Uncertainty propagation and best practice

Combining measurements with uncertainty

When a quantity is calculated from several measurements, uncertainty is propagated. Two simple rules are sufficient at A-level.

Rule for a sum or a difference

If z = x + y or z = x – y, the absolute uncertainties are always added together (even for a subtraction, as errors can accumulate):

delta z = delta x + delta y

Example: x = 5.0 ± 0.1 cm and y = 3.0 ± 0.2 cm give z = x + y = 8.0 ± 0.3 cm.

Rule for a product or a quotient

If z = x * y or z = x/y, it is the relative uncertainties that are added together:

delta z / z = delta x / x + delta y / y

Example: for an area A = L × l, where L = 5.0 ± 0.1 cm and l = 3.0 ± 0.1 cm, the relative uncertainty is 0.1/5.0 + 0.1/3.0 = 0.02 + 0.033 = 0.053, or approximately 5.3 per cent. As A = 15.0 cm², delta A = 0.053 × 15.0 = 0.8 cm² (rounded to 1 significant figure).

Consistency between value and uncertainty

The uncertainty is generally rounded to 1 or 2 significant figures, and the central value is then rounded to the same decimal place as the uncertainty. Writing A = 15.03 ± 0.8 cm² is incorrect; it should be written as A = 15.0 ± 0.8 cm².

Pitfalls to avoid

• Subtracting uncertainties instead of adding them together when there is a difference.
• Forgetting to convert back to absolute uncertainty after a relative calculation.
• Giving an uncertainty to 3 or 4 significant figures: this suggests a precision that does not exist.