10 



BULLETIN 529, U. . S. DEPARTMENT OF AGRICULTURE. 



Abundant study of the law of error has shown that large errors occur 

 less often than small ones, and if bias is absent plus errors of any 

 magnitude occur just about as often as minus errors of similar mag- 

 nitude. This is well illustrated in Table VIII, which shows the dis- 

 tribution of errors in 354 separate measurements of an area. 



Table VIII. — Distribution of errors. 



Magnitude of error. 



Number 

 of plus 

 errors. 



Number 



of minus 



errors. 



0to0.3 



.3110 .6 



.61 to .9 



.91 to 1.2 



1.21 to 1.5 



Total number 



89 

 51 

 26 



8 

 2 



93 



55 

 22 



8 

 



176 



+ 178=354 



In these measurements there were in all 176 plus and 178 minus 

 errors. Furthermore, of the errors of any given magnitude there 

 are about as many plus as minus. 



In so far as we have been able to test the matter, the errors arising 

 in securing data from farm experience distribute themselves about 

 the true value in approximately the manner above illustrated. It is 

 therefore possible, by securing large numbers of estimates, to get 

 averages of a very satisfactory degree of accuracy. 



The third factor governing the accurac}^ of an average is the ac- 

 curacy of the individual items averaged. Inaccuracies in these items, 

 if bias is absent, tend to eliminate each other because of the manner 

 in which errors group themselves about the true mean, provided the 

 number of items is large enough. For this reason inaccuracies in 

 the original measurements are less important than either absence of 

 bias or number of items averaged. 



Pearl and others have shown by actual count that an average is 

 more accurate than the data on which it is based. This fact has in- 

 deed long been known. The relation of the accuracy of an average 

 to that of the items averaged is given by the well-known formula 

 e 



E=-yjn 



» where E is the probable error of the mean, e the probable 



error of a single observation, and n the number of observations aver- 

 aged. Thus it might be said that an average based on, say, 40 ob- 

 servations of a variable quantity is twice as reliable as one based on 

 10, and an .average based on 100 observations is 10 times as trust- 

 worthy as a single observation. Even if the probable error of the 

 individual estimates is as much as 25 per cent, the probable error of 

 the average of 100 such estimates is only 24 per cent. Hence, even 

 if the farmer's knowledge of the details of his business were even 

 less definite than experience has shown it to be it would still be 



