1918] Waynick: A Statistical Study of Nitrification in Soil 263 



and since the standard deviation in this instance is 0.7 gram, hence 



.6745 X 0-7 

 y/n 



We can make Z? M of any dimension desired and since our method of 



determining nitrates allows of direct determinations only to 0.1 milli- 

 gram, we will use this number for the probable error of the mean of 

 the desired number of determinations so that 



.6745 X 0.70 

 ±0.10 = — = 

 \/n 



from which n = 22. It must be remembered, however, that the labora- 

 tory error increases as we decrease the number of determinations in 

 the same manner as the probable error of our sampling, so that this 

 increased error must be taken into account in estimating the number 

 of samples necessary to ensure any desired degree of accuracy. Refer- 

 ring to table 7. we find that the probable error of making the readings 

 on the colorimeter for twenty-five samples is .017 milligram or very 

 nearly .018 milligram for twenty-two samples, so that for a probable 

 error of 0.1 milligram we have 



.6745 X 0.1 3 

 ±0.1= = 



y » 



and n — 1. Thus twenty-three samples are sufficient so that the 

 probable error in the mean is 0.10 milligram. The taking into 

 account of the laboratory error involves an extra calculation, which. 

 for all practical purposes, may be avoided by the use of a table, such 

 as shown by table 12 (represented graphically in figure 2), which has 

 been calculated from the data given in tables 1, 2, 3, and 4, after 

 the manner already outlined. It will be noted that this table is of 

 limited range and accounts for numbers of samples less than eighty- 

 one, but may be extended for a range greater than the one given it' 

 desired. The approximate number of samples may be readily found 

 after the following manner. It is desired to determine the number of 

 ammonium sulfate samples necessary to be taken to ensure the same 

 degree of accuracy as for twenty-three residual nitrate samples, of which 

 the probable error of the mean was calculated to be ±0.10 milligram. 

 By reference to table 12. it is found that about eighty-one samples arc 

 required without taking the laboratory error into account. This error 

 is .009 milligram (table 7) for eighty-one samples, an amount less 

 than one per cent of the error which we are allowing for and hence 



