Propagation of error

In statistics, propagation of uncertainty (or propagation of error) is the affect of variables' uncertainties (or errors) on the uncertainty of a function based on them. Mainly, the variables are measured in an experiment, and have uncertainties due to measurement limitations (e.g. instrument precision) which propagate to the result.

The uncertainty is usually defined by the absolute error — a variable that is probable to get the values x&plusmn;&Delta;x is said to have an uncertainty (or margin of error) of &Delta;x. In other words, for a measured value x, the true value is probable to be in [x&minus;&Delta;x, x+&Delta;x]. Uncertainties can also be defined by the relative error, &Delta;x/x, and then it is usually written as percentage. It is assumed that the probability of the true value to be in distinct distances from the measured value is normally distributed, with the uncertainty being the standard deviation.

This article explains how to calculate the uncertainty of a function, if the variables' uncertainties are known.

General formula
Let $$f(x_1,x_2,...,x_n)$$ be a function which depends on $$n$$ variables $$x_1,x_2,...,x_n$$. The uncertainty of each variable is given by $$\Delta x_j$$:


 * $$x_j \pm \Delta x_j\, .$$

If the variables are uncorrelated, we can calculate the uncertainty &Delta;f of f that results from the uncertainties of the variables:


 * $$\Delta f = \Delta f \left(x_1, x_2, ..., x_n, \Delta x_1, \Delta x_2, ..., \Delta x_n \right) = \left( \sum_{i=1}^n \left(\frac{\partial f}{\partial x_i}\Delta x_i \right)^2 \right)^{1/2} \, ,$$

where $$\frac{\partial f}{\partial x_j}$$ designates the partial derivative of $$f$$ for the $$j$$-th variable.

If the variables are correlated, the covariance between variable pairs, Ci,k := cov(xi,xk), enters the formula with a double sum over all pairs (i,k):


 * $$\Delta f = \left( \sum_{i=1}^n \sum_{k=1}^n \left(\frac{\partial f}{\partial x_i}\frac{\partial f}{\partial x_k}C_{i,k} \right) \right)^{1/2}\, ,$$

where Ci,i = var(xi) = &Delta;xi&sup2;.

After calculating $$\Delta f$$, we can say that the value of the function with it's uncertainty is:


 * $$f \pm \Delta f \, .$$

Example formulas
This table shows the uncertainty of simple functions, resulting from uncorrelated variables A, B, C with uncertainties &Delta;A, &Delta;B, &Delta;C, and a precisely-known constant c.

Example application: Resistance measurement
A practical application is an experiment in which one measures current, I, and voltage, V, on a resistor in order to determine the resistance, R, using Ohm's law, $$R = V / I.$$

Given the measured variables with uncertainties, I±&Delta;I and V±ΔV, the uncertainty in the computed quantity, ΔR is


 * $$\Delta R = \left( \left(\frac{\Delta V}{I}\right)^2+\left(\frac{V}{I^2}\Delta I\right)^2\right)^{1/2} = R\sqrt{\left(\frac{\Delta V}{V}\right)^2+\left(\frac{\Delta I}{I}\right)^2}.$$

Thus, in this simple case, the relative error ΔR/R is simply the square root of the sum of the squares of the two relative errors of the measured variables.