Propagation of Uncertainty I: The First-Order Formula

The Core Idea

Imagine you are standing on a hill that is slightly tilted. If you take a small step in any direction, your change in height is approximately proportional to the size of your step and the steepness of the slope in that direction. This is the essence of linearization. In our context, the "height" is the derived quantity \( f \), and the "steps" are the small deviations of the measured variables \( x_i \) from their mean values. Because the deviations are small, we can ignore the curvature of the function and treat it as a flat plane locally. The total uncertainty in \( f \) is then determined by how sensitive \( f \) is to each \( x_i \) (the slope) and how uncertain each \( x_i \) is (the step size). If the variables are independent, the uncertainties add in quadrature (like the sides of a right-angled triangle). If they are correlated, we must also account for the fact that when one variable is high, the other tends to be high or low, which adds or subtracts from the total uncertainty.

The General Formula

Let \( f(x_1, \dots, x_n) \) be a differentiable function of \( n \) measured quantities. Let \( \mu_i \) be the mean of \( x_i \), \( \sigma_i^2 \) be the variance of \( x_i \), and \( \mathrm{Cov}(x_i, x_j) \) be the covariance between \( x_i \) and \( x_j \). The variance of \( f \), denoted \( \sigma_f^2 \), to first order in the deviations is given by the sum of the variances weighted by the squared partial derivatives, plus the sum of the covariances weighted by the product of the partial derivatives.

This formula is the most general form. It accounts for all possible correlations between the input variables. The partial derivatives are evaluated at the mean values \( (\mu_1, \dots, \mu_n) \). Note that the double sum \( \sum_{i \neq j} \) includes both \( i < j \) and \( i > j \). Since \( \mathrm{Cov}(x_i, x_j) = \mathrm{Cov}(x_j, x_i) \), this is often written as \( 2 \sum_{i < j} \dots \) for clarity, but the summation notation above is standard.

The Independent Case

In many introductory experiments, we assume that the measurement errors in the different variables are independent. This means that the error in measuring \( x_i \) gives no information about the error in measuring \( x_j \). Mathematically, this implies that the covariance is zero for all \( i \neq j \): \( \mathrm{Cov}(x_i, x_j) = 0 \). The general formula simplifies significantly.

This is the form you will use most often. It states that the variance of the result is the sum of the variances of the inputs, each scaled by the square of the sensitivity of the function to that input. The standard uncertainty \( \sigma_f \) is the square root of this sum. This "root-sum-square" rule is the standard for independent errors.

Interpreting the Terms

Each term in the sum \( \left( \frac{\partial f}{\partial x_i} \right)^2 \sigma_i^2 \) represents the contribution to the total variance from the uncertainty in \( x_i \). The factor \( \left| \frac{\partial f}{\partial x_i} \right| \) is the sensitivity coefficient. If \( f \) changes rapidly with \( x_i \), a small error in \( x_i \) leads to a large error in \( f \). If \( f \) is insensitive to \( x_i \), the error in \( x_i \) contributes little to the total uncertainty. It is good practice to calculate each term separately to identify which measurement dominates the uncertainty. This tells you where to focus your efforts to improve the precision of the final result.