Suppose we calculate a quantity Q based on the measured values x,y, and z. Then Q is a function of three variables.
Each of the measured values will have an associated uncertainty 
,
, and 
. Then the formal uncertainty in Q is given by the expression
This expression may seem intimidating at first but is rather easy to interpret. The “curly d” derivative used, is called the partial derivative of f with respect to x. It is easy to compute since you simply pretend that y and z are constant and find the ordinary derivative with respect to x. The following example will illustrate the method and all of the details.
Suppose that Q is given by the function shown, , where k = 0.3872 is a constant. Also assume that we have the measured values
.
First compute Q
(We will round after the uncertainty has been found)
Now find the partial derivatives
and finally compute the uncertainty in Q using the expression above.
So rounding to one significant figure. Combining with the computed value for Q and rounding so that it agrees with the uncertainty gives .
If you apply the simpler rules described in section G, you will find to give for the final result. You can see that the simpler rules slightly overestimate the uncertainty.
Last Updated November 13, 2014
