Error Propagation Reference Page

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Intro to Error Propagation

All experimental measurements contain uncertainty. When measured quantities are used in calculations, the uncertainties in the inputs affect the uncertainty of the final result. The process of determining how measurement uncertainties influence calculated results is known as error propagation or uncertainty propagation.

Error propagation is widely used in engineering to estimate the reliability of calculated results and to determine which measurements contribute most strongly to uncertainty.

The videos on this page introduce the principles of error propagation and demonstrate how uncertainties propagate through equations, derivatives, and integrals.

Error Propagation

ImportantNot a DOFPro video

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This video introduces the basic principles of error propagation and derives the commonly used formula for estimating how measurement uncertainties affect calculated quantities.

Integrals and Derivatives Part 1

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This video explains how uncertainties propagate through numerical derivatives and integrals, which commonly appear in engineering calculations involving experimental data.

Integrals and Derivatives Part 2

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This video continues the discussion of error propagation in numerical calculations and examines how uncertainty affects computed derivatives and integrals in practical engineering problems.

Examples and Definitions

Definitions

Error Propagation
The process of determining how uncertainties in measured quantities affect the uncertainty in a calculated result.

For a function

\[ f(x_1,x_2,\dots,x_n) \]

with uncorrelated uncertainties \(S_{x_i}\), the estimated uncertainty in \(f\) can often be approximated by

\[ S_f = \sqrt{ \left(\frac{\partial f}{\partial x_1}S_{x_1}\right)^2 + \left(\frac{\partial f}{\partial x_2}S_{x_2}\right)^2 + \dots + \left(\frac{\partial f}{\partial x_n}S_{x_n}\right)^2 } \]

Uncorrelated Errors
Measurement uncertainties that are statistically independent of each other. When errors are uncorrelated, their contributions to total uncertainty combine as the square root of the sum of the squares of the individual contributions.
Correlated Errors
Measurement uncertainties that are related to one another. In this case the covariance between measurements must be included when calculating the propagated uncertainty.
Numerical Error Propagation
The process of estimating uncertainty in calculated quantities using numerical methods such as spreadsheets, simulation tools, or numerical evaluation of derivatives.
Analytical Error Propagation
The process of determining uncertainty using analytical formulas derived from calculus, typically involving partial derivatives of the function being evaluated.