When can you ignore the smaller relative uncertainty in error propagation?

Conditions

• The comparison is between relative uncertainties.
• One measurement is much more accurate relative to the other.
• The approximation is acceptable for the intended level of precision.

Reasoning, step by step

1. Write the relative uncertainty quadrature formula: $\frac{\delta z}{z} = \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left(\frac{\delta y}{y}\right)^2}$.
2. Compare the two relative uncertainties.
3. Identify if one is much smaller than the other (e.g., 0.03% vs 5.4%).
4. Drop the smaller squared term from the sum under the square root.
5. Conclude that $\frac{\delta z}{z} \approx \frac{\delta y}{y}$, meaning the smaller uncertainty is negligible.

Example

The video shows an animation where a 0.03% relative uncertainty term shrinks and disappears, leaving only the 5.4% term. For a product $z=xy$, this approximation simplifies the absolute uncertainty to $\delta z \approx x\delta y$, making $x$ behave like a constant.

Common misconceptions

• Assuming it is always valid to treat the more accurate measurement as exact without checking the relative magnitudes.
• Forgetting that this approximation applies to relative uncertainties, not absolute uncertainties, for products and quotients.
• Believing that dropping a small term changes the fundamental quadrature rule for independent errors.

Watch the explanation

Explore next

• How do you propagate uncertainty for products and quotients?
• What is the difference between absolute uncertainty and relative uncertainty?
• Why are independent uncertainties added in quadrature instead of simply added?