12 BULLETIN 897, U. S. DEPARTMENT OF AGRICULTURE. 
study of 736 normal frequency distributions made by commercial 
packers on the weight of tea, coffee, cocoa, spices, butter, cheese, 
condensed milk, starch, flour, and various other food products, 50 
observations being made on each distribution, showed that the most 
frequent chance of occurrence of the maximum error was 1 in 60; 
the average, 1 in 198; and that 80 per cent were | in 1,000 or less. 
Many of those greater than 1 in 1,000 were due to avoidable causes, 
such as inaccurate reweighing, and occasional slips of the packer. 
The probability of occurrence of errors within a certain limit is 
found by the formula f =— , where fis a factor whose probability value 
is found from a table of probability integrals,’ x is the limiting error 
whose probability is desired, and r is the probable error of a single 
observation of the distribution. The complement of the equivalent 
of f gives the probability that errors greater than x will occur, and the 
resulting fraction reduced until its numerator is 1 gives the chances 
of their occurrence. 
Tn the case of average weights, x represents the calculated maximum 
error on the average of a representative sample, and r for practical 
purposes is 0.8453 times the average deviation from 0 average minus 
aim, The resulting chances are given in Table 8. 
TABLE 8.—Chance of occurrence of variations greater than the calculated maximum error 
on the average of a representative sample. 



| 
Caleu- 
lated 
maxi- 
mum Average pe Zs Chance of occurrence 
error on deviation ea sree A of variations greater 
Size of package. average from 0 from 0 = _ than calculated 
of a rep- average average uy maximum, based 
nesente minus aim.) snus aim. on observed data. 
sample | 
(z). (7). (7). 
Ozs Oz. Oz. 
2-07 2 Ase eee 0. 097 0.014 0. 012 8.1 1 in more than 10,000. 
AROZ 2 UHR eRe . 097 . 032 . 027 3. DOs utneeGD: 
8207S ee eee . 097 . 023 - 019 5.10 | 1in 1,666 
LID ie ae ee - 106 032 . 027 Sogo ele eel 22 
O-lbEs see eee eens . 148 . 120 . 101 1.46 | lin 3. 
5) ORE Sac eee SRG . 209 . 005 . 004 52.2 | 1inmore than 10,000. 
BO be Sees eee 3. 57 - 70 - 59 6.0 | lin more than 10,000. 

With the exception of 3-pound packages, on which only one set of 
data was obtained, and 4-ounce packages where there is apparently 
a comparatively abnormal result, the chances in the last column of 
the table would indicate that the calculated maximum error will 
seldom if ever occur in actual practice. 
In the case of individual weights, x is the calculated maximum 
error on single packages. Since vis calculated from the aim, r is 0.8453 
times the average deviation from the aim. This average deviation is 
1 Merriman. Loc. cit., p. 221. 

