The big idea: Weigh yourself twice and the scale might read 71.2 kg, then 71.4 kg — no measurement is ever exact. That wiggle room is the uncertainty: the ± range the true value could lie in.
You can quote it three ways, all carrying the same information:
- absolute — a ± amount in the same unit (e.g. 12.4 ± 0.2 cm) - fractional — the absolute ÷ the value (a plain number, no unit) - percentage — the fractional × 100%
- the absolute uncertainty — a ± value in the same unit as x
- the fractional uncertainty — a plain number, no unit
- the percentage uncertainty
| Form | How to get it | Example (12.4 ± 0.2 cm) |
|---|---|---|
| Absolute, Δx | read it off / given directly | 0.2 cm |
| Fractional, Δx/x | absolute ÷ value | 0.2 ÷ 12.4 = 0.016 |
| Percentage | fractional × 100% | 1.6 % |
Spot it: convert in the right direction: Have absolute, want %? divide by the value, ×100.
Have %, want absolute? ×value, ÷100. So 1.6% of 12.4 cm gives back 0.2 cm.
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Before you can propagate anything you need the absolute uncertainty of each measurement. It comes from one of three places, depending on how the value was found.
Three sources of an absolute uncertainty
- Instrument resolution — ± half the smallest scale division. A ruler in mm → ±0.5 mm; a digital balance reading 0.01 g → ±0.005 g (half the last digit).
- Spread of repeated readings — ± half the range (½ × (largest − smallest)). This is the everyday method when you take several readings.
- A stated percentage — convert it back to absolute: absolute = percentage × value ÷ 100.
Watch: half the range, not the whole range: From a spread of readings the uncertainty is half the spread, because the true value sits in the middle.
Readings 4.6, 4.8, 4.9 mm → range = 0.3 mm → uncertainty = ±0.15 mm, rounded to ±0.2 mm.
A wire's diameter is measured four times with a micrometer: 0.52, 0.54, 0.55 and 0.51 mm. Find the mean diameter and its absolute uncertainty from the spread.
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When measurements are combined in a calculation, their uncertainties combine too. There are just three rules — and two of them are given in the data booklet.
The three rules: Adding or subtracting (+ , −): add the absolute uncertainties.
Multiplying or dividing (× , ÷): add the fractional (or percentage) uncertainties.
Powers (aⁿ): multiply the fractional uncertainty by the power |n|.
- a result found by adding or subtracting measurements
- the measured quantities added or subtracted
- their absolute uncertainties (same unit as the quantity)
- the absolute uncertainty in the result (same unit)
- the calculated result (e.g. a density, a resistivity)
- the measured quantities multiplied or divided to get y
- the absolute uncertainty in a (same unit as a)
- the fractional uncertainty in a (no unit)
- the fractional uncertainty in the result y
- the result is a raised to a power n (e.g. area = πr², so r²)
- the power (2 for a square, 3 for a cube, ½ for a square root)
- the fractional uncertainty in a (no unit)
- the fractional uncertainty in the result (no unit)
Match the form to the operation: + or − → work in absolute. × or ÷ or a power → work in fractional/percentage, then convert back to absolute at the very end (Δy = fractional × y).
A square plate has side L = 8.0 ± 0.1 cm. Its area is A = L². Find the percentage uncertainty in the area.
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How this is tested — an uncertainty calculation is the single most-tested experimental skill:
Paper 1B
- Find an absolute uncertainty from a spread or resolution.
- Propagate it through a formula (density, resistivity, area) and quote to a sensible precision.
Paper 1A
- A one-step multiple-choice: convert between forms, or pick which rule applies (add absolute vs add fractional).
The classic trap: Adding percentages where you should add absolutes (a + or − step), or forgetting to multiply by the power for a squared/cubed term.
Rounding the final answer: Round the uncertainty to 1 significant figure, then round the value to the same decimal place as the uncertainty.
So 2.643 ± 0.087 → 2.64 ± 0.09.
A small metal block has mass m = 240 ± 5 g and volume V = 30.0 ± 0.5 cm³. Its density is ρ = m ÷ V. (a) Calculate the density. (b) Find the absolute uncertainty in the density and quote ρ to a sensible precision.
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