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In statistics, propagation of uncertainty (or propagation of error) is the affect of variables' uncertainties (or errors) on the uncertainty of a function based on them. Mainly, the variables are measured in an experiment, and have uncertainties due to measurement limitations (e.g. instrument precision) which propagate to the result.
The uncertainty is usually defined by the absolute error — a variable that is probable to get the values x±Δx is said to have an uncertainty (or margin of error) of Δx. In other words, for a measured value x, the true value is probable to be in [x−Δx, x+Δx]. Uncertainties can also be defined by the relative error, Δx/x, and then it is usually written as percentage. It is assumed that the probability of the true value to be in distinct distances from the measured value is normally distributed, with the uncertainty being the standard deviation.
This article explains how to calculate the uncertainty of a function, if the variables' uncertainties are known.
If the variables are uncorrelated, we can calculate the uncertainty Δf of f that results from the uncertainties of the variables:
where designates the partial derivative of for the -th variable.
where Ci,i = var(xi) = Δxi².
After calculating , we can say that the value of the function with it's uncertainty is:
This table shows the uncertainty of simple functions, resulting from uncorrelated variables A, B, C with uncertainties ΔA, ΔB, ΔC, and a precisely-known constant c.
|X = A ± B||(ΔX)² = (ΔA)² + (ΔB)²|
|X = cA||(ΔX) = c(ΔA)|
|X = c(A×B) or X = c(A/B)||(ΔX/X)² = (ΔA/A)² + (ΔB/B)²|
|X = c(A×B×C) or X = c(A/B)×C||(ΔX/X)² = (ΔA/A)² + (ΔB/B)² + (ΔC/C)²|
|X = cAn||(ΔX/X) = |n| (ΔA/A)|
|X = ln cA||ΔX = (ΔA/A)|
|X = exp A</td>||(ΔX/X) = ΔA|
Example application: Resistance measurementEdit
Given the measured variables with uncertainties, I±ΔI and V±ΔV, the uncertainty in the computed quantity, ΔR is
Thus, in this simple case, the relative error ΔR/R is simply the square root of the sum of the squares of the two relative errors of the measured variables.
- An Informal Guide to Error Analysis from Union College.
- Uncertainties and Error Propagation, Appendix V from the Mechanics Lab Manual, Case Western Reserve University.
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