Uncertainties — OCR A Level Physics
Module 1 · Development of Practical Skills

Uncertainties

Specification: OCR A H556  |  Section: 1.2  |  Teaching time: ~3 hours

By the end of this topic you should be able to…
Part 1

Types of Uncertainty

Every measurement has an associated uncertainty — a range within which the true value is likely to lie. Understanding and communicating uncertainty is essential for valid scientific conclusions.

Absolute uncertainty (Δx) is the uncertainty in the same units as the measurement.

Absolute uncertainty x = xbest ± Δx

Percentage uncertainty expresses the uncertainty as a percentage of the measured value:

Percentage uncertainty % uncertainty = (Δx / x) × 100%

Fractional uncertainty is the ratio Δx/x (without multiplying by 100).

⚡ Key Point

For analogue instruments (rulers, thermometers, ammeters), the absolute uncertainty is typically half the smallest division. For digital instruments, it's ±1 in the last digit.

Common uncertainties:

  • Metre ruler (1 mm divisions): ±0.5 mm or ±0.0005 m
  • Thermometer (1°C divisions): ±0.5°C
  • Digital stopwatch (0.01 s): ±0.01 s (but human reaction time ~0.1–0.2 s dominates)
  • Digital multimeter: ±1 in last digit + manufacturer's % specification
Part 2

Combining Uncertainties

When you use measured values in calculations, you must propagate the uncertainties. The method depends on the mathematical operation.

Addition and Subtraction

For z = x + y or z = x − y, add absolute uncertainties:

Adding/subtracting measurements Δz = Δx + Δy
Worked Example: Length difference
Length A = 12.4 ± 0.1 cm
Length B = 8.7 ± 0.1 cm
Difference = A − B = 3.7 cm
Δ(Difference) = 0.1 + 0.1 = 0.2 cm
Result: 3.7 ± 0.2 cm

Multiplication and Division

For z = x × y or z = x ÷ y, add percentage uncertainties:

Multiplying/dividing measurements (Δz/z) × 100% = (Δx/x) × 100% + (Δy/y) × 100%
Worked Example: Area of rectangle
Length L = 5.0 ± 0.1 cm   →   % uncertainty = (0.1/5.0) × 100% = 2%
Width W = 3.0 ± 0.1 cm   →   % uncertainty = (0.1/3.0) × 100% = 3.3%
Area A = L × W = 5.0 × 3.0 = 15.0 cm²
% uncertainty in A = 2% + 3.3% = 5.3%
Absolute uncertainty = 15.0 × 0.053 = 0.8 cm²
Result: 15.0 ± 0.8 cm² (or 15 ± 1 cm² to 1 sf uncertainty)

Powers and Roots

For z = xn, multiply the percentage uncertainty by n:

Raising to a power (Δz/z) × 100% = n × (Δx/x) × 100%
Worked Example: Volume of sphere
Radius r = 2.0 ± 0.1 cm   →   % uncertainty = 5%
Volume V = (4/3)πr³   →   r³ has uncertainty = 3 × 5% = 15%
V = (4/3) × π × (2.0)³ = 33.5 cm³
Absolute uncertainty = 33.5 × 0.15 = 5.0 cm³
Result: 34 ± 5 cm³
💡 Exam Tip

For complex expressions involving multiple operations, work through step by step. Convert to percentage uncertainties for × and ÷, then convert back to absolute at the end.

Check your understanding

Knowledge Check

1
A student measures the time for 10 oscillations as 18.4 ± 0.2 s. Calculate the period T and its absolute uncertainty.
3 marks

T = 18.4 / 10 = 1.84 s (1 mark)

Since this is division by an exact number (10), absolute uncertainty also divides: ΔT = 0.2 / 10 = 0.02 s (1 mark)

Period = 1.84 ± 0.02 s (1 mark)

2
The resistance R = V/I is calculated from V = 6.0 ± 0.1 V and I = 0.50 ± 0.02 A. Calculate R with its absolute uncertainty.
4 marks

R = 6.0 / 0.50 = 12 Ω (1 mark)

% uncertainty in V = (0.1/6.0) × 100% = 1.7% (1 mark)

% uncertainty in I = (0.02/0.50) × 100% = 4.0%

% uncertainty in R = 1.7% + 4.0% = 5.7%

Absolute uncertainty = 12 × 0.057 = 0.68 ≈ 0.7 Ω (1 mark)

R = 12.0 ± 0.7 Ω (1 mark)

Part 3

Graphical Treatment of Uncertainties

Error bars on graphs show the uncertainty in each data point. For a point at (x, y), the error bars extend from (x, y − Δy) to (x, y + Δy) for vertical uncertainty.

Finding Uncertainty in Gradient

Draw the best-fit line through the data points. Then draw:

  • Worst acceptable line — the steepest/shallowest line that still passes through all error bars
  • Calculate gradient of best line (mbest) and worst line (mworst)
Uncertainty in gradient Δm = |mbest − mworst|

Finding Uncertainty in y-intercept

Use the same approach with the best and worst lines:

Uncertainty in y-intercept Δc = |cbest − cworst|
⚠️ Common Mistake

Don't just draw any steep line — the worst acceptable line must pass through (or touch) all the error bars. Use the outer edges of error bars to construct the steepest and shallowest possible lines.

Part 4

Significant Figures and Final Answers

The number of significant figures in your final answer should be consistent with the uncertainty:

  • Uncertainties are typically quoted to 1 significant figure (occasionally 2 if the first digit is 1)
  • The main value should be quoted to the same decimal place as the uncertainty
Correct Formatting
✅ 3.46 ± 0.02 s   (uncertainty to 1 sf, value to same decimal place)
✅ 1250 ± 40 Ω   (uncertainty to 1 sf, value to same decimal place)
✅ 0.00123 ± 0.00004 m   (uncertainty to 1 sf)
❌ 3.463 ± 0.02 s   (value has more decimal places than uncertainty)
❌ 3.5 ± 0.02 s   (value doesn't match uncertainty's decimal places)
❌ 1250 ± 3.82 Ω   (uncertainty has too many sf)
⚡ Exam Tip

Always state your final answer with uncertainty in the form: value ± uncertainty unit. For example: "The acceleration due to gravity is 9.81 ± 0.05 m s⁻²"

Exam Practice

Exam-Style Questions

1
A student investigates the relationship between force F and extension x for a spring. They measure:

Force F = 4.0 ± 0.1 N
Extension x = 0.025 ± 0.001 m

(a) Calculate the spring constant k = F/x with its absolute uncertainty. (4 marks)
(b) State the final answer with appropriate significant figures. (1 mark)
5 marks

(a) k = F/x = 4.0 / 0.025 = 160 N m⁻¹ (1 mark)

% uncertainty in F = (0.1/4.0) × 100% = 2.5%

% uncertainty in x = (0.001/0.025) × 100% = 4.0%

% uncertainty in k = 2.5% + 4.0% = 6.5% (1 mark)

Absolute uncertainty = 160 × 0.065 = 10.4 ≈ 10 N m⁻¹ (1 mark)

k = 160 ± 10 N m⁻¹ (1 mark)

(b) k = 160 ± 10 N m⁻¹ (uncertainty to 1 sf, value to same decimal place) (1 mark)

2
A student measures the diameter of a wire as d = 0.46 ± 0.02 mm using a micrometer.

(a) Calculate the cross-sectional area A = πd²/4 with its absolute uncertainty. (5 marks)
(b) The student uses this wire in an experiment to determine resistivity. Explain why the diameter measurement contributes the largest uncertainty to the final result. (2 marks)
7 marks

(a) % uncertainty in d = (0.02/0.46) × 100% = 4.3% (1 mark)

Since A ∝ d², % uncertainty in A = 2 × 4.3% = 8.6% (1 mark)

A = π × (0.46 × 10⁻³)² / 4 = 1.66 × 10⁻⁷ m² (1 mark)

Absolute uncertainty = 1.66 × 10⁻⁷ × 0.086 = 0.14 × 10⁻⁷ m² (1 mark)

A = (1.7 ± 0.1) × 10⁻⁷ m² (1 mark)

(b) The diameter is squared in the area formula, so its uncertainty is doubled. This 8.6% uncertainty in area is likely much larger than uncertainties in length or voltage/current measurements. (2 marks)

3
In an experiment to determine g using a pendulum, a student plots a graph of T² (y-axis) against l (x-axis), where T is period and l is length. The gradient should equal 4π²/g.

The student's best-fit line has gradient 4.05 s² m⁻¹. The steepest acceptable line has gradient 4.18 s² m⁻¹.

(a) Calculate g and its absolute uncertainty. (4 marks)
(b) Compare the result with the accepted value g = 9.81 m s⁻². (2 marks
6 marks

(a) g = 4π² / gradient

gbest = 4π² / 4.05 = 9.75 m s⁻² (1 mark)

gworst = 4π² / 4.18 = 9.45 m s⁻² (1 mark)

Δg = |9.75 − 9.45| = 0.30 m s⁻² (1 mark)

g = 9.8 ± 0.3 m s⁻² (1 mark)

(b) The accepted value (9.81 m s⁻²) falls within the uncertainty range (9.5 to 10.1 m s⁻²) (1 mark), so the result is consistent with the accepted value. (1 mark)

Topic Summary

Absolute vs %

Δx is absolute; (Δx/x)×100% is percentage

Combining

+/− : add absolute; ×/÷ : add percentage; powers : multiply % by n

Graphs

Draw error bars, best line, worst line. Δgradient = |mbest − mworst|

Key Rules
z = x ± y → Δz = Δx + Δy
z = xy → Δz/z = Δx/x + Δy/y
z = xⁿ → Δz/z = n(Δx/x)
Δm = |mbest − mworst|