Error Propagation: Multiplication & Division

For any measured quantity $x$, its absolute uncertainty $\Delta x$ has the same units as $x$ itself. The fractional (or relative) uncertainty is the dimensionless ratio $\frac{\Delta x}{x}$. For example, if a mass is $m = 2.50 \pm 0.05\,\text{kg}$, the absolute uncertainty is $\Delta m = 0.05\,\text{kg}$ and the fractional uncertainty is $\frac{0.05}{2.50} = 0.020$, i.e. 2.0%.

Suppose $Z = A \times B$, where $A = A_0 \pm \Delta A$ and $B = B_0 \pm \Delta B$. To find the maximum possible value of $Z$, we multiply the maximum values of both: $Z_{\max} = (A_0 + \Delta A)(B_0 + \Delta B) = A_0 B_0 + A_0 \Delta B + B_0 \Delta A + \Delta A \Delta B$. Since $\Delta A$ and $\Delta B$ are small, the product $\Delta A \Delta B$ is negligibly small (second-order term) and is dropped. The absolute uncertainty in $Z$ is therefore $\Delta Z \approx A_0 \Delta B + B_0 \Delta A$. Dividing both sides by $Z = A_0 B_0$ gives the fractional uncertainty rule: $$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$$.

Suppose $Z = \frac{A}{B}$. Using a similar approach (or a first-order Taylor expansion), the maximum fractional change in $Z$ is again the sum of the individual fractional uncertainties: $$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$$. The rule is identical to multiplication — uncertainties always add when quantities are multiplied or divided. This makes sense: dividing by a quantity whose value is uncertain contributes just as much imprecision as multiplying by that quantity.

If a quantity is raised to a power, e.g. $Z = A^n$, the fractional uncertainty is multiplied by the magnitude of that power: $$\frac{\Delta Z}{Z} = |n| \frac{\Delta A}{A}$$. For a general expression such as $Z = \frac{A^p \cdot B^q}{C^r}$, the rule extends to: $$\frac{\Delta Z}{Z} = p\frac{\Delta A}{A} + q\frac{\Delta B}{B} + r\frac{\Delta C}{C}$$, where $p$, $q$, $r$ are positive exponents. This powerful formula covers the vast majority of expressions encountered in physics experiments.

After computing the fractional uncertainty $\frac{\Delta Z}{Z}$, multiply it by the central value $Z$ to recover the absolute uncertainty $\Delta Z$. The final result is then quoted as $Z \pm \Delta Z$ with appropriate significant figures — conventionally, $\Delta Z$ is rounded to one or two significant figures, and $Z$ is quoted to the same decimal place.

• Z — Derived quantity (varies): The calculated result obtained by multiplying or dividing two measured quantities.
• \Delta Z — Absolute uncertainty in Z (same as Z): The uncertainty in the final result, found by multiplying the fractional uncertainty by Z.
• A,\, B — Measured quantities (varies): Individual physical quantities measured in the experiment, e.g. length, time, mass.
• \Delta A,\, \Delta B — Absolute uncertainties (same as respective quantity): The half-range or estimated uncertainty associated with each measured quantity.

Problem. A student measures the dimensions of a rectangular metal block to calculate its volume. She records: length $L = 12.4 \pm 0.1\,\text{cm}$, width $W = 5.2 \pm 0.1\,\text{cm}$, height $H = 3.0 \pm 0.1\,\text{cm}$. Find the volume and its absolute uncertainty.

Answer. Volume = 193 ± 12 cm³ (approximately 6% fractional uncertainty)

• Absolute and Fractional Uncertainty — Understanding the difference between absolute and fractional uncertainty is a prerequisite for applying the multiplication and division error propagation rule correctly.
• Error Propagation: Addition and Subtraction — In addition and subtraction, absolute uncertainties add — the complementary rule to the fractional-uncertainty rule used in multiplication and division.
• Significant Figures and Rounding — After propagating errors, results must be rounded correctly to match the precision indicated by the calculated uncertainty.
• Systematic and Random Errors — Error propagation applies to random (statistical) uncertainties; knowing the difference between systematic and random errors clarifies when these rules are appropriate.