248 University of California Publications in Agricultural Sciences [Vol.3 



coefficient of variability of the amount of nitrate as found in the field 

 soil is high, no less than 25.9 ± 2.1 per cent in the surface six inches 

 and 51.4 ± 3.3 per cent in the vertical section from six to twenty-four 

 inches. The range, therefore, within which two-thirds of the deter- 

 minations may be expected to fall is from 2.0 to 3.4 milligrams in the 

 surface soil and from 0.3 to 1.1 milligram in the subsoil. The extreme 

 range is, of course, much greater than this, but the bulk of the deter- 

 minations fall within the limits given. 



A single determination, or the mean of a series of determinations 

 can never be an absolute value and hence before any large degree of 

 confidence can be placed in any experimental result, its degree of 

 reliability must be known. The reliability of any determination is 

 expressed by the probable error (E) of the determination. This 

 figure is of such a magnitude that the probability of making an error 

 greater than it is equal to the probability of making an error less than 

 it, both probabilities being one-half. 



The probable error (E s ) of a single determination is calculated 

 by the formula 



E s = ±.614:5 X -7= 

 y/n 



where o- is the standard deviation, as given above, and n the number of 



determinations. Since with a single determination the V w is equal to 1 



#, = ±.6745 Xo-. 



or in other words, the probable error of a single variant is equal to 



approximately two-thirds of the standard deviation of the series in 



which it lies. The probable error of the mean (E^). is given by the 



formula 



Es .6745 X o- 



E M will vary as the square root of the number of determinations and 

 thus decreases but slowly as we increase the number of determinations. 

 The probable error (Eo-) of the standard deviation is calculated from 

 the formula 



Eo-=±. 614:5—^= 

 and for the coefficient of variability : 



Ec = .6745 — = ' 



[i+»(as)T 



\/2w 



when C.V. is greater than ten per cent as is the case with the results 

 recorded below. 



