Uncertainties — the skill Paper 1B is built on
Every measurement has an uncertainty, and a physics result quoted without one is incomplete rather than merely untidy. Paper 1B is 20 marks — the same 20 at SL and HL — and a large share of them are uncertainty marks.
What makes this awkward to revise is that uncertainties are no longer a chapter of their own. They are spread through the course as a skill you are assumed to pick up, which is exactly why most students arrive at the exam having never sat down and practised them. This page is that sit-down.
Three ways of writing the same uncertainty
Measured value = quantity + uncertainty
For a measurement: L = 2.34 ± 0.05 m
Absolute — ΔL, carrying the same units as the measurement itself:
ΔL = 0.05m
Fractional — ΔL / L, no units:
ΔL / L = 0.05 / 2.34 = 0.021
Percentage — ( ΔL / L ) × 100%, no units:
( 0.05 / 2.34 ) × 100% = 2.1%
All three exist because they are used at different moments. The combination rules below work in fractional or percentage form, but a final answer is always quoted as an absolute uncertainty in the same units as the value.
Converting back at the end is a step students forget.
Where the uncertainty comes from
From the instrument's resolution.
For a digital display, ± 1 in the last digit shown.
For an analogue scale, ± half the smallest division — that is the uncertainty in one reading of the scale.
Then count your readings.
A length measured with a ruler usually needs two, one at each end, and those two uncertainties add — so the length carries ± one whole division. On a millimetre ruler a single position is ± 0.5 mm, but a length is ± 1 mm.
If the object starts exactly at the zero mark you have only read the scale once, and ± 0.5 mm stands.
The rule worth remembering is not a number but a question: how many times did I read the scale?
From repeated readings.
Take half the range: Δx = ( Xmax − Xmin ) / 2
Note that this is the IB method. It is not the standard deviation, and you will not be asked for one.
From the manufacturer.
Some equipment states its own tolerance; if a question gives you one, use it in preference to the resolution.
Combining uncertainties — three rules, and that is all
1 · Adding or subtracting quantities: add the absolute uncertainties.
y = a ± b → Δy = Δa + Δb
Both cases add. Subtracting two measurements does not subtract their uncertainties — it makes the result less certain, not more, and this is one of the most reliable ways to lose a mark in this topic.
Example 1 — combining two masses
A = m₁ + m₂, m₁ = 120 ± 5 g, m₂ = 150 ± 5 g
Quantity: A = m₁ + m₂ = 120 + 150 = 270 g
Absolute Uncertainty: ΔA = Δm₁ + Δm₂ = 5 + 5 = 10 g
Total mass A = (120 + 150) ± (5 + 5) = 270 ± 10 g
Example 2 — subtracting two lengths
L = L₁ − L₂, L₁ = 55 ± 0.5 mm, L₂ = 35 ± 0.5 mm
Quantity: L = L₁ − L₂ = 55 − 35 = 20 mm
Absolute Uncertainty: ΔL = ΔL₁ + ΔL₂ = 0.5 + 0.5 = 1 mm
L = ( 55 − 35 ) ± ( 0.5 + 0.5 ) = 20 ± 1 mm
2 · Multiplying or dividing: add the fractional uncertainties.
y = ab or y = a/b → Δy/y = Δa/a + Δb/b
Example 3 — density of a cylinder
m = 2.00 kg ± 2.5 %, V = 0.00100 m³ ± 5 %
Quantity: ρ = m/V = 2.00 / 0.00100 = 2.00 × 10³ kg m⁻³
% uncertainty in ρ = % uncertainty in m + % uncertainty in V = 2.5 % + 5 % = 7.5 %
Δρ = ρ × % uncertainty in ρ = 2.00 × 10³ × 7.5 % = 0.15 × 10³ kg m⁻³
ρ = ( 2.00 ± 0.15 ) × 10³ kg m⁻³
By the rule "quote the absolute uncertainty to 1 significant figure", the final answer should be written as ρ = (2.00 ± 0.2) × 10³ kg m⁻³y = aⁿ → Δy/y = |n| × Δa/a
A square is doubled, a cube tripled. A square root is a power of ½, so its fractional uncertainty is halved — which surprises people the first time.
Example 4 — elastic potential energy
U = ½ k x², with k = 500 ± 10 N m⁻¹ and x = 0.25 ± 0.02 m
k = 500 ± 10 N m⁻¹ → Δk / k = 10 / 500 = 0.02
and x = 0.25 ± 0.02 m → Δx / x = 0.02 / 0.25 = 0.08
Quantity: U = ½ ( 500 )( 0.25 )² = 0.5 × 500 × 0.0625 = 15.625 J
Fractional uncertainty: ΔU / U = Δk / k + 2 Δx / x = 10/500 + 2 ( 0.02 / 0.25 ) = 0.18
Absolute uncertainty: ΔU = 15.625 × 0.18 = 2.8 J
U = 15.6 ± 2.8 J, or rounded properly 16 ± 3 J
The IB uses these simple additive rules throughout. You are not expected to combine uncertainties in quadrature, and doing so will not earn extra credit.
How to quote the answer
- Round the uncertainty to one significant figure.
- Round the value to the same decimal place as the uncertainty.
4.3671 ± 0.0234 → 4.37 ± 0.02
The precision of the answer is set by its uncertainty, not by how many digits the calculator was willing to show. Writing 4.3671 ± 0.02 claims a precision the measurement does not have; writing 4.37 ± 0.0234 quotes an uncertainty more precisely than you can know it.
Separately: a calculated result should carry no more significant figures than the least precise measurement that went into it.
Accuracy, precision, random and systematic
These four words have specific meanings and examiners mark them strictly.
TERM MEANS REDUCED BY
Random error Scatter of readings either side of the true value Repeating and averaging
Systematic error A consistent shift in one direction —zero error, Not reduced byrepeating
poor calibration, a background count left in
Precision How close repeated readings are to each other Reducing random error
Accuracy How close a reading is to the true value Removing systematic error
A set of readings can be tightly clustered and all wrong — precise but inaccurate, which is what a systematic error does to your data. They can also scatter widely around the right answer: accurate on average, imprecise.
Worth memorising for Paper 1B. On a graph, a random error shows as scatter of points about the line; a systematic error shows as a shifted intercept. When a question asks what a non-zero intercept suggests, when theory says the line should pass through the origin, the answer it wants is a systematic error — and then a plausible source of one.
Where marks are actually lost
- Subtracting two quantities and subtracting their uncertainties. Always add.
- Using percentage uncertainties in an addition step. Percentages are for multiplying and dividing only.
- Quoting an uncertainty to three significant figures.
- Giving a value to more decimal places than its uncertainty supports.
- Writing "precise" in an answer where the mark scheme wants "accurate".
- Using ± half a division for a ruler measurement that needed two readings.
- Losing the units when converting a fractional uncertainty back to an absolute one.
Notes and practice
Reading through a topic can make it feel clear; working through the questions is what tells you whether it really is.
Next
Units, prefixes and scientific notation ➡️
Vectors: components and resolving ➡️
Paper 1B: data analysis and graphs ➡️
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