error propagation

You measure the two sides of a rectangle, each with some uncertainty, and now you want the uncertainty on its area. The errors in the inputs do not vanish when you multiply; they flow through the calculation and reappear as an error on the output. Error propagation is the bookkeeping that carries measurement uncertainties through any formula to tell you how precisely you actually know the derived result.

For a quantity f(x, y, ...) computed from measured inputs with uncertainties sigma_x, sigma_y, ..., assuming the errors are independent and small enough that f is locally linear, the variance of f is sigma_f^2 = (df/dx)^2 sigma_x^2 + (df/dy)^2 sigma_y^2 + ..., where each input's uncertainty is weighted by the square of the partial derivative that measures f's sensitivity to it, and the contributions add in quadrature. Two everyday corollaries follow: for sums or differences f = x +/- y the absolute errors add in quadrature, sigma_f^2 = sigma_x^2 + sigma_y^2; for products or ratios f = x y or x / y the fractional errors add in quadrature, (sigma_f / f)^2 = (sigma_x / x)^2 + (sigma_y / y)^2.

Every derived physical result needs this, whether it is a resistance from V / I, a gravitational g from a pendulum's period, or a density from mass and volume; without propagating the uncertainty the number is not yet a measurement. The honest caveat is that the quadrature formula assumes the input errors are independent (uncorrelated) and small. If the inputs share a common source of uncertainty you must add covariance terms, + 2 (df/dx)(df/dy) cov(x, y); ignoring correlation can badly over- or under-state the final error.