Propagation
of Errors
In many cases our final results from an experiment will not be
directly measured, but will be some function of one or more other
measured quantities. For example, if we want to measure the density
of a rectangular block, we might measure the length, height, width,
and mass of the block, and then calculate density according to the
equation
Each of the measured quantities has an error associated
with it 
and these errors will be carried through in some way to the error
in our answer, .
Writing the equation above in a more general form,
we have:
The change in
for a small error in (e.g.) M is approximated by
where is
the partial derivative of
with respect to .
In the worstcase scenario, all of the individual errors would act
together to maximize the error in .
In this case, the total error would be given by
If the individual errors are independent of each other
(i.e., if the size of one error is not related in any way to the
size of the others), some of the errors in will
cancel each other, and the error in
will be smaller than shown above. For independent errors, statistical
analysis shows that a good estimate for the error in
is given by
Differentiating the density formula, we obtain the
following partial derivatives:
Substituting these into the formula for ,
Dividing by
to obtain the fractional or relative error,
This gives us quite a simple relationship between
the fractional error in the density and the fractional errors in
.
It may be useful to note that, in the equation above, a large error
in one quantity will drown out the errors in the other quantities,
and they may safely be ignored. For example, if the error in the
height is 10%
and the error in the other measurements is 1%,
the error in the density is 10.15%,
only 0.15% higher than the error in the height alone.
