Uncertainty: readings and propagation
Compare how input allowances affect three simplified measurement models using approximate propagation and separately varied contributions.
Learning goals
- Assess random, systematic and propagated uncertainties.
ρ = 4m / (πd²h), so %ρ ≈ %m + 2%d + %h. d readings: 12.7 mm; h readings: 30 mm; m readings: 31.9 g. Here % denotes percentage uncertainty. Each input uses the larger of its stated instrument allowance and half the reading range, not statistical uncertainty in a mean. Uniform-cylinder model with separately varying input allowances; product and power estimates are first-order. The approximate result is ρ = 8.4 ± 0.4 g cm⁻³.
- Result
- ρ = 8.4 ± 0.4 g cm⁻³
- Estimated percentage uncertainty
- 5.2 %
- Input contributions
- m: 0.31%; d²: 1.6%; h: 3.3%
- Rounded input estimates
- d: 12.7 ± 0.1 mm; h: 30 ± 1 mm; m: 31.9 ± 0.1 g
- Input allowance model
- Larger of stated instrument allowance and half the reading range
- Model assumptions
- Uniform cylinder; separately varying allowances; first-order estimate
Try this
0 of 4 doneMeasure the diameter with a metre rule, then with a micrometer. Watch the d² bar. (not done yet)
Squaring d doubles its fractional contribution. Compare every input contribution before choosing which measurement to improve.
Get the percentage uncertainty in the density below 1%. (not done yet)
Improve the biggest contribution first; making a small one smaller hardly helps.
Take five or more repeat readings of the diameter with the micrometer. (not done yet)
More readings can reveal a larger spread. This model uses the larger of the stated instrument allowance and half-range; it does not calculate statistical uncertainty in a mean.
Make the spring's extension large enough that its uncertainty is under 5%. (not done yet)
For separately varying input allowances, absolute uncertainties add in a difference. A larger extension reduces their percentage contribution; a shared additive offset would cancel.
You’ve tried every task.