Uncertainty Propagation for Multiplication and Division

Step 1 - Compute the result:

Step 2 - Compute relative uncertainties:

Step 3 - Combine in quadrature:

Step 4 - Convert to absolute uncertainty:

Step 5 - Round and report:

Compute the result

Substitute your measured values into the formula and calculate f.

Compute each relative uncertainty

For each measured variable, divide its absolute uncertainty by its value: Δxᵢ/xᵢ. Express as a decimal or percentage.

Combine and convert back

Add all relative uncertainties in quadrature to get Δf/f, then multiply by f to get the absolute uncertainty Δf.

Using absolute instead of relative uncertainties

For multiplication and division you must use relative uncertainties Δx/x. Writing Δf = √(Δx² + Δy²) is the rule for addition - applying it to a product gives a completely wrong answer. Always check which operation your formula involves before choosing the rule.

Forgetting to convert back to absolute uncertainty

The propagation formula gives you Δf/f - a relative uncertainty. Your final answer must report an absolute uncertainty Δf with units. Multiply Δf/f by the computed value of f to get Δf before writing your final result.

Including exact constants as variables

Constants like ½, 2, π, and g (if treated as exact) have no uncertainty. Including them as variables inflates your propagated uncertainty incorrectly. Only include quantities that were actually measured with an instrument.

Confusing multiplication rule with addition rule

This is the single most common error in physics lab reports. Remember: addition/subtraction → combine absolute uncertainties. Multiplication/division → combine relative uncertainties. When in doubt, apply the general formula using partial derivatives.