GlacierFilm, LLC. · English · ExplanationConnection evidence
Reviewed English material v1: card/concept_card "One-variable uncertainty propagation" starting at 2:03 — When only velocity is uncertain, the uncertainty in kinetic energy is estimated by multiplying the velocity uncertainty by the derivative dK/dv. Since dK/dv=mv, the rule becomes δK≈mv δv. δK≈dvdK\,δv=(mv)δv
Docs and Sheets Pro · English · ExplanationConnection evidence
Reviewed English material v1: card/concept_card "Definition of Error Propagation" starting at 0:05 — Error propagation in Excel refers to the process where small inaccuracies or uncertainties in initial input data spread through calculations, leading to larger errors in the final results. This is especially critical in complex financial or scientific models.
NEON Science · English · ExplanationConnection evidence
Reviewed English material v1: knowledge_item/formula "Direct Variance Component in Uncertainty Propagation" from 2:30 to 2:48 — The contribution of a single parameter's uncertainty to the total output variance is calculated as the product of the parameter's variance and the square of the partial derivative of the model output with respect to that parameter. This represents the sensitivity of the output to that specific input. var[f(x)] ≈(∂θi∂f)2 var[θi] Assumes first-order Taylor approximation.; Applies to independent parameters or isolates the direct effect of one parameter.
Questions and answers about Uncertainty propagation
For addition and subtraction, the absolute uncertainties of the inputs combine in quadrature. The plus or minus signs in the formula do not matter because they are squared in the propagation equation, resulting in the same rule for both operations.
The derived quantity is a sum or difference of two measured variables (z=±x±y).
The variables x and y are independent measured quantities.
Example: For z=x−y, the partial derivatives are 1 and −1. Squaring them gives 12 and (−1)2, both equal to 1. The final uncertainty is δz=(δx)2+(δy)2, exactly the same as for z=x+y.
You can sometimes ignore the smaller relative uncertainty when one measurement is much more accurate, relatively speaking, than the other. In this case, the combined relative uncertainty is approximately equal to the larger one, and the more accurate measurement behaves as if it were an exact constant.
The comparison is between relative uncertainties.
One measurement is much more accurate relative to the other.
The approximation is acceptable for the intended level of precision.
Example: The video shows an animation where a 0.03% relative uncertainty term shrinks and disappears, leaving only the 5.4% term. For a product z=xy, this approximation simplifies the absolute uncertainty to δz≈xδy, making x behave like a constant.
When both mass and velocity are measured, the uncertainty in kinetic energy is found by adding the two uncertainty sources in quadrature. You compute the partial derivative of kinetic energy with respect to each variable, multiply each derivative by the corresponding absolute uncertainty, square those terms, sum them, and take the square root.
The kinetic energy formula is K=21mv2.
Both mass m and velocity v carry independent absolute uncertainties δm and δv.
The uncertainties are small enough for a first-order linear approximation.
Example: For a mass of 10.12±0.11kg and a velocity of 2.34±0.04m/s, the propagated uncertainty is calculated as δK=(21(2.34)2(0.11))2+((10.12)(2.34)(0.04))2, which evaluates to 0.99J.
For products and quotients, the relative uncertainties of the inputs add in quadrature. You calculate the relative uncertainty of the result by taking the square root of the sum of the squared relative uncertainties of the individual variables.
The derived quantity is a product (z=xy) or a quotient (z=yx).
The variables x and y are nonzero so the relative uncertainty ratios are defined.
The uncertainties are treated as independent.
Example: For z=xy, the raw propagated form is δz=(yδx)2+(xδy)2. Dividing by z=xy and simplifying yields zδz=(xδx)2+(yδy)2. The same relative rule applies to z=yx.
Independent uncertainties are added in quadrature because they represent random effects that can push a measurement up or down. If the effects are random and independent, their combined influence is neither always cancelling nor always additive. The perpendicular geometric model represents an average relationship, giving a result that is less than the worst-case arithmetic sum but greater than either individual uncertainty.
The two uncertainty effects are random.
The two effects are independent from one another.
The combination is modeled by a perpendicular geometric construction.
Example: The video animates arrows e1 and e2 rotating from opposing to aligned to perpendicular. The screen explicitly shows the inequality: hypotenuse < e1+e2, illustrating that the quadrature combination is smaller than the fully additive worst case.
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