Maths bridge › Uncertainty arithmetic
Uncertainty arithmetic
Absolute against percentage uncertainty, and the three combination rules.
Three rules cover the course
An absolute uncertainty has units (±0.01 mm); a percentage uncertainty is that as a fraction of the value. For repeated readings, the absolute uncertainty is half the range of the repeats.
Combining: 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: squaring doubles it, a square root halves it.
The power rule is where marks are won: any formula with a d2 in it makes the diameter the quantity worth measuring most carefully, which is why the micrometer paragraph appears in practical after practical.
Worked and handed over
WORKED EXAMPLE
The diameter's double charge
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 nothing.
YOUR TURN
A density budget
A block's mass is measured to 1.0% and each side of its cube shape 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: 1.5% on V.
Density divides m by V, so add their percentages: 1.0 + 1.5 = 2.5% on ρ.
Where physics leans on this: Uncertainty and error · RP4: the Young modulus · RP5: resistivity. All eight skills: the maths bridge.