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 YIII. — Distribution of errors. 
Magnitude of error. 
Number 
of plus 
errors. 
Number 
of minus 
errors. 
0to0.3 
.31 to .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 accuracy 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 
w— rz? 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, 10 ob- 
servations of a variabfe 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 2 J 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 
