Propagation of Error

Propagation of error is simply figuring out how uncertain a calculated answer is, given how uncertain the measurements going into it were. The main formula, $\sigma_f^2 \approx \sum_i \left(\frac{\partial f}{\partial x_i}\right)^2\sigma_i^2$, has two small pieces per input: $\frac{\partial f}{\partial x_i}$ (how sensitive the answer is to that input) times $\sigma_i$ (how shaky the input is), squared and added up. The product-and-quotient version is the same thing in percentages, with each exponent as a weight. If you followed the last lesson's tolerance stacks, you already know this; the derivatives just replace the $\pm1$ coefficients.

One number, three measurements. A lab checks a copper rod by weighing it and measuring its diameter and length, then computing its density $\rho = \dfrac{4m}{\pi d^2L}$. Each measurement has its own small random error. How uncertain is the density? And if the lab can afford one upgrade, should it buy a better scale or a better micrometer?

Zoom in until it's linear. Near the true values, any smooth formula is nearly its tangent plane, exactly as in Differentials:

The right side is a constant plus a linear combination of the input errors, which is what Tolerance Stack-Up handled: coefficients get squared and independent variances add. So

Propagation of error is a tolerance stack whose coefficients are partial derivatives.

Calculus's worst case vs statistics' typical case. The differential bound $|df| \le \sum |f_i|\,|dx_i|$ assumes every error pushes the same way at full size: that's a worst-case stack. Treating the errors as independent random variables adds them in quadrature instead, an RSS stack. The quadrature answer is smaller, and it's what you'd actually see if you repeated the measurement many times.

Products and quotients: take logs. For $f = c\,x^ay^b$, $\ln f = \ln c + a\ln x + b\ln y$, so $\dfrac{df}{f} = a\dfrac{dx}{x} + b\dfrac{dy}{y}$. The relative errors form a linear combination with the exponents as coefficients, so

Division ($b = -1$) still adds. Squaring ($a = 2$) doubles a relative error, and a square root ($a = \tfrac12$) halves it.

The error budget. Each term $f_i^2\sigma_i^2$ is one input's share of the output variance. In the copper rod the diameter, squared in the formula, supplies about 75% of the variance, so a better micrometer helps far more than a better scale. Always attack the biggest term: shrinking a small one barely moves the square root.

Averaging is a special case. The mean of $n$ independent readings, $\bar x = \frac1n\sum x_i$, has every $f_i = \frac1n$, so $\sigma_{\bar x}^2 = n\cdot\frac{\sigma^2}{n^2}$ and $\sigma_{\bar x} = \sigma/\sqrt n$: the standard error from the central limit theorem, derived in one line.

A common mistake. It's tempting to treat the formula as always safe, since it came from calculus. It's only as good as the straight-line approximation behind it. When the linear approximation fails:

• Big errors on curved formulas. The second-order term shifts the mean: $E(X^2) = \mu^2 + \sigma^2$, not $\mu^2$. Harmless when $\sigma/\mu$ is a few percent, misleading at 20%.
• Subtracting nearly equal numbers. A difference of 5 °C between two readings near 80 °C keeps both readings' absolute errors but has a tiny value, so its relative error explodes.
• Correlated inputs, like two readings from the same instrument: add $2f_if_j\operatorname{Cov}(x_i, x_j)$.
• Flat spots, where a derivative is near zero: the first-order term vanishes and the formula underestimates.

When any of these bite, simulate instead: Monte Carlo Tolerance Analysis. And remember that propagation handles random error only. A bias in an input, from Measurement Error, passes through as a bias, untouched by averaging.

Every lab report's error bars, every sensor datasheet's accuracy figure and every calibration certificate come from propagation of error. Calibration labs follow an international measurement-uncertainty guide that is this lesson's formula plus careful bookkeeping. Flow meters, strain gauges and satellite navigation receivers publish uncertainty budgets built term by term, and engineers read those budgets to decide which sensor to upgrade, which is far cheaper than upgrading all of them.