Uncertainty & Error Propagation (IPhO)

Uncertainty work is mostly bookkeeping. If you are consistent about definitions, units, and rounding, you pick up a lot of easy marks.

1. Definitions (Must Know)

• Measured value: a reported number from an instrument, written as x.
• Absolute uncertainty: Δ x in the same units as x.
• Relative uncertainty: Δ x/x (dimensionless).
• Percentage uncertainty: (Δ x/x) × 100%.
• Random uncertainty: causes scatter; reduced by repeats.
• Systematic error: shifts all readings in a consistent direction; not reduced by repeats (zero error, calibration, method bias).
• Propagated uncertainty: the uncertainty in a calculated quantity Q = f(x,y,…) caused by uncertainties in the inputs.
• Reporting rule (typical): quote Δ Q to 1 significant figure (2 if it begins with 1 or 2), then round Q to the same decimal place.

2. Key Ideas (What Earns Marks)

3. Detailed Explanations

A. Assigning uncertainties

For a single reading, instrument resolution is your baseline. For a digital display, a common choice is half a least significant digit. For an analogue scale, use about half the smallest division, plus any reading uncertainty (for example parallax).

For repeated readings, separate:

• mean value x bar
• random uncertainty of the mean (from scatter)
• any known systematic contribution (for example a zero offset)

If N is small, a simple estimate like half the range can be acceptable, but state what you did.

B. Propagating uncertainties

Two styles are common. Worst case adds uncertainties linearly. Independent random uncertainties combine in quadrature (root-sum-square). Pick one and use it consistently.

When a result depends on several factors, working in relative uncertainties is usually fastest.

C. Derived parameters from graphs

If you extract a parameter from a gradient, propagate uncertainty from the gradient like any other input. See Data Fitting & Linearisation (IPhO) for a practical way to estimate Δ m using max and min acceptable lines.

4. Common Mistakes

• Mixing absolute and percentage uncertainties in the same line.
• Rounding intermediate values too early, then getting inconsistent final rounding.
• Writing Q = 12.345 ± 0.9 (value has more precision than uncertainty allows).
• Treating a systematic offset as a random uncertainty and claiming repeats fix it.
• Forgetting powers: if Q ∝ T⁻², the relative uncertainty contribution from T doubles.
• Using an uncertainty smaller than instrument resolution without justification.

5. Exam Tips

• Put uncertainties in your first data table, not as an afterthought.
• Show one clear propagation line, then a final statement: Q = (…) unit.
• If one term dominates, say so: “uncertainty is dominated by d measurement”.
• If comparing to a known value Q₀, compute a percentage difference and compare it with the percentage uncertainty.

6. Worked Examples

Example 1: Resistance from R = V/I

Suppose V = 2.50 V with Δ V = 0.02 V and I = 0.184 A with Δ I = 0.003 A.

Compute R = 2.50/0.184 = 13.6 Ω.

Relative uncertainties: (Δ V)/V = 0.02/2.50 = 0.008, (Δ I)/I = 0.003/0.184 ≈ 0.016.

Quadrature gives (Δ R)/R ≈ square root of (0.008² + 0.016²) ≈ 0.018. So Δ R ≈ 13.6 × 0.018 ≈ 0.25 Ω, and report R = (13.6 ± 0.3) Ω.

Example 2: g from g = (4π² L)/T²

Let L = 0.800 m, Δ L = 0.001 m, T = 1.79 s, Δ T = 0.02 s.

The relative uncertainty is (Δ g)/g ≈ square root of (((Δ L)/L)² + (2(Δ T)/T)²) ≈ 0.022. So if g ≈ 9.86 m s⁻², then Δ g ≈ 0.22 m s⁻² and g = (9.86 ± 0.22) m s⁻².

7. Mind Stretchers

• Two quantities come from the same instrument scale. Are their uncertainties independent?
• You measure x and compute y = 1/x. Does the absolute uncertainty in y grow or shrink as x increases?
• If your result disagrees with an accepted value by more than your quoted uncertainty, list two plausible systematic effects.

8. Practice

1. You measure r = 12.0 mm with Δ r = 0.1 mm. Find the percentage uncertainty in A = π r².
2. A lens formula gives f = uv/(u + v). Derive Δ f/f in terms of Δ u/u and Δ v/v (use either linear or quadrature, but state which).
3. A parameter is found from a gradient m = 3.20 ± 0.15. If k = 1/m, find k and Δ k.

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Answer key and feedback

• For y = 1/x, Delta y ≈ Δ x/x², so the absolute uncertainty shrinks as x grows if Delta x is fixed. The relative uncertainty remains approximately Delta x/x.
• Measurements read from the same scale can be correlated. Do not combine them as independent until the shared zero, calibration or reading error has been modelled.