文档介绍：Chapter 8
Error Analysis
Estimate of errors is fundamental to all branches of natural sciences that
deal with experiments. Very frequently, the final result of an experiment
cannot be measured directly. Rather, the value of the final result (u) will
be calculated from several measured quantities (x, y, z..., , each of which
has a mean value and an error):
u = f(x,y, z...) . ()
The goal is to estimate the error in the final result ufrom the errors in
measured quantities x , y , z . The errors can be random or systematic.
Random errors displace measurements in an arbitrary direction whereas
systematic errors displace measurements in a single direction. Systematic
errors shift all measurement in a systematic way so that their mean value
is displaced. For example, a miscalibrated ruler may yield consistently
higher values of length. Random errors fluctuate from one measurement
to the next. They yield results distributed about the same mean value.
Random errors can be treated by statistical analysis based on repeated
measurements whereas systematic errors cannot.
. Random Errors
Both random errors and systematic errors in the measurements of x, y, or
z lead to error in the determination of u in Eq. (). Since systematic
errors shift measurement in a single direction, we use du to treat
systematic errors. In contrast, since random errors can be both positive
and negative, we use (d~)~to treat random errors.
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Error Analysis 147
()
If the measured variables are independent (non-correlated), then the
cross-terms average to zero
dvdy = 0, dydz = 0, dvdz = 0, ()
as dr, dy, and dz each take on both positive and negative values.
Thus,
Equating standard deviation with differential, .,
a,,=du, ax=dv, 0,=dy,and a, =dz,
results in the error propagation formula for independent (non-correlated)
random errors
Equation () for independent random errors has been frequently used in
the un