Error Analysis and Propagation of Uncertainty

Every number a physicist writes down is really two numbers: the value and the uncertainty that shadows it. A voltmeter reads 1.503 V, but the true voltage lies in a band around it; a stopwatch clicks at 2.01 s, but your thumb was late. This would be a mere nuisance if measured quantities were the end of the story — but they never are. We measure a length and a period to get $g$ , a mass and a volume to get a density, counts and a luminosity to get a cross-section. The quantity we care about is computed from several measured inputs, and its uncertainty must be assembled from the uncertainties of all of them. Error propagation is the disciplined bookkeeping for doing exactly that. Underneath, the framework is pure probability: each input is a random variable with a mean, a variance, and — crucially — covariances with the other inputs. But the whole subject is dominated by one approximation. If the function is smooth and the input spreads are small next to the scale over which it curves, then over the width of the data the function looks like a flat ramp. Linearize it, and because variance is quadratic in fluctuations, the output variance becomes a tidy quadratic form built from the gradient of the function and the covariance matrix of the inputs. From that single law flow every rule you memorized in a first lab course: absolute errors in quadrature for sums, fractional for products, the exponent for powers, $1/\sqrt{N}$ when you average. And from its failure — strong nonlinearity, large errors, correlated systematics — flow the reasons a serious experiment reaches for a full covariance matrix or a Monte Carlo instead.

Key takeaways

Error propagation answers one question: given measured inputs with known uncertainties, what is the uncertainty in a quantity computed from them?
The engine is a first-order Taylor expansion about the input means, turning a curved relation into a straight one whose variance is trivial to compute.
The master result is a quadratic form: $\sigma_y^2 = \sum_{i,j}(\partial_i f)(\partial_j f)\,\mathrm{Cov}(x_i,x_j)$ , gradient contracted twice against the input covariance matrix.
For independent inputs it collapses to sum-in-quadrature; sums add absolute errors, products add fractional errors, powers multiply the fractional error by the exponent.
Correlation is the subtle part: a shared systematic makes inputs covary, and in a difference or ratio can cancel almost exactly — the basis of every differential measurement.
Linearization fails for strongly curved functions, large relative errors, or near a stationary point; then the mean is biased and you must sample the full distribution.
Key idea: linearize the function about the means, then propagate — the output variance is the quadratic form $\sigma_y^2 = (\nabla f)^{\mathsf T}\,\Sigma\,(\nabla f)$ in the gradient of $f$ and the input covariance matrix $\Sigma$ .

The core insight

Linearize the function about the input means and propagate: the output variance is a quadratic form $\sigma_y^2 = (\nabla f)^{\mathsf T}\,\Sigma\,(\nabla f)$ in the gradient of $f$ and the input covariance matrix $\Sigma$ , reducing to sum-in-quadrature when the inputs are independent.

Where this lives today

Every quantitative result in modern physics carries a propagated uncertainty, and the frontier is where the simple quadrature rule stops being enough. The Particle Data Group's world averages combine measurements from dozens of experiments that share systematic errors — common luminosity scales, shared theoretical inputs, correlated calibrations — so the combinations use full covariance matrices, with correlation-aware budgets running to hundreds of entries. CODATA's periodic adjustment of the fundamental constants is one enormous correlated least-squares fit, propagating the covariances among the electron mass, the fine-structure constant, and the Planck constant so the redefined SI kilogram inherits a consistent uncertainty. Gravitational-wave astronomy lives or dies on it: LIGO's strain-calibration uncertainty propagates directly into the inferred masses and distance of every merger, reported as full posterior covariances. And because so many of these functions are strongly nonlinear, the field increasingly bypasses linearization entirely — the GUM's Supplement 1 (2008) blesses Monte Carlo propagation, sampling the inputs and histogramming the output to capture the skew the tangent-plane approximation misses. The linear law is the everyday workhorse; the covariance matrix and the Monte Carlo are what you reach for when precision is the product.

The physical picture

Picture each measurement not as a point but as a probability cloud — a bell curve centered on the best estimate $\bar x$ , with width equal to the standard uncertainty $\sigma_x$ . Push that cloud through the function $f$ that computes your result. Because $f$ curves, it stretches and skews the cloud on the way through. But if the cloud is narrow compared with how fast $f$ bends, then over its width $f$ is indistinguishable from its tangent line: a straight ramp of slope $\partial f/\partial x$ . A ramp maps a bell curve to a bell curve and scales the width, so the output width is the input width times the slope magnitude, $\sigma_y = |\partial f/\partial x|\,\sigma_x$ . That one sentence — cloud times slope — is the whole of single-variable error propagation, and the first figure makes it literal.

Figure

Linearizing f over the error cloud: a narrow input distribution of width σ_x at x̄ is mapped through the local tangent of slope ∂f/∂x, so the output width is σ_y = ∂f/∂x × σ_x. With slope 2.0 and σ_x = 0.10 the output spread is σ_y = 0.20

Two things complicate the single-variable picture, and both live in the multivariable law. First, when several clouds feed one result, independent fluctuations partly cancel — positive and negative excursions are equally likely and uncorrelated — so the errors add in quadrature, as the legs of a right triangle add to a hypotenuse, rather than linearly (the same $\sqrt N$ economy that makes averaging $N$ readings cut the error by $\sqrt N$ , not $N$ ). Second, absolute and fractional errors behave differently: addition preserves absolute scatter, multiplication preserves fractional scatter. The second figure turns the pendulum error budget — length to 0.5%, period to 0.2% and entering doubled as 0.4% — into this Pythagorean picture, reading off 0.64%.

Figure

Errors add in quadrature like the legs of a right triangle. For g from a pendulum the length term σ_L/L = 0.5% and the period term 2σ_T/T = 0.4% combine as σ_g/g = √(0.5² + 0.4²)% = 0.64% — the doubled period leg is the shorter one here

The core equations

Everything starts with a functional relationship: the result you report is a known, smooth function of the quantities you measured, and nothing is possible until you write this line down explicitly.

y = f(x_1, x_2, \dots, x_n) \qquad (1)

where: $y$ — the derived result whose uncertainty we want; $x_i$ — the measured inputs, random variables with means $\bar x_i$ ; $f$ — the known relation, assumed smooth near the means. Reading: the physics lives in $f$ ; propagation is what you do to $f$ once you know it. The one approximation the whole subject rests on replaces $f$ by its tangent plane at the mean point — expand to first order in the deviations $x_i - \bar x_i$ and keep the linear term.

y \approx f(\bar x) + \sum_{i} \left.\frac{\partial f}{\partial x_i}\right|_{\bar x}\,(x_i - \bar x_i) \qquad (2)

where: $\bar x$ — the vector of input means; $\partial f/\partial x_i$ — the sensitivity of $y$ to input $i$ , evaluated at the means and therefore constant. Reading: over the width of the data $y$ is an affine function of the inputs — a constant plus a weighted sum of fluctuations, exactly the object whose variance we can compute in closed form. Taking the variance of that linear combination gives the master result: because variance is bilinear in fluctuations, every pair of inputs contributes a term weighted by both sensitivities and their covariance.

\sigma_y^2 = \sum_{i=1}^{n}\sum_{j=1}^{n} \frac{\partial f}{\partial x_i}\,\frac{\partial f}{\partial x_j}\,\mathrm{Cov}(x_i, x_j) \qquad (3)

where: $\sigma_y^2$ — variance of the derived quantity; $\mathrm{Cov}(x_i,x_j)$ — covariance of inputs $i$ and $j$ , equal to $\sigma_i^2$ on the diagonal. Reading: the diagonal terms are the squared-sensitivity contributions; the off-diagonal correlations can add to or subtract from the total. The double sum is a vector-matrix-vector product — the form used in any serious analysis with many correlated channels.

\sigma_y^2 = (\nabla f)^{\mathsf T}\,\Sigma\,(\nabla f) \qquad (4)

where: $\nabla f$ — the gradient of sensitivities; $\Sigma$ — the input covariance matrix, $\Sigma_{ij}=\mathrm{Cov}(x_i,x_j)$ . Reading: this compact form is what metrology software evaluates, and it generalizes to a vector of outputs via the Jacobian, $\Sigma_y = J\,\Sigma\,J^{\mathsf T}$ . When the inputs are independent the covariance matrix is diagonal and the cross terms vanish, leaving the sum-in-quadrature every lab teaches.

\sigma_y^2 = \sum_{i=1}^{n}\left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_i^2 \qquad (5)

where: $\sigma_i$ — the standard uncertainty of input $i$ . Reading: the largest squared term dominates — quadrature is forgiving of small contributions, unforgiving of the big one. Two special cases cover most hand calculations. For a sum or difference the sensitivities are $\pm1$ , so absolute uncertainties add in quadrature, the sign irrelevant.

\sigma_y^2 = \sigma_a^2 + \sigma_b^2 \qquad (y = a \pm b) \qquad (6)

where: $\sigma_a,\sigma_b$ — absolute uncertainties of the two terms. Reading: subtraction does not shrink the absolute error even as it shrinks the result — the origin of precision loss when subtracting nearly equal numbers. For a product or ratio, differentiate the logarithm to turn multiplication into addition; the fractional uncertainties then add in quadrature.

\left(\frac{\sigma_y}{y}\right)^2 = \left(\frac{\sigma_a}{a}\right)^2 + \left(\frac{\sigma_b}{b}\right)^2 \qquad (y = ab \text{ or } a/b) \qquad (7)

where: $\sigma_y/y$ — the relative uncertainty. Reading: for products and quotients, work in percentages; the magnitudes drop out. A power is the same rule with a repeated factor: the exponent multiplies the fractional error directly, a negative exponent counting as much as a positive one.

\frac{\sigma_y}{y} = |n|\,\frac{\sigma_x}{x} \qquad (y = x^n) \qquad (8)

where: $n$ — the exponent, possibly non-integer or negative. Reading: cubing a quantity triples its fractional error; a $1/T^2$ dependence makes the period twice as damaging as a linear term. Finally, averaging $N$ independent repeats is itself an application of the law, yielding the single most important formula in experimental practice.

\sigma_{\bar x} = \frac{\sigma}{\sqrt N} \qquad (9)

where: $\sigma$ — the spread of a single measurement; $N$ — the number of independent repeats. Reading: the mean is $\sqrt N$ times more precise than one shot, not $N$ times — the diminishing return that sets how long you take data.

Derivation AThe general propagation law from the definition of variance

The goal is to derive the master formula, equation (3), from nothing but the linearized expansion and the definition of variance as a mean square deviation, making explicit why the result is a quadratic form in the gradient and why the input covariances — not just the variances — are the objects that appear. We compute the mean of the linearized output, subtract it to isolate the fluctuation, square, and take the expectation term by term, watching the constant sensitivities factor out until only the covariance of the inputs remains inside.

Step 1. Start from the linearized output $y \approx f(\bar x) + \sum_i \partial_i f\,(x_i - \bar x_i)$ and take its expectation. Each $\langle x_i - \bar x_i\rangle = 0$ by the definition of the mean, so every fluctuation term averages away and $\langle y\rangle = f(\bar x)$ . Why: the approximation is unbiased — the derived quantity sits at $f$ of the input means, so the spread we compute next is about the right center.

Step 2. Subtract the mean to isolate the fluctuation. Only the linear term survives, giving $y - \langle y\rangle \approx \sum_i \partial_i f\,\delta x_i$ with $\delta x_i \equiv x_i - \bar x_i$ . Why: variance cares only about deviations from the mean, so discarding the constant $f(\bar x)$ leaves a clean linear combination of the input fluctuations.

Step 3. Square and expand the product of the two sums, $\big(\sum_i \partial_i f\,\delta x_i\big)\big(\sum_j \partial_j f\,\delta x_j\big) = \sum_{i,j}\partial_i f\,\partial_j f\,\delta x_i\,\delta x_j$ . Why: squaring produces every cross-pair, and those $i\neq j$ pairs carry the correlation information naive quadrature throws away.

Step 4. Take the expectation. The sensitivities are constants evaluated at $\bar x$ , so they pull outside the average: $\sigma_y^2 = \sum_{i,j}\partial_i f\,\partial_j f\,\langle \delta x_i\,\delta x_j\rangle$ . Why: linearity of expectation is the only probabilistic fact used, and it isolates the mean product of two input fluctuations.

Step 5. Identify that as the covariance, $\langle \delta x_i\,\delta x_j\rangle = \mathrm{Cov}(x_i,x_j)$ , the diagonal giving the variances $\sigma_i^2$ . Substituting yields equation (3), or in matrix form $(\nabla f)^{\mathsf T}\Sigma(\nabla f)$ . Why: the whole law is just the definition of variance applied to a linear function — the physics entered only through the gradient. *As a check: setting $\mathrm{Cov}(x_i,x_j) = \sigma_i^2\,\delta_{ij}$ collapses the double sum to $\sum_i(\partial_i f)^2\sigma_i^2$ , recovering equation (5) for independent inputs.*

Go deeper · Derivation B · 1 minPropagating into g from a pendulum's length and period

Derivation BPropagating into g from a pendulum's length and period

The aim is to turn the propagation law into the single most-taught experimental result — the uncertainty in the acceleration due to gravity from a simple pendulum — and to show why the period must be measured about twice as carefully as the length to matter equally. We invert the pendulum relation for $g$ , use logarithmic differentiation to read the sensitivities off the exponents, then combine the two independent fractional errors in quadrature with real numbers.

Step 1. The small-amplitude period is $T = 2\pi\sqrt{L/g}$ ; square and rearrange to isolate the target, $g = 4\pi^2 L / T^2$ . Why: propagation always begins by writing the derived quantity as an explicit function of exactly what you measured — here $L$ and $T$ — with constants in the prefactor.

Step 2. Take the logarithm, $\ln g = \ln(4\pi^2) + \ln L - 2\ln T$ . Why: for a product-and-power relation the logarithm linearizes the sensitivities for free, and the uncertainty-free constant $4\pi^2$ drops out on differentiation.

Step 3. Differentiate: $\mathrm{d}g/g = \mathrm{d}L/L - 2\,\mathrm{d}T/T$ . The sensitivities are $+1$ for $L$ and $-2$ for $T$ . Why: these coefficients are the $\partial f/\partial x$ of the general law, now in the fractional variables that products and powers make natural.

Step 4. Treat $L$ and $T$ as independent and combine in quadrature, the $-2$ entering squared: $(\sigma_g/g)^2 = (\sigma_L/L)^2 + 4(\sigma_T/T)^2$ . Why: the factor $4=2^2$ is the moral — because $g\propto T^{-2}$ , a given fractional error in the period does four times the damage in variance that the same error in length would, so the period deserves twice the care.

Step 5. Put in numbers: with $\sigma_L/L = 0.5\%$ and $\sigma_T/T = 0.2\%$ , the period term enters as $0.4\%$ , and $\sigma_g/g = \sqrt{0.5^2 + 0.4^2}\,\% = \sqrt{0.41}\,\% \approx 0.64\%$ . Why: the two contributions are comparable, so neither dominates — a well-balanced experiment. *As a check: the power-law rule $\sigma_y/y=|n|\sigma_x/x$ with $n=-2$ reproduces the factor of 2 on the period, and $\sqrt{0.25+0.16}=0.6403\%$ confirms the numerical result.*

Go deeper · Derivation C · 1 minHow correlation cancels a shared systematic error

Derivation CHow correlation cancels a shared systematic error

This derivation shows the most useful consequence of keeping the off-diagonal covariance terms: when two inputs share a common source of error, forming their difference or ratio can cancel that shared error almost entirely, driving the propagated uncertainty far below the naive sum-in-quadrature estimate. We take the difference of two correlated measurements, apply the full quadratic law, insert the correlation coefficient, and read off the perfect-correlation limit where the error vanishes outright.

Step 1. Consider $y = x_1 - x_2$ with gradient components $\partial y/\partial x_1 = +1$ and $\partial y/\partial x_2 = -1$ . Why: the opposite signs are the mechanism of cancellation — they make the cross term subtract, which is what we want for a shared error that moves both inputs the same way.

Step 2. Apply the general law (3): $\sigma_y^2 = \sigma_1^2 + \sigma_2^2 + 2(1)(-1)\,\mathrm{Cov}(x_1,x_2) = \sigma_1^2 + \sigma_2^2 - 2\,\mathrm{Cov}(x_1,x_2)$ . Why: the cross term now carries a minus sign, so positive covariance reduces the total variance instead of inflating it.

Step 3. Write the covariance through the correlation coefficient, $\mathrm{Cov}(x_1,x_2) = \rho\,\sigma_1\sigma_2$ with $\rho\in[-1,1]$ : $\sigma_y^2 = \sigma_1^2 + \sigma_2^2 - 2\rho\,\sigma_1\sigma_2$ . Why: $\rho$ measures how much of the inputs' scatter is common; at $\rho=0$ we recover ordinary quadrature, and away from zero the shared part is subtracted.

Step 4. Take the perfectly shared error, $\rho = 1$ and $\sigma_1 = \sigma_2 = \sigma$ : then $\sigma_y^2 = \sigma^2 + \sigma^2 - 2\sigma^2 = 0$ . Why: if both inputs always err identically, their difference has no error — the working principle of bridge circuits, lock-in detection, and ratio thermometry, which null out drifts common to both arms. *As a check: at $\rho=0$ the expression reduces to $\sigma_y^2=\sigma_1^2+\sigma_2^2$ , recovering ordinary quadrature, and the $\rho=1$ null is the differential-measurement limit.*

Limiting cases

The propagation law is an approximation; its four regimes are worth holding in the head at once — each the law with a different piece switched on or off.

Case	Result	Meaning
Small, independent errors	$\sigma_y^2 = \sum_i (\partial_i f)^2\,\sigma_i^2$	Textbook sum-in-quadrature; exact for linear $f$ and excellent whenever the spreads are small next to the curvature scale.
Correlated errors	$\sigma_y^2 = \sum_{i,j}\partial_i f\,\partial_j f\,\mathrm{Cov}(x_i,x_j)$	The full quadratic form; off-diagonal terms can dominate the answer and can push $\sigma_y$ below the independent estimate.
Strong nonlinearity / large errors	$\langle y\rangle \approx f(\bar x) + \tfrac12\sum_i \partial_i^2 f\,\sigma_i^2$	Curvature biases the mean and the output distribution turns skew; the linear variance is only approximate — use Monte Carlo or an exact change of variables.
Repeated measurements (CLT regime)	$\sigma_{\bar x} = \sigma/\sqrt N$	Standard error of the mean; follows from propagating through $\bar x = \tfrac1N\sum_i x_i$ , and the CLT makes $\bar x$ Gaussian regardless of the parent shape.

Common trapSum-in-quadrature works only when the inputs are independent

The formula $\sigma_y^2 = \sum_i(\partial_i f)^2\sigma_i^2$ is not the general law — it is the special case in which the covariance matrix is diagonal. The moment two inputs share a common source of error — the same calibration constant, reference standard, drifting thermometer, or fitted background — they are correlated, and dropping the off-diagonal terms can be badly wrong in either direction. If the shared error affects a sum, quadrature under-estimates; if it affects a difference or ratio, quadrature over-estimates, sometimes by an order of magnitude, throwing away the very cancellation that makes precision measurement possible. Two companion traps ride along: combine absolute errors for sums but fractional errors for products (mixing them is a classic slip), and never trust the tidy quadrature formula near a stationary point where $\partial_i f = 0$ and the second-order term is all that is left.

Worked examples

1 · Density of a machined cylinder — building an error budget (fully worked)

A small cylinder is measured as mass $m = 25.00 \pm 0.05$ g, diameter $d = 12.00 \pm 0.05$ mm, and height $h = 40.00 \pm 0.10$ mm. Find its density and uncertainty, and identify which measurement to improve first.

Step 1. Write the derived quantity explicitly. A cylinder's volume is $V = \pi(d/2)^2 h = \pi d^2 h/4$ , so $\rho = 4m/(\pi d^2 h)$ — a product of powers, so we work fractionally. The central value: $V = \pi(12.00\,\text{mm})^2(40.00\,\text{mm})/4 = 4.5239\ \text{cm}^3$ , giving $\rho = 25.00/4.5239 = \mathbf{5.526\ g/cm^3}$ .

Step 2. Read the sensitivities off the exponents. From $\rho \propto m\,d^{-2}\,h^{-1}$ , logarithmic differentiation gives $\sigma_\rho/\rho = \sqrt{(\sigma_m/m)^2 + (2\,\sigma_d/d)^2 + (\sigma_h/h)^2}$ — the diameter enters doubled because $\rho\propto d^{-2}$ .

Step 3. Evaluate each fractional term: $\sigma_m/m = 0.05/25.00 = 0.20\%$ ; $2\,\sigma_d/d = 2\times(0.05/12.00) = 0.833\%$ ; $\sigma_h/h = 0.10/40.00 = 0.25\%$ .

Step 4. Combine in quadrature: $\sigma_\rho/\rho = \sqrt{0.20^2 + 0.833^2 + 0.25^2}\,\% = \sqrt{0.797}\,\% = \mathbf{0.89\%}$ , so $\sigma_\rho = 0.0089 \times 5.526 = 0.049$ , and $\rho = \mathbf{5.53 \pm 0.05\ g/cm^3}$ .

Step 5. Build the error budget — each source's share of the variance: diameter $0.833^2/0.797 = \mathbf{87\%}$ , height $8\%$ , mass $5\%$ . The diameter dominates overwhelmingly — both because it enters squared and because it was measured coarsely — so a better micrometer for $d$ , not a better balance, shrinks the answer. Halving $\sigma_d$ nearly halves the total; halving $\sigma_m$ barely moves it.

2 · A ratio that beats its own inputs — correlated systematics

Two resistors are read on the same ohmmeter, which carries a 2% gain-calibration uncertainty common to every reading, plus 0.5% independent random scatter per reading. What is the fractional uncertainty in each resistance, and in their ratio $R_1/R_2$ ?

Step 1. Model the shared error. Each reading is $R_i = G\,r_i$ , where $G$ is the common calibration factor (2% uncertain, perfectly correlated across the readings) and $r_i$ is the true value seen through 0.5% independent noise. A single resistance is known to $\sqrt{2^2 + 0.5^2}\,\% = \mathbf{2.06\%}$ — dominated by the calibration.

Step 2. Naive (wrong) ratio. Treating the two resistances as independent, you would combine their fractional errors in quadrature: $\sqrt{2.06^2 + 2.06^2}\,\% = \mathbf{2.92\%}$ .

Step 3. Correct ratio. In $R_1/R_2 = (G r_1)/(G r_2) = r_1/r_2$ the common gain $G$ cancels exactly. Only the independent 0.5% pieces survive: $\sqrt{0.5^2 + 0.5^2}\,\% = \mathbf{0.71\%}$ .

Step 4. Read the moral. The ratio is known to 0.71% even though each resistance is known only to 2.06%, and the naive independent estimate (2.92%) is four times too pessimistic. This is why a Wheatstone bridge reports resistance ratios, and why so many precision experiments measure a ratio or difference: the expensive systematic cancels, and the measurement rides on the small random part alone.

3 · The cost of subtracting nearly equal numbers

Two path lengths are $L_1 = 5.000 \pm 0.003$ m and $L_2 = 4.980 \pm 0.003$ m. Compare the fractional uncertainty of their sum and their difference.

Step 1. Both combine absolute errors in quadrature, $\sigma = \sqrt{0.003^2 + 0.003^2} = 0.0042$ m alike — subtraction does not reduce the absolute error.

Step 2. The sum $9.980 \pm 0.004$ m carries $0.0042/9.980 = \mathbf{0.04\%}$ ; the difference $0.020 \pm 0.004$ m carries $0.0042/0.020 = \mathbf{21\%}$ . The same absolute error is negligible in the sum and catastrophic in the difference — the notorious loss of significance when subtracting nearly equal numbers, a standing warning for experimental design and numerical algorithms alike.

Physics of the portrait

Error analysis has many parents, but one figure sits at its center: the man who tied the business of combining measurements to the structure of probability, and whose name is stamped on the bell curve every uncertainty is quietly assumed to follow.

Origins · 1 minCarl Friedrich Gauss and how this was discovered

Carl Friedrich Gauss

1777–1855 · the least-squares theory of measurement and the propagation of error

Gauss did not invent the idea that measurements err, but he was the first to make the errors themselves an object of exact mathematics. Computing the orbit of the newly lost asteroid Ceres in 1801, he needed a best trajectory from scattered, uncertain observations — and so systematized the method of least squares, publishing it in the Theoria Motus of 1809 (Legendre had printed the recipe in 1805, and the two quarrelled over priority for years). Gauss's deeper move was to ask which error distribution makes the arithmetic mean the most probable estimate, and to show the answer is the bell-shaped curve now called Gaussian. In the Theoria Combinationis of 1823 he founded the theory on variance rather than any assumed distribution: least-squares estimates have minimum variance among linear unbiased estimators (now shared with Markov), and the variance of a computed quantity follows from the variances and covariances of its inputs — the propagation of error in all but name. The quadratic form $(\nabla f)^{\mathsf T}\Sigma(\nabla f)$ this article is built around is Gauss's insight in modern dress; Laplace supplied the central limit theorem that explains why so many errors are Gaussian, and Gauss the machinery for living with them.

History

1805

— Legendre publishes the method of least squares as a computational recipe

1809

— Gauss ties least squares to the normal law of errors in the Theoria Motus

1810

— Laplace proves the central limit theorem, explaining why errors tend to Gaussian

1823

— Gauss's Theoria Combinationis founds error propagation on variance and covariance

1993

— the ISO Guide to the Expression of Uncertainty in Measurement standardizes propagation for metrology

Going further

Texts. Taylor, An Introduction to Error Analysis, 2nd ed., Ch. 3 and 9 (the standard undergraduate treatment) · Bevington & Robinson, Data Reduction and Error Analysis for the Physical Sciences, Ch. 3–4 · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, Ch. 30–31 · Arfken & Weber, Mathematical Methods for Physicists, probability chapter · Stone & Goldbart, Mathematics for Physics · BIPM, Guide to the Expression of Uncertainty in Measurement (GUM) and its Supplement 1 on Monte Carlo methods. The key numerical results worked out in this article are: the pendulum $\sigma_g/g = \sqrt{(\sigma_L/L)^2 + 4(\sigma_T/T)^2} = 0.64\%$ from $0.5\%$ and $0.2\%$ ; the density budget $\rho = 5.53\pm0.05\ \text{g/cm}^3$ (total $0.89\%$ , diameter share $87\%$ ) from $m=25.00\pm0.05$ g, $d=12.00\pm0.05$ mm, $h=40.00\pm0.10$ mm; the correlated ratio, $0.71\%$ correct versus $2.92\%$ naive; the power-law result $6\%$ from $n=4$ and $1.5\%$ ; and the sum-versus-difference loss, $0.04\%$ against $21\%$ .

Prerequisites. Expectation, Variance and Moments · Random Variables and Probability Distributions · Joint, Marginal and Conditional Distributions · Cumulative Distribution Functions and Change of Variables. Leads to. The Multivariate Gaussian · The Central Limit Theorem · The Gaussian (Normal) Distribution.

Ask the tutor: Bring me any formula you measure a quantity with, and I will walk you through propagating its uncertainty, spotting hidden correlations, and building the error budget that tells you what to measure better.

Why do errors add in quadrature instead of just adding?

Because independent fluctuations partly cancel. If two inputs each wobble randomly, sometimes both are high, sometimes both low, but often one is high while the other is low — those opposite excursions partially offset in the result. Linear addition would assume they always conspire in the same direction (a worst case), massively over-estimating the typical spread. The variance is the quantity that adds cleanly for independent variables, and its square root gives the $\sqrt{\sigma_a^2+\sigma_b^2}$ rule. The one exception is fully correlated inputs, which really do move together; then, and only then, the errors add linearly.

When is the linear propagation formula not good enough?

Three situations. First, when the function is strongly curved over the width of the input distribution — an exponential, a resonance denominator, a ratio with a small denominator — so the tangent plane misjudges both the width and the center. Second, when the relative errors are large (tens of percent), so the "small deviation" premise fails. Third, near a stationary point where a first derivative vanishes and the leading behavior is quadratic. In all three, sample the inputs from their distributions and push them through the exact function — Monte Carlo propagation — which recovers the true, often skewed, output distribution and asymmetric error bars.

What is the difference between a random and a systematic error here?

A random error scatters independently from measurement to measurement, so averaging beats it down as $1/\sqrt N$ . A systematic error shifts many measurements the same way — a mis-calibrated ruler, a slow clock — and no averaging removes it, because it is perfectly correlated across your repeats. In the propagation framework the distinction is exactly correlation: random errors populate the diagonal of the covariance matrix and combine in quadrature, while shared systematics fill the off-diagonal terms. That is why differential and ratio measurements are so powerful — they are engineered so the correlated systematic cancels, leaving only the random part.

Do I need the full covariance matrix, or is sum-in-quadrature enough?

For a quick lab calculation with genuinely independent inputs, sum-in-quadrature is exactly the diagonal of the full law and all you need. You need the off-diagonal terms whenever inputs share a source of error: the same calibration constant feeding several channels, a fitted parameter used in more than one place, correlated theoretical inputs. The practical test: could a single mistake shift more than one of my inputs at once? If yes, they are correlated and the cross terms matter. Serious combinations — particle-physics averages, constant adjustments, gravitational-wave parameter estimates — always carry the full matrix, because ignoring correlations can be wrong by an order of magnitude.

Where does the standard error of the mean, σ/√N, come from?

It is error propagation applied to averaging itself. The mean is $\bar x = \tfrac1N\sum_i x_i$ , so its sensitivity to each of the $N$ independent readings is $\partial \bar x/\partial x_i = 1/N$ . Sum in quadrature: $\sigma_{\bar x}^2 = \sum_i (1/N)^2\sigma^2 = \sigma^2/N$ , hence $\sigma_{\bar x}=\sigma/\sqrt N$ . Variances of independent terms add, $N$ of them each scaled by $1/N^2$ . This is why the improvement is only $\sqrt N$ : to halve your error you must take four times as much data — the diminishing return that governs every long measurement campaign.

Is uncertainty the same thing as the standard deviation?

Closely related, but not identical in usage. The standard uncertainty $\sigma$ is a standard deviation — of the distribution describing your knowledge of the quantity. Two subtleties matter. First, uncertainties come in types: Type A are evaluated statistically from repeated data, Type B from other information such as a tolerance or prior calibration (converted to an equivalent standard deviation). Second, results are often quoted with a coverage factor — " $\pm2\sigma$ " for roughly 95% confidence — so check whether a stated $\pm$ is one standard uncertainty or an expanded one. Propagation combines standard uncertainties; apply the coverage factor only at the very end.