How do you propagate uncertainty for products and quotients?

Conditions

• The derived quantity is a product ($z=xy$) or a quotient ($z=\frac{x}{y}$).
• The variables $x$ and $y$ are nonzero so the relative uncertainty ratios are defined.
• The uncertainties are treated as independent.

Reasoning, step by step

1. Identify the function as a product $z=xy$ or a quotient $z=\frac{x}{y}$.
2. Write the general two-variable absolute uncertainty propagation formula using partial derivatives.
3. Compute the partial derivatives for the product or quotient.
4. Divide the resulting absolute uncertainty expression by $z$ to convert it into relative form.
5. Simplify the expression to reveal that the relative uncertainties add in quadrature: $\frac{\delta z}{z} = \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left(\frac{\delta y}{y}\right)^2}$.

Example

For $z=xy$, the raw propagated form is $\delta z = \sqrt{(y\delta x)^2 + (x\delta y)^2}$. Dividing by $z=xy$ and simplifying yields $\frac{\delta z}{z} = \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left(\frac{\delta y}{y}\right)^2}$. The same relative rule applies to $z=\frac{x}{y}$.

Common misconceptions

• Trying to add the absolute uncertainties directly for products and quotients, which is mathematically incorrect.
• Stopping at the raw derivative form for products and quotients without dividing by $z$ to find the simpler relative-uncertainty pattern.
• Assuming the minus sign in a quotient affects the final quadrature formula (it disappears after squaring).

Watch the explanation

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