Error Propagation and Uncertainty Calculations Reference Cheat Sheet

Key Facts

• A measured value should be reported as $x \pm \Delta x$, where $x$ is the best estimate and $\Delta x$ is the absolute uncertainty.
• Fractional uncertainty is $\frac{\Delta x}{|x|}$, and percent uncertainty is $\frac{\Delta x}{|x|} \times 100\%$.
• For addition or subtraction, $q = a \pm b$, the maximum absolute uncertainty is $\Delta q = \Delta a + \Delta b$.
• For multiplication or division, $q = ab$ or $q = \frac{a}{b}$, the maximum fractional uncertainty is $\frac{\Delta q}{|q|} = \frac{\Delta a}{|a|} + \frac{\Delta b}{|b|}$.
• For a power, $q = x^n$, the fractional uncertainty is $\frac{\Delta q}{|q|} = |n|\frac{\Delta x}{|x|}$.
• For independent random uncertainties in addition or subtraction, quadrature gives $\Delta q = \sqrt{(\Delta a)^2 + (\Delta b)^2}$.
• For independent random uncertainties in multiplication or division, quadrature gives $\frac{\Delta q}{|q|} = \sqrt{\left(\frac{\Delta a}{a}\right)^2 + \left(\frac{\Delta b}{b}\right)^2}$.
• A final answer should usually round the uncertainty to one significant figure, or two if the first digit is $1$ or $2$, and round the measured value to the same decimal place.

Vocabulary

Absolute uncertainty

The amount $\Delta x$ by which a measured value $x$ may reasonably vary, written in the same units as the measurement.

Fractional uncertainty

The ratio $\frac{\Delta x}{|x|}$ that compares the absolute uncertainty to the size of the measured value.

Percent uncertainty

The fractional uncertainty written as a percentage, calculated by $\frac{\Delta x}{|x|} \times 100\%$.

Propagation of uncertainty

The process of finding the uncertainty in a calculated result from the uncertainties in the measured quantities.

Quadrature

A method for combining independent random uncertainties by adding their squares and taking the square root.

Significant figures

The meaningful digits in a measured or calculated value that reflect the precision of the data.