# Volltext: On mean values for direct measurements (2)

```18
E. W. Scripture,
we introduce an uncertainty corresponding to a mean error of 0.25/?
in the whole numbers, whereas by retaining the 6, the uncertainty
corresponds to only 0.08/3 in the unit-place or to 0.8/3' in the first
decimal place, being a gain corresponding to 0.17/8. Rounding
off to 212.7 adds an uncertainty of 0.25/3' in the first decimal place,
giving a total of 0.83/8' + 0.25/3'=1.08/8' for that place or 0.108/8 for
unit-place, being a loss corresponding to 0.028. Since under any
circumstances the loss would have to be 0.025/8, the writing of 212.67
has practically no advantage over 212.7.
The case is different when the uncertainty of the values is not due
simply to the omission of decimal places. Let the measure of the
uncertainty of a value x be denoted by ± Aæ. Then the uncertainty
of the average of n results will be given by
± Aæ
Vn
The mean error is the most convenient measure of uncertainty.
Under very favorable conditions the record of the Hipp chrono-
scope1 * is liable to a mean error of A*=1.5'7. The average of 9
records is reliable to 0.5^. To obtain a result numerically precise
to 1°, i. e. with a mean error of 0.25^, it would be necessary to have
36 original records.
Thus, if a body, e. g. a control-hammer, were known to fall with
perfect constancy, 36 records with the chronoscrope would be re¬
quired to determine its time of fall to lff.
Dependence on characteristic variations. The result of a set of
direct measurements is stated to be A±d, A±m or A±r. The
quantity d is the mean variation or mean error, m is the mean-
square-error and r is the half probable variation or probable error.
These characteristic variations are determined from the formulas
d=
m-
r—0.674y/t)'’1 + w
>+(«,)+ • • •
+ (0,
Vw(w— i)
, 1
V+<+• •
. +v„3
n—1
/«,• + «,■+ • •
. +ü„s
0.708
0.708
n— 1
V<
n-
In these formulas the signs ± have the meanings usually given
1 K/ülpe and Kirscbmann, Ein neuer Apparat zum Contrôle zeitmessender Instru¬
mente, Phil. Stud., 1892 VII 145.
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