How do you propagate uncertainty for addition and subtraction?

Conditions

• The derived quantity is a sum or difference of two measured variables ($z = \pm x \pm y$).
• The variables $x$ and $y$ are independent measured quantities.

Reasoning, step by step

1. Start with the function $z = \pm x \pm y$.
2. Write the general two-variable quadrature propagation formula.
3. Compute the partial derivatives: $\frac{\partial z}{\partial x} = \pm 1$ and $\frac{\partial z}{\partial y} = \pm 1$.
4. Substitute the derivatives into the propagation formula.
5. Square the terms, which removes the plus or minus signs.
6. Simplify to get the final rule: $\delta z = \sqrt{(\delta x)^2 + (\delta y)^2}$.

Example

For $z = x - y$, the partial derivatives are $1$ and $-1$. Squaring them gives $1^2$ and $(-1)^2$, both equal to $1$. The final uncertainty is $\delta z = \sqrt{(\delta x)^2 + (\delta y)^2}$, exactly the same as for $z = x + y$.

Common misconceptions

• Thinking that subtraction propagates differently from addition because of the minus sign.
• Forgetting to square the absolute uncertainties before adding them together.
• Assuming the plus/minus signs affect the final magnitude of the uncertainty.

Watch the explanation

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