Error Propagation
Combine measurement uncertainties through a calculation, rounded the way a lab report wants them.
Relative uncertainty 2.968% · unrounded 50 ± 1.484082
Every measurement carries an uncertainty, and calculations carry it through. Absolute uncertainties add in quadrature when you add or subtract; relative uncertainties add in quadrature when you multiply or divide. So 12.5 ± 0.2 times 4.0 ± 0.1 gives 50.0 ± 1.5.
Why quadrature rather than plain addition
The obvious guess is that uncertainties simply add, and that is what a worst-case analysis would say. But independent uncertainties are unlikely to both land at their extreme in the same direction at the same time — one measurement reading a little high is as likely to meet another reading a little low. Adding them in quadrature, √(δx² + δy²), accounts for that partial cancellation and gives the realistic combined spread rather than the worst imaginable one.
The consequence is that the largest uncertainty dominates. Combining 0.5 with 0.1 gives √(0.25 + 0.01) ≈ 0.51 — the small one barely registers. That is worth knowing before you spend an afternoon improving the precision of a quantity that was never the limiting factor.
Absolute for sums, relative for products
Which quantity goes into the quadrature depends on the operation, and mixing them up is the usual mistake. Adding or subtracting works on absolute uncertainties in the original units. Multiplying or dividing works on relative uncertainties — the fractions δx/x — and the result is converted back to an absolute figure at the end. Raising to a power is the same relative rule with the exponent as a multiplier, which is why squaring a measurement doubles its relative uncertainty.
absolute uncertainties combine for sums; relative uncertainties combine for products
- 1 Identify the operation. Multiplying 12.5 ± 0.2 by 4.0 ± 0.1 is a product, so the relative rule applies.
- 2 Convert each uncertainty to a fraction. 0.2 ÷ 12.5 = 0.016 and 0.1 ÷ 4.0 = 0.025.
- 3 Combine them in quadrature. √(0.016² + 0.025²) = 0.0297, so the result carries a 2.97% relative uncertainty.
- 4 Convert back to absolute. 0.0297 × 50 = 1.48.
- 5 Round the uncertainty, then match the value. One significant figure would give 1, but the leading digit is 1 so two are kept: 50.0 ± 1.5.
The rule for each operation
δ is the uncertainty in each quantity; z is the result.
| Operation | Rule | Which uncertainty |
|---|---|---|
| z = x + y | δz = √(δx² + δy²) | Absolute |
| z = x − y | δz = √(δx² + δy²) | Absolute — subtraction adds them too |
| z = x × y | δz/|z| = √((δx/x)² + (δy/y)²) | Relative |
| z = x ÷ y | δz/|z| = √((δx/x)² + (δy/y)²) | Relative — same as multiplication |
| z = xⁿ | δz/|z| = |n| × (δx/x) | Relative, scaled by the exponent |
| z = c × x (c exact) | δz = |c| × δx | Absolute — an exact constant adds nothing |
Reporting the answer
Uncertainties are quoted to one significant figure, with the common refinement of keeping two when the leading digit is 1 — the difference between 1 and 1.5 is too large to throw away. The value is then rounded to the same decimal place: 50.0 ± 1.5, never 50.0 ± 1.4841 or 49.97 ± 1.5. A value quoted to more places than its uncertainty claims a precision the measurement does not have.
Two limits are worth knowing. These rules assume the uncertainties are independent; if two quantities share a systematic error, they do not partially cancel and quadrature understates the result. And subtracting two nearly equal numbers is dangerous — the absolute uncertainty stays the same while the value shrinks, so the relative uncertainty can explode. If a result comes out as 0.2 ± 0.3, the experiment, not the arithmetic, is what needs revisiting.