Chapter 3: Propagation of Uncertainties

The chapter covers both digital and analog measurement contexts and introduces the square root rule for counting experiments, establishing that uncertainty in counted random events equals the square root of the count itself. Two complementary approaches to error propagation are presented: provisional rules that represent worst-case scenarios where absolute uncertainties add in sums and differences, while fractional uncertainties add in products and quotients; and the more realistic quadrature method that accounts for independent random errors by combining uncertainties as the square root of the sum of squares, yielding smaller and more probable final estimates. Special cases are highlighted, including multiplication by exact constants and raising measurements to fixed powers, where fractional uncertainty is multiplied by the exponent's absolute value. For single-variable functions such as trigonometric or radical expressions, derivatives determine how input uncertainties translate to output uncertainties. The chapter emphasizes a critical distinction between stepwise propagation through sequential operations and the general formula using partial derivatives, explaining how stepwise methods overestimate final uncertainty when variables appear multiple times in an equation due to compensating errors. The general formula, combining partial derivatives of all variables as a quadrature sum, provides the most comprehensive and accurate approach to uncertainty propagation in complex multi-variable functions without overestimation.