 Probable nncertaintv of the average = ± — 7^. 



vn 



This latter formula is to be interpreted as meaning that there is 

 an even chance that the computed average is too large or too small 

 by this probable uncertainty. Applying these principles to the meas- 

 ures of plats C and E, I obtain the figures 31.3 and 22.9 as the indices 

 of variability and 8.33 and 5.26 as the probable errors of the two 

 averages. That is to say, so far as any internal evidence is given 

 l\y the discrepancies between the measurements of the plats them- 

 selves, there is an even chance that the crop from a plat in series C is 

 betw^een the limits 212.9 and 196.3 or outside of these limits; simi- 

 larly, for series E there is an even chance that the crop from any 

 plat is within the limits 188.9 and 177.4 or outside of these limits. 

 But the numbers within each of these two series overlap each other so 

 much that it is perfectly possible that if we could increase the number 

 of plats in each series sufficiently, all other conditions remaining the 

 same, we should eventually arrive at very nearly the same average 

 value for each. In other words, the mere difference of the two aver- 

 ages 201.6 and 182.7 is no evidence that in this particular case there 

 was any important constant difference between the plats of series C 

 and those of series E, but that, on the contrary, unknown sources of 

 influence are at w^ork in each series and in all the plats that are more 

 important than any that were thought of when the experimenter 

 endeavored to make these 36 plats perfect duplicates of each other. 



Professor Plumb shows that this difference did not depend upon the 

 previous crops or treatment of the j^lats during the previous five 

 years. It certainly did not depend on the meteorological climate, 

 the mechanical condition of the soil, nor on the seeds, nor on injury 

 by insects and animals. We may possibly fiiKl a partial explanation 

 in the irregular distribution of microbic life in the soil, but it is 

 more likely that it depended upon the inherent variability of the 



