Propagate Measurement Uncertainty Across Formulas

Add variances, not standard deviations. For y = a ± b, σ_y = √(σ_a² + σ_b²) when a and b are independent. Quadratic addition (in quadrature) reflects the fact that random errors don't generally cancel. Example: L₁ = 10.0 ± 0.2 cm, L₂ = 5.0 ± 0.1 cm. The sum L₁ + L₂ = 15.0 ± √(0.04 + 0.01) = 15.0 ± 0.224 cm. The difference L₁ − L₂ has the same uncertainty as the sum because the variance formula doesn't care about the sign. If a and b are correlated, add a covariance term: σ_y² = σ_a² + σ_b² + 2·Cov(a, b).

Add relative uncertainties in quadrature. For y = a · b or y = a / b, (σ_y / y)² = (σ_a / a)² + (σ_b / b)². Example: L = 10.0 ± 0.2, W = 5.0 ± 0.1. Area = L · W = 50.0. Relative uncertainty: √((0.2/10)² + (0.1/5)²) = √(0.0004 + 0.0004) = 0.0283. Absolute: σ_A = 50.0 · 0.0283 ≈ 1.4 cm². Note: σ_A is NOT just σ_L + σ_W = 0.3. Adding absolute uncertainties for products is the most common error in error propagation. Always convert to relative first, add in quadrature, convert back.

Random errors are random; they sometimes cancel and sometimes accumulate. Linear addition assumes worst-case alignment (all errors push the same way), which over-estimates the true spread. Quadratic addition (the variances add) is what falls out of statistical theory for independent random variables: Var(X + Y) = Var(X) + Var(Y), so SD(X + Y) = √(Var(X) + Var(Y)). For systematic errors that always push the same direction, linear addition is correct. For independent random errors, quadrature is correct. Most lab uncertainties are random, hence the standard quadrature rule.

For y = x^n, σ_y / y = |n| · σ_x / x. The exponent multiplies the relative uncertainty linearly, not in quadrature. Example: volume of a sphere V = (4/3)·π·r³. If r = 5.0 ± 0.1 cm (2% relative), then σ_V / V = 3 · (0.1/5) = 6%. So V = 523.6 ± 31.4 cm³. The cube amplifies relative uncertainty threefold. This is why high-power dependencies are unforgiving: a 2% error in r becomes a 6% error in V. For y = x^(1/2), the half-exponent halves the relative uncertainty, which is why averaging reduces uncertainty.

Absolute uncertainty (σ) carries the units of the measurement: 5.0 ± 0.1 cm. Relative uncertainty (σ/x) is dimensionless and useful for comparing precision across scales: a 0.1 cm error on a 5 cm measurement (2%) is much worse than a 0.1 cm error on a 100 cm measurement (0.1%). Add absolute uncertainties in quadrature for sums and differences. Add relative uncertainties in quadrature for products and quotients. Always report the final result with absolute uncertainty matching the unit of the measured quantity, but the intermediate computation should use the form that fits the formula.

When the relative uncertainty σ/x is large (above roughly 10%) or when the function is strongly nonlinear in the range of the inputs. The linear approximation u²(y) = Σ (∂f/∂xᵢ)² · u²(xᵢ) treats f as locally linear around the central value; for big σ relative to the curvature scale, that's wrong. Fix: Monte Carlo. Sample each input from its distribution thousands of times, evaluate f for each sample, take the standard deviation of the resulting f values. Python's uncertainties package handles linear propagation; numpy.random and scipy.stats handle Monte Carlo.

GUM is the Guide to the Expression of Uncertainty in Measurement (JCGM 100:2008), the international metrology standard for reporting measurement uncertainty. It distinguishes Type A uncertainty (statistical, from repeated measurements) and Type B (other sources: calibration certificates, manufacturer specs). It introduces the combined standard uncertainty u_c, the expanded uncertainty U = k·u_c with coverage factor k (typically 2 for ~95% confidence), and effective degrees of freedom via the Welch-Satterthwaite formula. For published measurements in physics, chemistry, and engineering, GUM-compliant reporting is the expected format.

Quote uncertainty to 1 or 2 significant figures, then round the value to the same decimal place. NIST recommends 2 sig figs for the uncertainty by default. Example: x = 12.3456 ± 0.0234 reports as 12.346 ± 0.023, or 12.35 ± 0.02 to 1 sig fig. Don't write 12.3456 ± 0.02. The mismatch (4-decimal value, 2-decimal uncertainty) overstates precision. The rule of thumb: the last digit of the value should be at the decimal place of the uncertainty's first significant figure.