Uncertainty Propagation

All observations have associated uncertainties which must be propagated through your analysis to a an uncertainty on a final result. The principle of uncertainty propagation is fairly simple:

The first two terms on the right are the averages of the squares of the deviations in x produced by the uncertainties in observables u and v respectively. The third term on the right is the average of the cross terms, which cancel out if u and v are uncorrelated. Thus in most situations, a reasonable approximation is:


A typical situation is the sum of two observables each with a multiplicative factor:

which is the oft used "summing in quadrature."

Perhaps you have the product of two observables:

Finally, perhaps you have the inverse of one observable times the exponential of another:


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