Propagation of Uncertainty through Mathematical Operations

M. Palmer1 Propagation of Uncertainty through Mathematical OperationsSince the quantity of interest in an experiment is rarely obtained by measuring that quantitydirectly, we must understand how error propagates when Mathematical Operations are performedon measured quantities. Suppose we have a simple experiment where we want to measurevelocity, V, by measuring distance, d, and time, t. We take measurements and come up withmeasured quantities d d and t t. We can easily estimate V by dividing d by t, but we alsoneed to know how to find V. Below we investigate how error propagates when mathematicaloperations are performed on two quantities x and y that comprise the desired quantity and SubtractionIf we are trying to find the Uncertainty , q, associated with q = x + y, we can look at what thehighest and lowest probable values would be.

M. Palmer 4 Since b is assumed less than 1, b2 and all of the higher order terms will all be <<1. These can be neglected and we can say that: b b ≈+ − 1 1 1. (21) Then, (19) becomes ()()a b a b ab b a ≈ + + =+++ − + 1 1 1 1 1 Once again we eliminate ab because it is the product of two small numbers. We substitute the

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