Uncertainty

Key idea: Learn how to estimate measurement uncertainties, distinguish random vs systematic errors, and propagate uncertainties for +, −, ×, ÷ and powers.

GCE A-Level H2 Physics 2027

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Learning objectives

Assess random, systematic and propagated uncertainties.

These “propagation rules” are a fast, conservative way to estimate uncertainties. In university labs you’ll also meet statistical approaches (standard deviation, standard error), but the logic here is still a useful baseline.

1. Definitions (Must Know)

Error: the difference between a measured value and the true value (true value is usually unknown).
Uncertainty, Δ x: an estimate of the possible range of values around a measurement x.
Absolute uncertainty, Δ x: uncertainty written in the same unit as x (e.g. 2.50 ± 0.05 m).
Fractional uncertainty: (Δ x)/x (no unit).
Percentage uncertainty: (Δ x)/x × 100%.
Random error: unpredictable variation between repeated readings; causes scatter. Repeats reveal the spread, and averaging reduces the random contribution to the mean.
Systematic error: consistent bias (e.g. zero error, calibration error); affects accuracy; not reduced by averaging.
Precision: how close repeated readings are to each other (small scatter).
Accuracy: how close a measurement is to the true value.

2. Key Ideas (What Earns Marks)

Quote readings with sensible uncertainty:
- analogue scale: ± half the smallest division,
- digital reading: typically ± 1 in the last displayed digit (unless stated otherwise),
- repeated readings: random uncertainty ≈ half the range.
For a derived quantity, propagate uncertainties using the standard A Level rules (worst-case approach):
- If Y = a ± b: Δ Y = Δ a + Δ b
- If Y = ab or Y = a/b: (Δ Y)/Y = (Δ a)/a + (Δ b)/b
- If Y = aⁿ: (Δ Y)/Y = |n| (Δ a)/a
If the expression is awkward, use numerical substitution (max–min):
- compute Yₘₐₓ and Yₘᵢₙ using the extreme values,
- then Δ Y ≈ (Yₘₐₓ-Yₘᵢₙ)/2.
Quote the final answer so the value and uncertainty match:
- uncertainty: usually 1 s.f. (2 s.f. is common if it starts with 1 or 2),
- value: same decimal place as the uncertainty.

3. Detailed Explanations

A. Estimating uncertainty in a measurement

Use the information the question gives you.

Common defaults:

Analogue scale: uncertainty ≈ ± half the smallest division.
Digital display: uncertainty ≈ ± 1 in the last displayed digit.
Repeats (random uncertainty): uncertainty ≈ half range.

Example (percentage given): If a timer is specified as ± 0.1% and you measure t = 638.5 ms, then Δ t = 0.001 × 638.5 ≈ 0.6 ms So quote t = 638.5 ± 0.6 ms.

B. Random vs systematic errors (what examiners want)

Random error mainly affects precision. Repeating reveals its size; averaging reduces its effect on the mean but does not remove the scatter in the measurements.
Systematic error mainly affects accuracy. Averaging does not remove it (you must correct/calibrate).

If you see “zero error”, “calibration error”, “parallax always in the same direction”, treat it as systematic.

C. Propagating uncertainties: + and − (absolute uncertainties)

If Y = a + b or Y = a - b, add absolute uncertainties: Δ Y = Δ a + Δ b

Reason: you are estimating a worst-case uncertainty range.

D. Propagating uncertainties: × and ÷ (fractional/percentage uncertainties)

If Y = ab or Y = a/b, add fractional uncertainties: (Δ Y)/Y = (Δ a)/a + (Δ b)/b

You can do this in percentages as well: %Δ Y = %Δ a + %Δ b

E. Powers (including roots)

If Y = aⁿ: (Δ Y)/Y = |n|(Δ a)/a

Examples:

Y = a²: percentage uncertainty doubles.
Y = square root of a = a^(1/2): percentage uncertainty halves.

F. Uncertainty budget (spot the dominant term)

A quick way to decide what to improve is to break the result into percentage contributions from each measurement.

Example (simple pendulum): g = 4π²L/T² So, %Δ g ≈ %Δ L + 2%Δ T

Here

%Δ L = 0.25% and

2%Δ T = 2.2%, so improving the timing method gives the biggest gain.

Data table

Category	Contribution
Length, L	0
Time, T (×2)	2

G. Numerical substitution (max–min method)

Use this when the expression is awkward.

Workflow:

Compute the best estimate Y using best values.
Compute Yₘₐₓ using values that make Y as large as possible.
Compute Yₘᵢₙ using values that make Y as small as possible.
Estimate: Δ Y ≈ (Yₘₐₓ-Yₘᵢₙ)/2 and %Δ Y ≈ 100% × (Δ Y)/Y.

4. Common Mistakes

Adding percentage uncertainties for addition/subtraction (wrong). For + and -, add absolute uncertainties.
Adding absolute uncertainties for multiplication/division (usually wrong). For × and ÷, add fractional/percentage uncertainties.
Rounding too early (causes noticeable drift in uncertainty questions). Keep guard digits until the end.
Quoting an answer with inconsistent precision (e.g. 2.345 ± 0.1). Match decimal places.
Treating a systematic error as “reduced by repeats”.

5. Exam Tips

Write the rule you are using before you substitute (it signals method marks).
If one measurement has a much larger percentage uncertainty, it usually dominates the final uncertainty.
If you are asked for an “uncertainty in Y”, state whether you are giving absolute or percentage.
A good final line format is:
- Y = (value ± Δ Y) unit
- or Y = value unit (± percentage)
Related: Dimensional Analysis (unit checks) and the A Level Practical Hub (evaluation + uncertainties in Paper 4).

Practical Lab

Use this workflow-first lab if you want the Paper 4 version of this topic rather than another static summary:

Practical Lab: Uncertainty & Error Propagation

Work through instrument readings, repeated data, propagation, and final reporting with an exam-style uncertainty workflow.

BetaA LevelPractical SkillsBest for: A Level Paper 4 uncertainty handling

Absolute vs Percentage Uncertainty
Repeated-Reading Estimate
Propagation Rules
Final Answer Reporting

Open the full interactive simulation on its own page

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6. Worked Examples

Problem

For

L = x + y,

x = 12.4±0.1 cm and

y = 8.2±0.1 cm. Find

L with uncertainty.

Study the worked solution

Find the best estimate
Method
Add measured values.
Reason
The derived length is defined by addition.
Working
L = 12.4 + 8.2 = 20.6 cm
Propagate absolute uncertainties
Method
Add the absolute uncertainties.
Reason
The conservative addition/subtraction rule operates in the quantity’s units.
Working
Δ L = 0.1 + 0.1 = 0.2 cm
Report consistently
Method
State 20.6±0.2 cm.
Reason
Value and absolute uncertainty share units and decimal place.
Working
L = 20.6±0.2 cm.

Problem

V = 10.0±0.3 V and

I = 1.3±0.2 A. Find

R = V/I with uncertainty.

Try this before viewing the solution

Best estimate, percentage and absolute uncertainty

Hints

Hint 1: division uses relative uncertainty

Convert each input uncertainty to a percentage, then add them.

View solution step by step

Find resistance
Method
Calculate the best estimate.
Reason
R = V/I.
Working
R = 10.0/1.3 = 7.7 Ω
Add percentage uncertainties
Method
Combine 3.0% and approximately 15%.
Reason
For division, conservative fractional uncertainties add.
Working
%Δ R = 3.0% + 15% = 18%
Convert back to absolute
Method
Obtain approximately 1.4 Ω.
Reason
The final result is reported in ohms.
Working
Δ R = 0.18(7.7) = 1.4 Ω, R = 7.7±1.4 Ω

Learner claim

For

h = 2.00±0.01 m and

v ∝ h^(1/2), a learner assigns

v the same

0.5% uncertainty as

h. Diagnose the claim.

Try this before viewing the solution

Percentage uncertainty in v

0.25%0.50%1.0%

View solution step by step

Find height percentage uncertainty
Method
Divide absolute uncertainty by value.
Reason
Power rules act on fractional or percentage uncertainty.
Working
%Δ h = (0.01/2.00)100% = 0.5%
Apply the exponent
Method
Multiply by 1/2.
Reason
For v ∝ h^(1/2), the relative uncertainty factor is the exponent magnitude.
Working
%Δ v = (1/2)(0.5%) = 0.25%

Examination question

An athlete covers

100.00±0.01 m in

9.63±0.01 s. Find speed and absolute uncertainty. [4 marks]

Try this before viewing the solution

Best estimate, input percentages and final uncertainty

View solution step by step

Find speed
1 mark
Method
Divide distance by time.
Reason
v = d/t.
Working
v = 100.00/9.63 = 10.38 m s⁻¹
Find input percentages
1 mark
Method
Obtain 0.01% and 0.10%.
Reason
Each is absolute uncertainty divided by its value.
Working
%Δ d = 0.01%; %Δ t ≈ 0.10%.
Propagate
1 mark
Method
Add to approximately 0.11%.
Reason
Speed is a quotient.
Working
%Δ v ≈ 0.11%.
Report absolute uncertainty
1 mark
Method
State 10.38±0.01 m s⁻¹.
Reason
0.0011(10.38) ≈ 0.01 m s⁻¹.
Working
v ≈ 10.38±0.01 m s⁻¹.

Self-mark with the mark scheme

Compare your response with each mark point. Select a point only when your response contains that evidence.

Self-mark estimate, percentages, propagation and reported result.

Mark points

1 mark: Obtains 10.38 m s⁻¹.
1 mark: Calculates both input percentage uncertainties.
1 mark: Adds percentages for the quotient.
1 mark: Reports approximately 10.38 ± 0.01 m s⁻¹.

Independent graph transfer

For a

V-against-

I graph, best-fit points are

(0.20±0.01 A,2.10±0.05 V) and

(0.80±0.01 A,8.40±0.05 V). Estimate resistance and gradient uncertainty using maximum and minimum acceptable gradients.

Try this before viewing the solution

Best, maximum and minimum gradients with final uncertainty

Hints

Hint 1: choose opposite uncertainty-box corners

For the maximum gradient, maximise the vertical interval and minimise the horizontal interval; reverse both choices for the minimum.

View solution step by step

Method

Calculate the best gradient, then the steepest and shallowest acceptable lines.

Reason

The uncertainty boxes bound plausible endpoint positions, and half the extreme-gradient range estimates gradient uncertainty.

Working

Best estimate gradient: R = (Δ V)/(Δ I) = (8.40-2.10)/(0.80-0.20) = 6.30/0.60 = 10.5 Ω

To estimate uncertainty, find extreme gradients.

Maximum gradient: maximise numerator and minimise denominator. Δ Vₘₐₓ = (8.40 + 0.05)-(2.10-0.05) = 6.40 V Δ Iₘᵢₙ = (0.80-0.01)-(0.20 + 0.01) = 0.58 A Rₘₐₓ ≈ 6.40/0.58 ≈ 11.0 Ω

Minimum gradient: minimise numerator and maximise denominator. Δ Vₘᵢₙ = (8.40-0.05)-(2.10 + 0.05) = 6.20 V Δ Iₘₐₓ = (0.80 + 0.01)-(0.20-0.01) = 0.62 A Rₘᵢₙ ≈ 6.20/0.62 = 10.0 Ω

So, Δ R ≈ (Rₘₐₓ-Rₘᵢₙ)/2 = (11.0-10.0)/2 = 0.5 Ω

Final: R ≈ 10.5 ± 0.5 Ω

Best-fit line and the steepest/shallowest plausible gradients based on the stated uncertainties. — The error bars show each stated uncertainty box. The steepest acceptable line joins (0.21 A, 2.05 V) to (0.79 A, 8.45 V); the shallowest joins (0.19 A, 2.15 V) to (0.81 A, 8.35 V).

Values and uncertainty for Max–min gradients from the same two points
Series	Current (A)	Current uncertainty	Potential difference (V)	Potential difference uncertainty
Measured points	0.2	0.01	2.1	0.05
Measured points	0.8	0.01	8.4	0.05
Best-fit gradient	0.2		2.1
Best-fit gradient	0.8		8.4
Max gradient	0.21		2.05
Max gradient	0.79		8.45
Min gradient	0.19		2.15
Min gradient	0.81		8.35

Values and uncertainty for Max–min gradients from the same two points
Series	Current (A)	Current uncertainty	Potential difference (V)	Potential difference uncertainty
Measured points	0.2	0.01	2.1	0.05
Measured points	0.8	0.01	8.4	0.05
Best-fit gradient	0.2		2.1
Best-fit gradient	0.8		8.4
Max gradient	0.21		2.05
Max gradient	0.79		8.45
Min gradient	0.19		2.15
Min gradient	0.81		8.35

Next step

Attempt numerical max–min substitution independently.

7. Mind Stretchers

Mind stretcher 1: Numerical substitution (max–min)Extension

A student finds density from ρ = m/V where m = 52.3 ± 0.1 g and V = 19.5 ± 0.5 cm³. Use max–min to estimate ρ and Δ ρ.

Show Answer

Best estimate: ρ = 52.3/19.5 = 2.68 g cm⁻³

Extremes:

ρₘₐₓ = 52.4/19.0 = 2.76
ρₘᵢₙ = 52.2/20.0 = 2.61

So, Δ ρ ≈ (2.76-2.61)/2 = 0.075 g cm⁻³ ≈ 0.08 g cm⁻³

Final: ρ ≈ 2.68 ± 0.08 g cm⁻³

If you need SI units: 1 g cm⁻³ = (10⁻³ kg)/(10⁻⁶ m³) = 10³ kg m⁻³ So, ρ ≈ (2.68 ± 0.08) × 10³ kg m⁻³ ≈ (2680 ± 80) kg m⁻³

Mind stretcher 2: Uncertainty in g from a pendulum (power rule + multiplication)Extension

For a simple pendulum, g = 4π²L/T².

If L = 0.800 ± 0.002 m and T = 1.80 ± 0.02 s, estimate the percentage uncertainty in g.

Show Answer

%Δ g = %Δ L + 2%Δ T

%Δ L = 0.002/0.800 × 100% = 0.25% %Δ T = 0.02/1.80 × 100% ≈ 1.1%

So, %Δ g ≈ 0.25% + 2(1.1%) ≈ 2.5%

Mind stretcher 3: Optional (Enrichment)Extension

A. Standard deviation (beyond syllabus)

In the A Level syllabus, you do not need a rigorous statistical treatment.

If you meet standard deviation in other contexts: it is a measure of the spread of repeated readings around the mean. It is useful when you have many repeats and want a more formal uncertainty estimate than “half range”.

Continue with the next resource in this course.

Course and syllabus information

Course: GCE A-Level H2 Physics
Edition: GCE A-Level H2 Physics 2027

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Learning objectives

1. Definitions (Must Know)

2. Key Ideas (What Earns Marks)

3. Detailed Explanations

A. Estimating uncertainty in a measurement

B. Random vs systematic errors (what examiners want)

C. Propagating uncertainties: + and − (absolute uncertainties)

D. Propagating uncertainties: × and ÷ (fractional/percentage uncertainties)

E. Powers (including roots)

F. Uncertainty budget (spot the dominant term)

G. Numerical substitution (max–min method)

4. Common Mistakes

5. Exam Tips

Practical Lab

Open the full interactive simulation on its own page

6. Worked Examples

7. Mind Stretchers

Mind stretcher 1: Numerical substitution (max–min)Extension

Mind stretcher 2: Uncertainty in g from a pendulum (power rule + multiplication)Extension

Mind stretcher 3: Optional (Enrichment)Extension

A. Standard deviation (beyond syllabus)