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.