4 BULLETIN 897, U. S. DEPARTMENT OF AGRICULTURE. 
3. That scales meet the tolerances prescribed in Bureau of Standards Circular 61 
and be used at 20 per cent or more of their capacity for packages of less than 
25 pounds, and at 10 per cent or more of their capacity for larger packages. 
Tare weights may be made on the same scale as the gross weights. 
4, That the exact balance be estimated to within one-half of the sensibility reciprocal 
of the scale, and the poise be set to within one-fourth of the minimum gradua- 
tion on the beam. 
These specifications are drawn up in this form for the particular 
purpose of having something specific on which to base calculations. 
The study of factories has shown them to be in general use. ” 
CALCULATED MAXIMUM ERRORS OF GOOD COMMERCIAL PRACTICE. 
The calculation of the maximum errors of good commercial 
practice is based on the law of propagation of errors, the Bureau of 
Standards tolerances on scales, and the specifications of good com- 
mercial practice. 
The law of propagation of errors, as found in standard text books ! 
on the methods of least squares, is— 


B= 4] (a)?+ 6)? + (0)? + @? 
where (a), (b), (ec), and (d) are constituent errors which combine to 
form the resultant error /, all errors having the same probability of 
occurrence and the same chance of being plus or minus. 
The Bureau of Standards tolerances on scales, as found in Circular 
61 of that Bureau, are given in Tables 1, 2, 3, and 4. A general 
attempt is being made by weights and measures officials to eliminate 
all scales which fail to meet these tolerances, thereby leaving in 
general use scales which do meet them. 
The specifications of good commercial practice (p. 3) designate the 
figures to be selected from the tolerance tables (constituent errors), 
which in turn are combined according to the law of propagation of 
errors to give the resultant calculated maximum error. This calcu- 
lated maximum error will have the same probability of occurrence as 
the constituent errors. It is scarcely conceivable that all the various 
tolerances have the same probability of occurrence, but their long use 
by weights and measures officials for the purpose of condemning 
scales makes them a stable basis of a limiting probability. Other 
figures, such as one-fourth of a minimum graduation, used in deter- 
mining constituent errors, are assumed to have probabilities of occur- 
rence about equivalent to the tolerance figures. No attempt has 
been made to weight the constituent errors in order to correct for 
their unequal probabilities. The calculated maximum error, there- 
1 Merriman, Mansfield. A Text-Book on the Method of Least Squares, 8th ed. New York, 1913. 



