Uncertainty Propagation (Multiplication/Division/Powers) Calculator

Formula first

Overview

This equation provides rules for propagating uncertainties in calculations involving multiplication, division, or powers. For products or quotients (R=AB or R=A/B), the fractional uncertainties add. For powers (R=$A^n$), the fractional uncertainty of the base quantity is multiplied by the power 'n'. These rules are crucial for determining the reliability and precision of experimental results.

Symbols

Variables

\Delta A = Absolute Uncertainty in A, A = Value of A, \Delta B = Absolute Uncertainty in B, B = Value of B, n = Power (exponent)

Apply it well

When To Use

When to use: Apply this formula when you need to determine the uncertainty in a calculated quantity (R) that is derived from measurements (A, B) which themselves have uncertainties. It's used after performing the primary calculation (e.g., R=AB) to find the corresponding uncertainty in R, expressed as a fractional uncertainty.

Why it matters: Accurate uncertainty analysis is fundamental in experimental science, ensuring that conclusions drawn from data are statistically sound and reliable. It allows scientists and engineers to quantify the precision of their measurements and calculations, which is vital for comparing results, validating theories, and making informed decisions in research and industry.

Avoid these traps

Common Mistakes

• Adding absolute uncertainties instead of fractional uncertainties for multiplication/division.
• Forgetting to multiply by 'n' for the power rule, or applying it incorrectly.
• Confusing this rule with uncertainty propagation for addition/subtraction (where absolute uncertainties add).

One free problem

Practice Problem

A student measures the length of a rectangle as $A=2.5\pm 0.1$ m and its width as $B=4.0\pm 0.2$ m. Calculate the fractional uncertainty in the area $R=A\times B$. Round your answer to two significant figures.

Solve for: $delta{R}_{o}ve{r}_R$

Hint: For multiplication, add the fractional uncertainties of the individual measurements.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Halliday, Resnick, Walker, Fundamentals of Physics, 10th ed.
2. John R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd ed.
3. Wikipedia: Propagation of uncertainty
4. J. R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 3rd ed., University Science Books
5. NIST Engineering Statistics Handbook, Section 1.4.4.2: Combining Standard Uncertainties, National Institute of Standards and Technology
6. John R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd Edition
7. Halliday, Resnick, Walker, Fundamentals of Physics, 10th Edition
8. AQA A-level Physics — Practical Skills (3.2.1.4)