Combining uncertainties
When quantities are added or subtracted their absolute uncertainties add; when they are multiplied or divided their relative uncertainties add.
Practise this conceptWhen quantities with their own uncertainties get combined, the rule for combining the uncertainties depends on what you did with the quantities. Adding or subtracting two measured lengths adds their absolute uncertainties directly: (10.0 ± 0.1) cm + (5.0 ± 0.1) cm gives 15.0 ± 0.2 cm.
Multiplying or dividing works differently — there, it's the relative (or percentage) uncertainties that add. A length known to ±2% multiplied by a length known to ±3% gives an area known to about ±5%, not ±2% or ±3% individually.
The two rules are easy to mix up, and mixing them up is one of the most common mistakes in this area — subtracting instead of adding uncertainties, for example, would suggest a calculation somehow got more certain by combining two measurements, which is never how it works. Uncertainties only ever grow (or, at best, stay comparable) when quantities are combined; they never shrink.
Key ideas
Requires: Uncertainty and error
Common misconceptions
- When two measured quantities are multiplied, their absolute uncertainties are added directly.
- Uncertainties always cancel out when quantities are subtracted.
- A quantity with a smaller absolute uncertainty always has a smaller relative uncertainty.