Physics › Maths bridge › Combining uncertainties
Combining uncertainties
Adding or subtracting: add absolute uncertainties. Multiplying or dividing: add percentage uncertainties. For a power n, multiply the percentage by n. An absolute uncertainty has the measurement's units, and a percentage uncertainty is the absolute uncertainty divided by the value, times 100.
Three rules cover the course
An absolute uncertainty has units (±0.01 mm); a percentage uncertainty is the absolute uncertainty divided by the value, multiplied by one hundred. For repeated readings, A-level questions take the absolute uncertainty as half the range of the repeats. That is an exam estimate, not a universal statistical definition of uncertainty.
The three combination rules below are the maximum-uncertainty convention commonly used in A-level questions, and they hold to first order for small uncertainties. When quantities add or subtract, add their absolute uncertainties. When they multiply or divide, add their percentage uncertainties. When a quantity is raised to a power, multiply its percentage uncertainty by that power, so squaring doubles it and a square root halves it.
Independent random standard uncertainties are normally combined in quadrature instead, by adding their squares and taking the square root, which gives a smaller result than adding them. Follow the convention stated by your exam board or by the question.
The power rule decides which measurement to take most care over. In any formula containing d2, a percentage uncertainty in the diameter counts twice in the area, which is why a micrometer paragraph turns up in practical after practical.
The rules in practice
WORKED EXAMPLE
A diameter that counts twice
A wire's diameter is measured as 0.36 ± 0.01 mm, and the area comes from A = πd2/4. Find the percentage uncertainty in the area.
The diameter's percentage uncertainty is 0.01/0.36 × 100 = 2.8%.
The area depends on d2, so its percentage uncertainty is double, 5.6%. The π and the 4 are exact and contribute no uncertainty.
GUIDED PRACTICE
Density, revisited under uncertainty
You met a version of this calculation in the measurements unit; this time every input carries its own uncertainty. A cubic block's mass is measured to 1.0% and its side length to 0.5%. Find the percentage uncertainty in its density ρ = m/V, thinking first about what power the side length carries, before opening the working.
Show the working
The volume is the side cubed, so the side's 0.5% counts three times, giving 1.5% on V.
Density divides m by V, so add the percentages. 1.0 + 1.5 = 2.5% on ρ, under the maximum-uncertainty convention.
Combining the same two figures in quadrature instead gives √(1.02 + 1.52) = 1.8%. Use whichever convention the question or your exam board asks for.
Where physics leans on this: Uncertainty and error · RP4: the Young modulus · RP5: resistivity. All eight skills: the maths bridge.