INTERPRETING THE VARIATION OF PLAT YIELDS. 31 



plat, a total variation of 160 pounds, or 55.7 per cent of the mean yield. It will be 

 noted that a high extreme range is closely associated with a high probable error. 



(4) This item includes the mean of the deviations from the mean yield of each group 

 expressed as a percentage of the mean yield. It will be noted that this is closely 

 associated with and slightly greater than the probable error of a single result in each 

 case. 



(5) The probable error of a single result is first given in terms of yield. The prob- 

 able error of ±23.9 pounds in the case of group A means that if the yield of any plat 

 in the group is taken at random the chances are even that the yield of such plat will 

 be within 23.9 pounds of the mean yield of the group. It should be noted that the 

 figure expressing the probable error is always plus or minus. Hence, while a result 

 may vary from the mean by less than the probable error, it may be either greater 

 or less than the mean. 



(6) In order to facilitate comparisons the probable error is here stated as a percentage 

 of the mean yield. The probable error in group A is ±23.9 pounds of oats per plat, 

 or 8.3 per cent of the mean yield. The probable error of ±49 pounds of alfalfa per plat 

 in group E is 8.6 per cent of the mean yield of that group. When reduced to a per- 

 centage basis the probable error of group A and that of group E are seen to be very 

 nearly the same. The probable error of group C and that of group D are also extremely 

 close. This fact is of special interest, because the fields used were the same in both 

 cases — fields B II and B III, Huntley — while the crops were different and were, 

 of course, produced in different seasons. 



So far, the probable error of a single result, the yield of a single plat, has been con- 

 sidered. When the average of several results is taken, the probable error of that 

 average is, of course, less than the probable error of one result. In other words, the 

 average yield of 10 plats is likely to represent the mean yield of the group more nearly 

 than does the yield of any single plat taken at random. The average yield of any 10 

 plats taken at random in group A will have a probable error of ±7.59 pounds per plat, 

 or 2.6 per cent of the mean yield, as compared with 8.3 per cent of the mean yield, 

 the probable error of a single result. If, then, the number of results averaged be 

 increased to 20, the probable error is still smaller — 1.9 per cent of the mean yield in 

 group A ; and the probable error of the average grows continually smaller as the num- 

 ber of results averaged is increased, as is shown in the table. The fact that the prob- 

 able error of the average of several results is less than that of a single result empha- 

 sizes the desirability of conducting field experiments in multiplicate. The probable 

 error of an average can be obtained by dividing the probable error of a single result 

 by the square root of the number of results averaged. 



(7) Under this heading are given the number of plat yields differing from the mean 

 by more or by less than the probable error or one of its multiples. Under a and b is 

 given the number of results which, theoretically, should differ from the mean by 

 more or by less than the probable error, as well as the actual number found. Theo- 

 retically, half the results should differ from the mean by less and half by more than 

 the probable error. The table shows that the actual results are in the main reasona- 

 bly close to the theoretical requirements. They are as close as can be expected 

 where such small numbers of results are considered. Under d, e, and fit is seen that 

 comparatively few plat yields differ from the mean by more than twice the probable 

 error, still fewer by more than three times, and none by more than four times the 

 probable error. Hence, it may be expected that on these fields accidental variations 

 as great as four times the probable error are not likely to occur; so that when different 

 treatments are applied and variations greater than four times the probable error are 

 obtained, such variations can be attributed to the treatments with a high degree of 

 certainty. 



[Cir. 109] 



