The big idea: Every measurement has an uncertainty — a ± range it could really be in.
You can quote that uncertainty in three ways, and they all carry 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.
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.
IB-style question — absolute uncertainty from a spread
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.
Solution
- Mean = add the readings and divide by how many there are:
- Absolute uncertainty = half the range (half of largest − smallest). Formula first:
- Put in the largest and smallest readings:
Final answer
d = 0.53 ± 0.02 mm.
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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).
IB-style question — percentage uncertainty of an area
A square plate has side L = 8.0 ± 0.1 cm. Its area is A = L². Find the percentage uncertainty in the area.
Solution
- Area uses a power (A = L²), so use the power rule. Formula first:
- Fractional uncertainty in L, with n = 2:
- As a percentage (×100):
Final answer
the area has a percentage uncertainty of 2.5%.
How this is tested: Paper 1B almost always opens with an uncertainty calculation — it is the single most-tested experimental skill.
- Paper 1B: find an absolute uncertainty from a spread or resolution, then propagate it through a formula (density, resistivity, an area) and quote the answer to a sensible precision. - Paper 1A: a one-step multiple-choice — convert between forms, or pick which rule applies (add absolute vs add fractional).
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.
IB-style question — (a) density from mass and volume
A small metal block has mass m = 240 ± 5 g and volume V = 30.0 ± 0.5 cm³. Its density is ρ = m ÷ V. Calculate the density.
Part (a) — the value
- Density is mass ÷ volume. Formula first:
- Put in the values:
- Work it out — keep the unit:
Final answer
ρ = 8.0 g cm⁻³ (before the uncertainty).
IB-style question — (b) its absolute uncertainty
For the same block (m = 240 ± 5 g, V = 30.0 ± 0.5 cm³, ρ = 8.0 g cm⁻³), find the absolute uncertainty in the density and quote ρ to a sensible precision.
Part (b) — the uncertainty
- Density is a division, so add the fractional uncertainties. Formula first:
- Put in the numbers:
- Convert back to an absolute uncertainty (× the density):
- Round the uncertainty to 1 s.f., then match the value's precision:
Final answer
ρ = 8.0 ± 0.3 g cm⁻³.