Uncertainty Propagation

Uncertainty propagation using sensitivities and covariances. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Many physical quantities are not measured directly. They are calculated from lengths, times, masses, voltages, or other inputs that already carry uncertainty. That uncertainty does not disappear when a formula is applied. It is transformed according to the sensitivity of the output to each input. Propagation studies that transformation. The key question is not only the value of a derived quantity, but which variables dominate its uncertainty, how correlations act, and when linear approximation breaks down. Absolute uncertainties, relative uncertainties, and covariance must remain dimensionally coherent.

🟢 Essential level

A derived quantity depends on several inputs, and each transmits uncertainty differently. If an input changes slightly while output responds strongly, that input has high sensitivity and may dominate the result.

$c_x$ and $c_z$ describe this local response. $u_x$ and $u_z$ are first weighted by those sensitivities. Independent contributions combine in quadrature to produce $u_y$. Related inputs require $\sigma_{xz}$, which may increase or reduce total variance. Products, quotients, and powers are often clearer through $u_rel_x$, $u_rel_z$, and $u_rel_y$. A derivative sign indicates change direction, but uncertainty remains a nonnegative amplitude. Correct propagation identifies which measurement should be improved and prevents false precision in the derived result. Comparing weighted contributions therefore guides efficient experimental improvement: only the input that truly dominates the uncertainty budget deserves attention, not the one that happens to be easiest to improve in the laboratory, since improving an already-small contribution barely changes the total, no matter how much effort is spent on it.

🔵 Formal level

Basic formulas

For one input, output standard uncertainty follows:

Single-variable propagation

This relation multiplies $u_x$ by the absolute $c_x$. For two independent inputs, weighted contributions combine through:

Propagation for independent inputs

When statistical dependence exists, $\sigma_{xz}$ must be included through:

Propagation with covariance

These three relations cover the additive core of first-order propagation, from a single input through independent and correlated multi-input cases, before the sum/product-specific rules below refine them further.

Real student questions

Why are derivatives used? They describe how strongly output changes locally when an input changes.

Does a negative derivative give negative uncertainty? No. It gives direction, while uncertainty remains nonnegative.

When should relative uncertainty be used? For products, quotients, and powers, provided nominal values are not zero.

What does covariance do? It describes joint fluctuation and modifies total variance.

Does linear propagation always work? No. It fails near singularities, with large uncertainties, or with strong curvature.

How do I know which variable to improve? Compare sensitivity-weighted contributions and act on the dominant one.

Final synthesis

Uncertainty in a derived quantity comes from input uncertainties, but model sensitivity decides their weight. Independence permits quadrature; correlation requires covariance. Validity depends on coherent units, local linearization, and physical interpretation of the result.

A complete interpretation must also state which assumptions control the result, which contribution dominates, and whether correlation or nonlinearity could change the reported uncertainty. This check keeps the numerical answer connected to the physical model and prevents unsupported precision.