Error Propagation

Differential error propagation rules for systematic and random uncertainties in derived physical quantities.

What it is

Uncertainty propagation is the rigorous mathematical framework used to determine how experimental variations, tolerances, or standard uncertainties in directly measured variables transfer to a derived physical quantity computed through a functional relation $y = f(x_1, x_2, \dots, x_n)$.

It is grounded in the first-order multivariate Taylor expansion evaluated around the expected values or sample means of the input variables. When random perturbations are small compared to central values, the deviation of the dependent output is accurately approximated by a linear combination weighted by the partial derivatives of the model function: $\Delta y \approx \sum_{i=1}^n \frac{\partial f}{\partial x_i} \Delta x_i$

Taking the root-mean-square expectation of this deviation under independent (uncorrelated) random variables yields the classic Gauss uncertainty propagation formula: $\sigma_y = \sqrt{\sum_{i=1}^n \left( \frac{\partial f}{\partial x_i} \right)^2 \sigma_{x_i}^2}$

If statistical correlation exists between input quantities (such as common instrument calibration or shared thermal drift), cross-covariance terms must be strictly included: $\sigma_y = \sqrt{\sum_{i=1}^n \left( \frac{\partial f}{\partial x_i} \right)^2 \sigma_{x_i}^2 + 2 \sum_{i < j} \frac{\partial f}{\partial x_i} \frac{\partial f}{\partial x_j} \operatorname{cov}(x_i, x_j)}$

Why it appears in physics

Almost no fundamental constant or engineering quantity is measured in isolation. Physicists calculate mass densities $\rho = m / V$, elastic moduli $E = (F L)/(A \Delta L)$, gravitational acceleration via pendulums $g = 4\pi^2 L / T^2$, or kinetic energies $E_k = \frac{1}{2} m v^2$.

In each case, primary measurement uncertainties (from balances, calipers, timers) interact nonlinearly. Without uncertainty propagation:

1. One cannot evaluate whether experimental discrepancies refute theoretical hypotheses or merely reflect instrument noise.
2. Experimentalists cannot identify which measurement tool dominates the overall variance budget, making experimental optimization impossible.
3. Naively summing absolute upper bounds leads to unrealistically pessimistic error bars or, worse, ignores correlation effects.

Visual or geometric idea

Geometrically, a multivariable function $f(x_1, \dots, x_n)$ forms a hypersurface in Euclidean space. The input uncertainties define an uncertainty hyperellipsoid or dispersion box centered at $(\bar{x}_1, \dots, \bar{x}_n)$ with semi-axes given by $\sigma_{x_i}$.

The gradient $\nabla f = (\partial f/\partial x_1, \dots, \partial f/\partial x_n)$ defines the normal vector and path of steepest change. Linear propagation projects the input hyperellipsoid onto the local gradient line. A steep partial derivative strongly stretches the uncertainty along that axis. Under nonzero covariance, the hyperellipsoid tilts relative to coordinate axes, significantly shifting the length of its projection.

Minimal symbolic language

1. General Gauss formula (uncorrelated): $\sigma_f^2 = \sum_{i=1}^n \left( \frac{\partial f}{\partial x_i} \right)^2 \sigma_i^2$
2. Linear sum and difference ($f = a x + b y$): $\sigma_f = \sqrt{a^2 \sigma_x^2 + b^2 \sigma_y^2}$ Absolute uncertainties add in quadrature weighted by constant scale factors.
3. Power-law products and quotients ($f = c x^a y^b$): $\frac{\sigma_f}{|f|} = \sqrt{a^2 \left(\frac{\sigma_x}{x}\right)^2 + b^2 \left(\frac{\sigma_y}{y}\right)^2}$ Relative fractional uncertainties add in quadrature weighted by absolute power exponents.
4. Covariance term with correlation coefficient $r_{xy}$: $\operatorname{cov}(x, y) = r_{xy} \sigma_x \sigma_y$ $\sigma_f^2 = \left(\frac{\partial f}{\partial x}\right)^2 \sigma_x^2 + \left(\frac{\partial f}{\partial y}\right)^2 \sigma_y^2 + 2 \frac{\partial f}{\partial x} \frac{\partial f}{\partial y} r_{xy} \sigma_x \sigma_y$

What the result means

The calculated standard uncertainty $\sigma_y$ represents the standard deviation of the output probability distribution. Under approximate Gaussian inputs or asymptotic Central Limit Theorem behavior, the coverage interval $[\bar{y} - \sigma_y, \bar{y} + \sigma_y]$ carries an approximate $68.3\%$ level of confidence ($k=1$), while $[\bar{y} - 2\sigma_y, \bar{y} + 2\sigma_y]$ offers approximately $95.4\%$ confidence ($k=2$).

Additionally, the relative variance budget $\left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_{x_i}^2 / \sigma_y^2$ reveals the exact fractional contribution of each instrument to overall experimental error.

When not to use it

1. Large relative uncertainties: If $\sigma_x / \bar{x} > 15\%-20\%$ and $f$ displays strong local curvature (e.g. $f(x)=1/x^2$ or $\exp(x)$), first-order Taylor truncation fails, producing asymmetric distributions that require Monte Carlo sampling.
2. Near-singularities or division by zero: If an interval $x \pm \sigma_x$ approaches zero in a reciprocal or logarithmic argument, the theoretical variance diverges and first-order formulas give completely spurious numbers.
3. Overlooking strong correlations: Treating correlated calibration constants or shared measurement channels as independent can grossly underestimate final uncertainty or miscalculate differential cancellations.

Operational summary

• Write the explicit model equation $y = f(x_1, x_2, \dots, x_n)$.
• Compute analytical partial derivatives $\frac{\partial f}{\partial x_i}$ and evaluate them at the central mean values $(\bar{x}_1, \dots, \bar{x}_n)$.
• Check whether parameters are statistically independent or correlated.
• Combine variances in quadrature according to the Gauss formula.
• Report the final value rounded to consistent significant digits matching standard uncertainty precision.

Error and Uncertainty Propagation Procedure

Before calculating

Before manipulating formulas, ensure that all input variables have defined central values $\bar{x}_i$ and standard uncertainties $\sigma_{x_i}$ stated at the same confidence level (typically $1\sigma$ standard uncertainty according to the GUM / ISO guide). Verify unit consistency across all parameters and check whether cross-correlations exist among measurement channels.

Step 1: identify the mathematical object

Explicitly write down the physical functional model connecting input parameters to the output variable: $y = f(x_1, x_2, \dots, x_n)$ Inspect the algebraic structure:

• Is it a pure linear combination ($y = a x_1 + b x_2$)?
• Is it a power-law monomial ($y = C x_1^a x_2^b$)?
• Is it a complex transcendental relation ($y = \ln(x) + \sin(k x) / z$)?

Step 2: choose the operation

Select the most efficient evaluation path:

• For monomials and power products: Use fractional relative uncertainty in quadrature: $\frac{\sigma_y}{|y|} = \sqrt{\sum a_i^2 (\sigma_i/x_i)^2}$, which avoids tedious partial differentiation.
• For sums and differences: Sum absolute variances directly: $\sigma_y = \sqrt{\sum a_i^2 \sigma_i^2}$.
• For general multivariable equations: Compute analytical partial derivatives $\frac{\partial f}{\partial x_i}$ explicitly.

Step 3: preserve units and signs

Evaluate each partial derivative at the sample mean coordinate $(\bar{x}_1, \dots, \bar{x}_n)$.

• Each partial derivative $\frac{\partial f}{\partial x_i}$ carries physical dimensions $[y] / [x_i]$.
• Multiplying by $\sigma_{x_i}$ (with units $[x_i]$) ensures the contribution $(\partial f / \partial x_i) \sigma_{x_i}$ strictly matches output dimensions $[y]$.
• Preserve algebraic signs of derivatives if covariance terms exist, because conflicting signs with positive covariance reduce total output variance.

Step 4: calculate

Substitute values into the Gauss propagation formula: $\sigma_y = \sqrt{\sum_{i=1}^n \left( \left. \frac{\partial f}{\partial x_i} \right|_{\bar{x}} \cdot \sigma_{x_i} \right)^2 + 2 \sum_{i < j} \left. \frac{\partial f}{\partial x_i} \right|_{\bar{x}} \left. \frac{\partial f}{\partial x_j} \right|_{\bar{x}} \operatorname{cov}(x_i, x_j)}$ Evaluate the central output value: $\bar{y} = f(\bar{x}_1, \bar{x}_2, \dots, \bar{x}_n)$

Step 5: interpret the result

• Uncertainty budget: Identify which quadratic component $\left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_i^2$ provides the largest percentage to the total variance. That term indicates the experimental bottleneck.
• Significant figures: Round standard uncertainty $\sigma_y$ to one or at most two significant figures (following standard metrological conventions).
• Match the decimal place of the central estimate $\bar{y}$ to the precision of $\sigma_y$.
• Format the final statement: $y = (\bar{y} \pm \sigma_y) \text{ [units] (coverage probability 68.3\%)}$

Self-check checklist

• Do all terms inside the square root have identical physical dimensions $[y]^2$?
• Did I verify independence before assuming cross-covariances are zero?
• Are relative input uncertainties small enough ($< 15\%$) to justify the first-order Taylor approximation?
• Does the decimal precision of the central value match the rounded uncertainty?
• Did I remember to square both the partial derivatives and standard uncertainties within the radical?

Guided example: Density of a solid sphere

Physical situation

An experimentalist seeks to determine the mass density $\rho$ of a precision metal sphere by independently measuring its mass $m$ on a digital scale and its outer diameter $d$ using a Vernier caliper:

• Mass: $m = (145.20 \pm 0.05) \text{ g}$
• Diameter: $d = (3.240 \pm 0.005) \text{ cm}$

The two measurements are independent (separate instruments).

Mathematical translation

Density is defined through volume and diameter as: $\rho(m, d) = \frac{m}{V} = \frac{m}{\frac{4}{3}\pi \left(\frac{d}{2}\right)^3} = \frac{6 m}{\pi d^3} = \frac{6}{\pi} m d^{-3}$ For a pure power-law monomial, relative quadrature propagation is preferred: $\left(\frac{\sigma_\rho}{\rho}\right)^2 = \left(1 \cdot \frac{\sigma_m}{m}\right)^2 + \left(-3 \cdot \frac{\sigma_d}{d}\right)^2 = \left(\frac{\sigma_m}{m}\right)^2 + 9 \left(\frac{\sigma_d}{d}\right)^2$

Step-by-step calculation

1. Central density value: $\bar{\rho} = \frac{6 \cdot 145.20}{\pi \cdot (3.240)^3} = \frac{871.20}{\pi \cdot 34.012224} = \frac{871.20}{106.8524} \approx 8.1533 \text{ g/cm}^3$
2. Individual relative standard uncertainties: $\frac{\sigma_m}{m} = \frac{0.05}{145.20} \approx 0.0003443 \quad (0.0344\%)$ $\frac{\sigma_d}{d} = \frac{0.005}{3.240} \approx 0.0015432 \quad (0.1543\%)$
3. Quadrature weighting and sum: $\left(\frac{\sigma_\rho}{\rho}\right)^2 = (0.0003443)^2 + 9 \cdot (0.0015432)^2 = 1.185 \times 10^{-7} + 9 \cdot 2.381 \times 10^{-6}$ $\left(\frac{\sigma_\rho}{\rho}\right)^2 = 1.185 \times 10^{-7} + 2.143 \times 10^{-5} = 2.155 \times 10^{-5}$ $\frac{\sigma_\rho}{\rho} = \sqrt{2.155 \times 10^{-5}} \approx 0.004642 \quad (0.464\%)$
4. Absolute standard uncertainty: $\sigma_\rho = \bar{\rho} \cdot 0.004642 = 8.1533 \cdot 0.004642 \approx 0.03785 \text{ g/cm}^3$

Reading the result

Rounding uncertainty to one significant figure: $\sigma_\rho = 0.04 \text{ g/cm}^3$ (or to two figures: $0.038 \text{ g/cm}^3$). The formatted physical result is: $\rho = (8.15 \pm 0.04) \text{ g/cm}^3 \quad [\text{or } (8.153 \pm 0.038) \text{ g/cm}^3]$ The variance decomposition shows that diameter accounts for $99.4\%$ of the total variance ($2.143 \times 10^{-5}$ vs $1.185 \times 10^{-7}$). The exponent factor of 3 squares to 9, magnifying relative uncertainty. Upgrading the scale is futile; precision can only be enhanced by using a micrometer screw on the diameter.

Common mistake avoided

Adding relative uncertainties linearly: $\sigma_\rho / \rho = (\sigma_m/m) + 3(\sigma_d/d) = 0.00034 + 0.00463 = 0.00497$ instead of quadrature summation. Linear addition falsely assumes perfect positive correlation under worst-case limits.

Inverse example: Permissible tolerance for measuring gravity with a simple pendulum

Physical situation

An experimental laboratory requires measuring local gravitational acceleration $g$ using a simple pendulum with a strict relative uncertainty limit $\sigma_g / g \leq 0.2\%$ ($0.002$). The wire length is nominally $L = 1.000 \text{ m}$ measured with standard uncertainty $\sigma_L = 0.001 \text{ m}$ ($0.1\%$, $\sigma_L / L = 0.001$).

Given result

Target global relative uncertainty: $\frac{\sigma_g}{g} \leq 0.002$. Measured length relative uncertainty: $\frac{\sigma_L}{L} = 0.001$.

Analytical reconstruction

The period for small pendulum oscillations is $T = 2\pi \sqrt{L/g}$, which solves for gravity: $g = 4\pi^2 \frac{L}{T^2} = 4\pi^2 L T^{-2}$ Quadrature propagation gives: $\left(\frac{\sigma_g}{g}\right)^2 = \left(\frac{\sigma_L}{L}\right)^2 + 4 \left(\frac{\sigma_T}{T}\right)^2$ Isolating the required timing tolerance: $4 \left(\frac{\sigma_T}{T}\right)^2 = \left(\frac{\sigma_g}{g}\right)^2 - \left(\frac{\sigma_L}{L}\right)^2$ $\frac{\sigma_T}{T} = \frac{1}{2} \sqrt{(0.002)^2 - (0.001)^2} = \frac{1}{2} \sqrt{3 \times 10^{-6}} \approx 0.000866$

For a nominal period $T \approx 2\pi \sqrt{1.0 / 9.81} \approx 2.006 \text{ s}$: $\sigma_T \leq 2.006 \cdot 0.000866 \approx 0.00174 \text{ s} = 1.74 \text{ ms}$

What physical question it answers

How many consecutive oscillations $N$ must be timed if a manual human stopwatch reaction time carries an uncertainty $\sigma_{\text{stopwatch}} \approx 0.15 \text{ s}$? Since the average period uncertainty over $N$ oscillations is $\sigma_T = \frac{\sigma_{\text{stopwatch}}}{N}$: $\frac{0.15}{N} \leq 0.00174 \implies N \geq \frac{0.15}{0.00174} \approx 86.2$ A minimum of 90 to 100 complete oscillations must be timed to meet the $0.2\%$ target without requiring electronic photogates.

Limit verification

If $\sigma_L / L \geq \sigma_g / g$, the radicand turns zero or negative, demonstrating that it is physically impossible to achieve the required accuracy on $g$ without first upgrading the length measurement, even with a hypothetical atomic clock with zero timing error.

Common Mistakes in Error and Uncertainty Propagation

1. Adding uncertainties linearly instead of in quadrature

It seems reasonable because one intuitively thinks of a "worst-case scenario" where all experimental errors conspire in the same unfavorable direction. It fails because for independent random variables, the probability of every measurement simultaneously hitting its extreme limit in the same destructive direction is virtually zero; positive and negative fluctuations partially cancel out in the root-mean-square average. It is detected when reported error bars are noticeably larger than the actual scatter observed across repeated experimental trials. It is corrected by always combining independent standard uncertainties in quadrature using the square root of the sum of squared partial contributions. Mini-example: for $z = x + y$ with $\sigma_x = 3$ and $\sigma_y = 4$, the true combined standard uncertainty is $\sqrt{3^2 + 4^2} = 5$, not $3 + 4 = 7$.

2. Treating dependent variables as independent

It seems reasonable because breaking a lengthy expression into simple modular sub-blocks and applying standard formulas to each block appears convenient. It fails because if the same measured quantity appears in multiple terms of an equation (such as $f = x / (x + y)$ or differences between positions on the same ruler), the terms are statistically correlated; neglecting cross-covariance leads to massive errors. It is detected whenever a variable bearing the same physical symbol appears in multiple places without computing analytical derivatives directly with respect to original independent inputs. It is corrected by differentiating the full function with respect to primary independent variables, or by including the complete covariance matrix. Mini-example: for $f = x - x$, treating the two terms as independent yields $\sigma_f = \sqrt{\sigma_x^2 + \sigma_x^2} = \sigma_x \sqrt{2}$, whereas physically $x - x \equiv 0$ identically, carrying exactly zero uncertainty.

3. Forgetting to square partial derivatives or exponents in relative formulas

It seems reasonable because the mathematical notation is dense, making it easy to mistake $\left(\frac{\partial f}{\partial x}\right)^2 \sigma_x^2$ for $\frac{\partial f}{\partial x} \sigma_x^2$. It fails because it destroys dimensional homogeneity: the term under the radical loses dimensions of $[y]^2$. It is immediately detected by carrying out a dimensional check inside the radical. It is corrected by verifying that every sensitivity coefficient enters the radical as a squared term. Mini-example: in $y = x^3$, fractional relative uncertainty is $\sigma_y / y = 3 (\sigma_x / x)$, so its variance contribution is $3^2 (\sigma_x / x)^2 = 9 (\sigma_x / x)^2$, never $3 (\sigma_x / x)^2$.

4. Using first-order Taylor expansion when relative uncertainties are large

It seems reasonable because Gauss's equation is universally taught as "the propagation formula". It fails because Gauss's formula truncates Taylor series at first order $\Delta f \approx f' \Delta x$. If $\sigma_x / x > 20\%$ and the second derivative $f''(x)$ is large, the quadratic term $\frac{1}{2} f''(x) (\Delta x)^2$ significantly biases both the expected value and the variance. It is detected by testing $\frac{\frac{1}{2} |f''(\bar{x})| \sigma_x^2}{|f'(\bar{x})| \sigma_x}$. If this ratio exceeds $5\%$, linear propagation is invalid. It is corrected by employing Monte Carlo simulations or analytical transformation of probability density functions. Mini-example: if $x = 1.0 \pm 0.8$ and $y = 1/x$, linear propagation yields $\bar{y} = 1.0$ and $\sigma_y = 0.8$, but values near zero push $y$ toward infinity, generating a heavily skewed distribution whose true mean is far greater than 1.

5. Mismatched decimal precision between central estimate and uncertainty

It seems reasonable because digital calculators show 8 or 10 digits and users reflexively transcribe every digit onto laboratory reports. It fails because reporting $g = (9.814352 \pm 0.2) \text{ m/s}^2$ is scientific nonsense: if there is uncertainty in the tenths ($0.2$), digits in the thousandths and millionths carry zero physical significance. It is detected by checking the final decimal place of the central value against that of the uncertainty. It is corrected by rounding standard uncertainty to one significant figure (or two if starting with 1 or 2) and adjusting the central estimate to match that exact decimal rank. Mini-example: $12.3456 \pm 0.4321$ rounds to $12.3 \pm 0.4$ (or $12.35 \pm 0.43$).

Quick detection rule

Before finalizing any uncertainty calculation, apply this three-point sanity filter:

1. Dimensions: Does the calculated uncertainty have the exact same physical units as the central value?
2. Dominance: Does the parameter with the largest relative uncertainty and highest exponent dominate the total variance?
3. Limiting cases: If one input uncertainty tends to zero, does the output uncertainty smoothly reduce to the expected single-variable formula?

Physical Applications of Error and Uncertainty Propagation

1. Elastic modulus determination in tensile testing

Appears in materials science and mechanical engineering when measuring Young's modulus $E = \frac{\sigma}{\epsilon} = \frac{F \cdot L_0}{A_0 \cdot \Delta L}$, using load cells for tension force $F$, gauge length $L_0$, specimen cross-section $A_0 = \frac{\pi d^2}{4}$ from micrometer readings, and elongation $\Delta L$ from laser extensometers. It is calculated using power-law propagation: $\left(\frac{\sigma_E}{E}\right)^2 = \left(\frac{\sigma_F}{F}\right)^2 + \left(\frac{\sigma_{L_0}}{L_0}\right)^2 + 4\left(\frac{\sigma_d}{d}\right)^2 + \left(\frac{\sigma_{\Delta L}}{\Delta L}\right)^2$. The result provides the certified tolerance interval for Young's modulus to assess whether aerospace titanium or steel alloys meet structural safety standards. The critical danger is ignoring the quadratic exponent weighting on diameter ($4 (\sigma_d/d)^2$): a mere $1\%$ uncertainty on cylindrical rod diameter escalates into a $2\%$ uncertainty in Young's modulus, potentially concealing material fatigue limits. As a final sanity check, extensometer resolution is verified to dominate for small strains and measurements are benchmarked against standard reference specimens.

2. Thermodynamics: specific heat capacity in calorimetry

Appears when measuring the specific heat capacity $c_x$ of an unknown sample via the method of mixtures: $c_x = \frac{(m_w c_w + C_{\text{cal}})(T_{\text{eq}} - T_w)}{m_x (T_x - T_{\text{eq}})}$. It is calculated by evaluating partial derivatives with respect to masses $m_x, m_w$, calorimeter water equivalent $C_{\text{cal}}$, and temperature readings $T_{\text{eq}}, T_w, T_x$. The result gives the material's specific heat with rigorous confidence intervals to distinguish between metals with close heat capacities (such as lead vs gold). The critical danger is assuming temperatures taken with the same probe are independent without assessing systematic calibration drift, or failing to realize that if $(T_{\text{eq}} - T_w)$ is tiny, temperature subtraction catastrophically amplifies relative error. As a final sanity check, experiments are designed such that $(T_{\text{eq}} - T_w) \geq 10\text{ K}$, keeping relative thermal error below $2\%$.

3. Astrophysics: stellar luminosity from trigonometric parallax

Appears when determining the absolute luminosity $L = 4\pi d^2 F$ of a distant star by measuring apparent radiative flux $F$ via photometry and distance $d = 1/p$, where $p$ is trigonometric parallax from astrometry satellites (e.g. Gaia). It is calculated via $L = 4\pi F p^{-2}$, yielding: $\frac{\sigma_L}{L} = \sqrt{\left(\frac{\sigma_F}{F}\right)^2 + 4 \left(\frac{\sigma_p}{p}\right)^2}$. The result establishes the admissible bolometric energy output used to position stars on the Hertzsprung-Russell diagram and constrain stellar mass and age. The critical danger is applying linear propagation when parallax is comparable to its own uncertainty ($p / \sigma_p < 5$, known as the nonlinear Lutz-Kelker bias), where asymmetric distributions cause first-order Taylor expansions to grossly underestimate true luminosity. As a final sanity check, astrometric sources are filtered to require $p / \sigma_p \geq 10$ prior to adopting linear Gaussian propagation.

4. Optics: focal length from thin lens conjugate points

Appears on optical benches when determining the focal length $f$ of a converging lens by measuring object distance $s$ and sharp image distance $s'$ on a viewing screen: $\frac{1}{f} = \frac{1}{s} + \frac{1}{s'}$, so $f = \frac{s s'}{s + s'}$. It is calculated by taking implicit derivatives of the reciprocal form: $-\frac{\sigma_f}{f^2} = \sqrt{\left(-\frac{1}{s^2}\right)^2 \sigma_s^2 + \left(-\frac{1}{s'^2}\right)^2 \sigma_{s'}^2}$, giving $\sigma_f = f^2 \sqrt{\frac{\sigma_s^2}{s^4} + \frac{\sigma_{s'}^2}{s'^4}}$. The result supplies manufacturing tolerances for compound achromatic lenses and microscope objectives. The critical danger is calculating propagation on $f = \frac{s s'}{s + s'}$ assuming numerator and denominator are independent, which overlooks strong algebraic covariance and falsely inflates estimated error bars. As a final sanity check, for symmetric conjugate points ($s = s' = 2f$), the expression collapses cleanly to $\sigma_f = \frac{1}{4}\sqrt{\sigma_s^2 + \sigma_{s'}^2}$.

5. Electromagnetism: electrical resistivity via four-point probe

Appears in solid-state physics when measuring resistivity $\rho$ of a prismatic conductor: $\rho = R \cdot \frac{w \cdot t}{L}$, where resistance is $R = V/I$, $w$ is width, $t$ is thickness, and $L$ is voltage contact separation. It is calculated by combining fractional uncertainties: $\frac{\sigma_\rho}{\rho} = \sqrt{\left(\frac{\sigma_V}{V}\right)^2 + \left(\frac{\sigma_I}{I}\right)^2 + \left(\frac{\sigma_w}{w}\right)^2 + \left(\frac{\sigma_t}{t}\right)^2 + \left(\frac{\sigma_L}{L}\right)^2}$. The result provides the intrinsic material resistivity for comparison with quantum Drude-Sommerfeld conduction models. The critical danger is failing to recognize that in thin films, thickness $t$ is often sub-micron with relative uncertainties that far exceed all other instrumentation channels combined. As a final sanity check, a variance decomposition table is constructed to confirm what fraction of experimental variance originates from thickness metrology before reporting resistivity values.

Guided Practice: Error and Uncertainty Propagation

Level 1: recognition

Identify the correct propagation formula for the following functions of independent random variables $x$ and $y$ with standard uncertainties $\sigma_x$ and $\sigma_y$:

1. $f(x, y) = 3 x - 5 y$
2. $g(x, y) = x^2 \sqrt{y}$
3. $h(x) = \ln(x)$

Solution:

1. By linear combination: $\sigma_f = \sqrt{3^2 \sigma_x^2 + (-5)^2 \sigma_y^2} = \sqrt{9 \sigma_x^2 + 25 \sigma_y^2}$.
2. By power monomial ($g = x^2 y^{1/2}$): $\frac{\sigma_g}{g} = \sqrt{4 \left(\frac{\sigma_x}{x}\right)^2 + \frac{1}{4} \left(\frac{\sigma_y}{y}\right)^2}$.
3. Differentiating $h'(x) = 1/x$: $\sigma_h = \left|\frac{1}{x}\right| \sigma_x = \frac{\sigma_x}{x}$ (the absolute uncertainty of a natural logarithm equals the relative uncertainty of its argument).

Level 2: direct application

Ohm's law gives the power dissipated in an electrical resistor as $P = I^2 R$. Laboratory measurements yield:

• Current: $I = (2.00 \pm 0.04) \text{ A}$
• Resistance: $R = (50.0 \pm 0.5) \,\Omega$

Calculate the central dissipated power $\bar{P}$ and its standard uncertainty $\sigma_P$.

Solution:

1. Central estimate: $\bar{P} = (2.00)^2 \cdot 50.0 = 4.00 \cdot 50.0 = 200.0 \text{ W}$.
2. Fractional uncertainties: $\sigma_I / I = 0.04 / 2.00 = 0.02$ ($2\%$); $\sigma_R / R = 0.5 / 50.0 = 0.01$ ($1\%$).
3. Relative propagation: $\left(\frac{\sigma_P}{P}\right)^2 = (2 \cdot 0.02)^2 + (1 \cdot 0.01)^2 = (0.04)^2 + (0.01)^2 = 0.0016 + 0.0001 = 0.0017$.
4. $\frac{\sigma_P}{P} = \sqrt{0.0017} \approx 0.04123$ ($4.12\%$).
5. Absolute standard uncertainty: $\sigma_P = 200.0 \cdot 0.04123 \approx 8.25 \text{ W}$.
6. Final statement: $P = (200 \pm 8) \text{ W}$ or $(200.0 \pm 8.2) \text{ W}$.

Level 3: tool selection

The thermoelectric voltage generated by a thermocouple is $V = \alpha (T_2 - T_1)$, with Seebeck coefficient $\alpha = (40.0 \pm 0.2) \,\mu\text{V/K}$, $T_2 = (373.15 \pm 0.10) \text{ K}$ and $T_1 = (273.15 \pm 0.10) \text{ K}$. Which propagation route is preferred: full multivariable expansion or a two-stage modular reduction?

Solution: The two-step modular method is clean and robust:

1. Define intermediate temperature difference $\Delta T = T_2 - T_1$. Because $T_1$ and $T_2$ are independent: $\sigma_{\Delta T} = \sqrt{0.10^2 + 0.10^2} = \sqrt{0.02} \approx 0.1414 \text{ K}$. Central value: $\Delta T = 100.00 \text{ K}$.
2. Now $V = \alpha \cdot \Delta T$ is a product of two independent variables: $\frac{\sigma_V}{V} = \sqrt{\left(\frac{0.2}{40.0}\right)^2 + \left(\frac{0.1414}{100.0}\right)^2} = \sqrt{(0.005)^2 + (0.001414)^2} = \sqrt{2.5 \times 10^{-5} + 2.0 \times 10^{-6}} \approx 0.005196$ Central value: $\bar{V} = 40.0 \times 100.0 = 4000 \,\mu\text{V}$. Absolute uncertainty: $\sigma_V = 4000 \times 0.005196 \approx 20.8 \,\mu\text{V} \approx 21 \,\mu\text{V}$. Result: $V = (4000 \pm 21) \,\mu\text{V}$.

Level 4: physical interpretation

Two identical nominal resistors $R_1 = (100 \pm 1)\,\Omega$ and $R_2 = (100 \pm 1)\,\Omega$ are wired in parallel: $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$. Compute the standard uncertainty of $R_{eq}$ assuming independent measurements. Explain why parallel combination uncertainty is lower than that of either individual resistor.

Solution: Central value: $R_{eq} = \frac{100 \times 100}{200} = 50\,\Omega$. Partial derivatives: $\frac{\partial R_{eq}}{\partial R_1} = \frac{R_2^2}{(R_1 + R_2)^2} = \frac{100^2}{200^2} = 0.25$. By symmetry, $\frac{\partial R_{eq}}{\partial R_2} = 0.25$. Gauss formula: $\sigma_{R_{eq}} = \sqrt{(0.25 \times 1)^2 + (0.25 \times 1)^2} = \sqrt{0.0625 + 0.0625} = \sqrt{0.125} \approx 0.354\,\Omega$ Result: $R_{eq} = (50.0 \pm 0.4)\,\Omega$. Physical interpretation: In parallel circuits, current splits between paths. If one branch resistance fluctuates upwards while the other fluctuates downwards, the total conductance variation partially self-compensates. The sensitivity derivatives are $0.25$, dampening single-component fluctuations.

Level 5: intentional mistake

A student evaluates the uncertainty of $f = \frac{x}{x + y}$ for $x = (10.0 \pm 0.5)$ and $y = (10.0 \pm 0.5)$. The student writes: "Relative error of $x$ is $0.5/10 = 0.05$. The denominator is $z = x + y = 20.0 \pm \sqrt{0.5^2 + 0.5^2} = 20.0 \pm 0.707$. Treating $f = x/z$ as a quotient of independent variables, I sum relative errors in quadrature: $\frac{\sigma_f}{f} = \sqrt{(0.05)^2 + (0.707/20)^2} = \sqrt{0.0025 + 0.00125} \approx 6.12\%$". Identify the fatal conceptual error and compute the exact result.

Conceptual failure: The parameter $x$ appears in both the numerator and the denominator. Thus, $x$ and $z = x + y$ are not statistically independent; they share a strong positive covariance. Applying the independent quotient formula ignores covariance and grossly overestimates uncertainty.

Rigorous correction: Differentiate $f(x, y)$ directly with respect to original independent variables $x$ and $y$: $\frac{\partial f}{\partial x} = \frac{y}{(x + y)^2} = \frac{10.0}{400} = 0.025, \quad \frac{\partial f}{\partial y} = -\frac{x}{(x + y)^2} = -0.025$ Since $x$ and $y$ are independent: $\sigma_f = \sqrt{(0.025 \cdot 0.5)^2 + (-0.025 \cdot 0.5)^2} = \sqrt{2 \cdot (0.0125)^2} = 0.0125 \sqrt{2} \approx 0.01768$ With central estimate $\bar{f} = 0.5$, the true relative uncertainty is $\sigma_f / \bar{f} = 0.01768 / 0.5 = 3.54\%$, significantly smaller than the student's erroneous $6.12\%$.