68 DONALD C. BARTON 
are 50 chances out of 100 that algebraically, the actual error is less 
than —o.oo1 or greater than +o0.0o01 and that there are 50 chances 
out of 100 that algebraically, the actual error will be between —o.oo1 
and +0.001. According to the theory of the “probable error,” the 
actual error will be five times the “probable error” 1 time in 1,000, 
four times the “‘probable error” 7 times in 1,000, three times the 
‘“‘probable error” 43 times in 1,000, twice the “‘probable error” 177 
times in 1,000. The ‘‘probable error,” therefore, affords a method of 
comparing the precision of different measurements and of determin- 
ing the probable range within which the actual error of a measure- 
ment of quantity may lie. 
The “‘probable error’ of a series of measurements depends upon 
the ‘‘probable error’? of each measurement, the number of measure- 
ments, and the character of the series of measurements. If PE is the 
“probable error” of the series, if pe—1, pe—2-- - pe—m are the re- 
spective “‘probable errors” of m measurements numbered, 1, 2, - - - to 
m and if the series consists of repeated measurements of a single 
quantity, the ‘probable error”’ of the arithmetic mean of the measure- 
ments is: 
pe 
(1) Je, = = 
Vm 
if the pe’s are all the same. But if the series consists of a series of con- 
secutive applications of the measuring device, as for example, the 
consecutive use of a surveyor’s tape, in measuring the length of a line 
which is longer than the tape: 
(2) PE = V(pes)? + (pea)? + -- > (pen)? 
or if the pe’s are all the same: 
(3) PE =~/n pe 
The “‘probable error” (pe) of a mean in terms of the actual error 
(ae) of each measurement is given by the formula: 
(gre) sePE = 0.61454/ 
The accuracy of torsion balance surveys can be expressed by a 
probable error which can be calculated by the formula: 
1 Mansfield Merriman, A Textbook on the Method of Least Squares, 8th ed. (1911), 
pp. 66-79. 
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