Uncertainties
Specification: OCR A H556 | Section: 1.2 | Teaching time: ~3 hours
- Define and calculate absolute and percentage uncertainties
- Combine uncertainties in calculations (addition, multiplication, powers)
- Plot error bars and determine uncertainty in gradients
- Express final answers with appropriate significant figures and uncertainty
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.
Percentage uncertainty expresses the uncertainty as a percentage of the measured value:
Fractional uncertainty is the ratio Δx/x (without multiplying by 100).
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
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:
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:
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:
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
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.
Knowledge Check
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)
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)
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)
Finding Uncertainty in y-intercept
Use the same approach with the best and worst lines:
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.
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
✅ 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)
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-Style Questions
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)
(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)
(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)
(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)
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
(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
Δx is absolute; (Δx/x)×100% is percentage
+/− : add absolute; ×/÷ : add percentage; powers : multiply % by n
Draw error bars, best line, worst line. Δgradient = |mbest − mworst|