Intermediate · 12 min
Measurement uncertainty
Interpret uncertainty intervals and compare absolute and percentage uncertainty.
A reading is not an exact boundary
A measurement written as x ± u reports a value together with an uncertainty. For these exercises, treat u as a stated bound: the value lies between x − u and x + u. An uncertainty is not the same as a mistake. Its interpretation depends on the measurement method; in other contexts it may describe a statistical interval rather than a strict bound. Always use the definition given.
Worked example
A length reported as 8.0 ± 0.2 cm with stated bounds spans 7.8 cm to 8.2 cm.
Compare and combine stated bounds
Percentage uncertainty is absolute uncertainty divided by the magnitude of the measured value, multiplied by 100. For sums or differences of quantities with stated bounds, add the absolute bounds to find the largest possible deviation. Do not use this worst-case rule as a universal statistical uncertainty formula. Repeated readings can reveal variation but do not automatically remove systematic errors.
Worked example
Two bounded lengths are 5.0 ± 0.1 cm and 7.0 ± 0.2 cm. Their sum is 12.0 ± 0.3 cm. The first length's percentage uncertainty is 0.1 ÷ 5.0 × 100 = 2%.
Use an interval, not just a midpoint
- Write the lower and upper bounds for 15.0 ± 0.3 cm and calculate its percentage uncertainty.
- Add a second length of 5.0 ± 0.1 cm. Check the uncertainty of the sum by adding the two lowest and the two highest possible values.