Propogation of Uncertainty through Calculations

Suppose the "absolute" uncertainty (or error) in A is the quantity a, and in B it is b. Here a has the same units as A and b has the same units as B. We then say that the "relative" error (or "fractional error") in A is a/A, and, obviously, the "percentage error" in A is 100a/A.

Now, we'd like to be able to state the error in combinations like A+B or A-B or AB or A/B. This is done as follows:

the absolute error in a sum or difference, A+B or A-B, is the sum of the absolute errors, a+b. Thus
(A ± a) + (B ± b) = (A+B) ± (a+b), and,

(A ± a) - (B ± b) = (A-B) ± (a+b).
however to get the error in A*B or A/B we must add the "relative errors" to get the fractional error in the product or quotient (an alternative, given below, is to "add the relative errors in quadrature"). The relative error in A*B or A/B or B/A is (a/A+ b/B). And if we want the absolute error in A*B or A/B we simply multiply the result by the fractional error. So, for example
(A ± a) × (B ± b) = AB × [1 ± (a/A + b/B)].
You can equally well combine percentage errors instead of "relative" errors... a percentage error of 10% is a relative error of 0.1.
So if the percentage error in A is 10% and in B is 25%, the percentage error in A*B or B/A or A/B is 35%.
an alternative estimate for the fractional error in a product or quotient adds the fractional errors in quadrature, i.e.
(A ± a) × (B ± b) = AB × [1 ± ( (a/A)² + (b/B)²)^0.5 ].

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Propogation of Uncertainty through Calculations

Last Modified: 27 Mar., 2017