Propagation of uncertainty estimates how measurement errors in inputs affect a calculated result. For small input errors with covariance matrix , linearizing a differentiable function gives the displayed approximation for its variance. For independent errors in a product , the squared relative standard deviations add. Correlated errors need the covariance terms; systematic offsets require separate assessment.
If measured radiative flux is and with known , first-order uncertainty propagation gives the displayed result. Independent measurements have zero covariance; strict error bounds combine differently. Even a small fractional background contribution can matter if its relative uncertainty is large. Inferring the covariance or an unknown calibration factor from precision percentages alone is unjustified.
For , the sensitivity coefficients are and . The first-order uncertainty propagation law gives the displayed variance; independent inputs remove the covariance term. Subtracting a known constant does not change absolute uncertainty, but can greatly change relative uncertainty when the resulting value is small. The result is a standard uncertainty estimate, not automatically a worst-case error bound or a confidence interval.
For with reactance , linear differentiation gives . Independent standard uncertainties give the title formula; correlated inputs add . Hard error bounds instead add absolute linear contributions. The meaning of an error bar is an additional measurement assumption, not a consequence of the impedance formula.
For air approximated by index one, , so . Taking the variance gives the displayed first-order error formula, with angles and their errors in radians. Independence removes the covariance term; hard error bounds require a different combination rule. Common angular-zero errors can produce covariance and should not automatically be added in quadrature.
For axial moment of inertia , linear sensitivity to mass and outer/inner radius is . Independent standard uncertainties give . Correlated measurements instead require the full covariance quadratic form. For deterministic bounds, add the absolute linear contributions rather than their squares.
For with independent standard uncertainties, linear propagation gives . Angular uncertainty must be in radians when using this derivative. Covariance adds cross terms for correlated inputs. Strict error bounds instead add absolute first-order contributions; they do not justify Gaussian significance tests.
For positive measured inputs and , logarithmic differentiation gives . Applying propagation of uncertainty yields the displayed covariance formula. For independent input errors it reduces to . If the input uncertainties are strict bounds rather than standard deviations, a first-order conservative bound is . These are different statistical interpretations; a product formula alone does not choose between them.
For the same massless spring obeying Hooke's law, the frequency is . The displayed mass ratio follows. For small independent frequency errors with common fractional standard uncertainty , monomial uncertainty propagation gives . If instead describes individual measurements and each sample mean uses independent trials, its standard error is smaller by , giving . Four times as many trials halve this statistical uncertainty; a common calibration error needs its covariance included.
For simple pendulum oscillation period in the small-angle approximation, . If the length measurement and identically precise timing measurements have independent errors, the timing sample mean has standard error . Propagation of uncertainty gives the title formula. A target fractional standard uncertainty requires , rounded upward to an integer. No amount of repeated timing can beat the floor without improving the length measurement; correlated errors or a timing systematic error require a different model.
Articles by others on the same topic
Propagation of uncertainty, also known as uncertainty propagation or error propagation, refers to the process of assessing how uncertainties in measurements or input variables affect the uncertainty of a derived quantity. When calculating a result based on multiple measured or estimated quantities, each of these inputs may have a certain degree of uncertainty. Understanding how these uncertainties combine is crucial in fields such as experimental physics, engineering, and statistics. ### Key Concepts 1.