Page 137 - Mechanical Engineers' Handbook (Volume 2)

126 Measurements
(e ) 2
F
2
u
X i e xi (22)
Since F/ X 1/n,
i
e 1 / 2
2
1
2
u
n e xi (23)
or
e 2 (e ) 1 / 2
1
2
u
n xi (24)
from the definition of
2
(e ) n 2 (25)
xi
and

e
u
n
This equation must be corrected because the real errors in X are not known. If the
number n were to approach infinity, the equation would be correct. Since n is a finite number,
the corrected equation is written as

e (26)
u
(n 1) 1/2
and

q U (27)
(n 1) 1/2
This says that if one reading is likely to differ from the true value by an amount ,
then the average of 10 readings will be in error by only /3 and the average of 100 readings
will be in error by /10. To reduce the error by a factor of 2, the number of readings must
be increased by a factor of 4.

3.3 External Estimates

In almost all experiments several steps are involved in making a measurement. It may be
assumed that in each measurement there will be some error, and if the measuring devices
are adequately calibrated, errors are as likely to be positive as negative. The worst condition
insofar as accuracy of the experiment is concerned would be for all errors to have the same
sign. In that case, assuming the errors are all much less than 1, the resultant error will be
the sum of the individual errors, that is,

C C C C (28)
E
1
3
2
It would be very unusual for all errors to have the same sign. Likewise it would be very
unusual for the errors to be distributed in such a way that
C 0
E
A general method follows for treating problems that involve a combination of errors to
determine what error is to be expected as a result of the combination.