Scope and assumptions
Percentage uncertainty scales the absolute uncertainty of a measurement to the measured value. Illustrative; basic (not full propagation) treatment.
A reading of "50 cm" hides a quiet admission: it was really 50, plus or minus something. That spread, scaled to the value itself, is percentage uncertainty — and unlike percent error, it doesn't need a true value to exist, only the wobble of your own instrument.
Scaling the wobble
Percentage Uncertainty = (Absolute Uncertainty ÷ Measured Value) × 100
The absolute uncertainty is the ± spread of your measurement; dividing by the value and × 100 makes it a relative percent.
| Measured | ± Uncertainty | % Uncertainty |
|---|---|---|
| 50 cm | ±1 cm | (1/50) × 100 = 2% |
| 100 g | ±0.5 g | 0.5% |
| 20 L | ±2 L | 10% |
| 5 m | ±0.1 m | 2% |
The line between error and uncertainty
- Percent error compares your value to a known true value.
- Percent uncertainty describes the spread of your measurement when the true value is unknown.
They answer different questions; never substitute one for the other.
Combining uncertainties (basic)
When you add or subtract quantities, absolute uncertainties add: (a ± δa) + (b ± δb) = (a+b) ± (δa+δb). For multiplication, percentage uncertainties roughly add: 2% and 3% → about 5% relative.
Reading a reported value
"50 ± 1 cm (2%)" tells two things: the best estimate (50) and how much to trust it (±1 cm, or 2%). A 2% uncertainty on 50 cm is tighter than 10% on 20 L even though the absolute spread (1 vs 2) is smaller in the second.
Reading Uncertainty Ranges Wrong
- Using the wrong base — divide by the measured value, not the uncertainty.
- Confusing with percent error — uncertainty has no known true value to compare to.
- Dropping ± when reporting — give both the value and its spread.
- Skipping × 100 — the ratio is a decimal until scaled.
- Adding percentage uncertainties for sums — percentages add for products; absolute spreads add for sums.
- Treating % uncertainty as a hard bound — it's a spread estimate, often ±1σ, not a guarantee.
Related advanced operations
For the rest of the advanced percentage operations, start at the hub: Percentage Difference Formula. Adjacent: Percentage Error Calculation and Reverse Percentage Increase.
FAQ
How is percentage uncertainty different from accuracy? Accuracy describes how close you are to the true value (needs a standard); percentage uncertainty describes your spread without one. A device can be precise (low uncertainty) but inaccurate (systematically off).
Should I report absolute or percentage uncertainty? Report both — "50 ± 1 cm (2%)". Percentage communicates relatability across scales; absolute tells you the real-world margin. One alone can mislead (a 2% uncertainty on 50 cm vs on 5 m means very different absolute spreads).
What does '±1σ' mean and does it guarantee the value? ±1σ covers about 68% of readings if the spread is normal. It's not a hard bound — values outside happen. Don't present uncertainty as a guaranteed range.
How do I convert percentage uncertainty back to absolute? Multiply by the measured value: 2% of 50 cm = 1 cm. The inverse of the formula, handy when a spec gives you a percent tolerance.
Can percentage uncertainty be zero? Only for a perfect, infinitely precise measurement — in practice never. Even the best instrument has some finite spread, so report the smallest meaningful figure rather than 0%.
Sources: Khan Academy, Measurement uncertainty; NIST, TN 1297.
Disclaimer: Illustrative, basic treatment. Full uncertainty analysis uses standard deviation and propagation rules; consult your field's standard. Informational only.
Sources and references
- Khan Academy — Measurement uncertaintyhttps://www.khanacademy.org/science
- NIST — TN 1297https://www.nist.gov/pml/nist-technical-note-1297
Maintenance and corrections
The named author or publishing brand is responsible for this article. Worked examples are illustrative, not user testimonials, professional advice or approval by the institutions cited. Maintenance responsibility and review methods · Report an error