Once the uncertainty in a measured quantity x has been found, it is often
necessary to calculate the consequential uncertainty in some other quantity
which depends on x. Consider a function f which depends on x and/or y:
f=f(x) or f=f(x,y).
A series of measurements of x and/or y will yield 
. We will assume that the
best estimate of f is 
or
. (But note that 
defined this way is
not always the same as the mean of f obtained from the individual values
.)
Examples of the way to find 
, the standard deviation of f, are given
below:
The standard deviation for f is obtained by adding the standard deviations for x and y ``in quadrature."
The relative error for f is obtained by adding the relative errors for x and y ``in quadrature."
where A and b are precisely-known constants and b is positive. The relative error for f is b times the relative error for x.
need some explanation. Let df be the
small change in f which results from small changes dx, dy in the
quantities x, y. One can then make the identifications:
, 
, 
,
, and 
. For
example, let 
. Then,
With the above identifications, we obtain
. As another example, let
f=x+y. Then df=dx +dy. We might conclude that
.
However, during half of the time, the deviations in x and y will be in
opposite directions (as long as x and y are measured independently) so that
one expects to be less than 
. A careful
statistical
analysis shows that 
and 
should be added ``in quadrature."
all the rules above can easily be obtained by identifying differentials with
standard
deviations and by replacing addition or subtraction by addition ``in
quadrature."