Combining Uncertainties

In practice you rarely measure a single quantity in isolation. Resistance R is calculated from V and I; density ρ is calculated from m and V; the Young modulus E is calculated from F, L, A and x. In each case you must start with uncertainties on the raw measurements and end with an uncertainty on the final result. The process of doing this is called propagating uncertainty, and it is governed by a small set of simple rules.

These rules are on every OCR A-Level Physics paper that involves practical work. You must know them fluently — they are among the highest-yielding pieces of knowledge in the specification.

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The Three Rules

The rules at A-Level (OCR and AQA both) are simplifications of the rigorous statistical rules used at university. They are conservative — they give a slightly larger uncertainty than the strict statistical combination — but they are fast, easy to apply by hand, and match the mark-scheme expectations.

Rule 1 — Addition and Subtraction: Add Absolute Uncertainties

If Z = X + Y or Z = X − Y, then:

ΔZ = ΔX + ΔY

Absolute uncertainties add — never subtract, even when the quantities are subtracted. This is because the uncertainty of a subtraction is the worst-case spread of possibilities, which is the sum.

Rule 2 — Multiplication and Division: Add Percentage Uncertainties

If Z = X × Y or Z = X / Y, then:

(ΔZ/Z) × 100% = (ΔX/X) × 100% + (ΔY/Y) × 100%

Or in shorthand:

%Z = %X + %Y

Percentage uncertainties add. Note that this rule holds regardless of whether the operation is multiplication or division.

Rule 3 — Powers: Multiply Percentage Uncertainty by the Power

If Z = X^n, then:

%Z = |n| × %X

This follows from Rule 2: squaring is multiplying X by itself, so %X + %X = 2 × %X. Similarly for cubes, square roots (n = 1/2), and reciprocals (n = −1).

Combining the Rules

For a general formula Z = k × X^a × Y^b × W^c (where k is a dimensionless constant and a, b, c are powers), the rules combine cleanly:

%Z = |a| × %X + |b| × %Y + |c| × %W

Dimensionless constants (like ½, π, g) carry no uncertainty and do not appear in the formula.

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A Mermaid Flowchart — Which Rule to Use?

graph TD
    A[Start: Z depends on X, Y, ...] --> B{Are X, Y added or subtracted?}
    B -->|Yes| C[Add ABSOLUTE uncertainties<br/>ΔZ = ΔX + ΔY]
    B -->|No| D{Are X, Y multiplied or divided?}
    D -->|Yes| E[Add PERCENTAGE uncertainties<br/>%Z = %X + %Y]
    D -->|No| F{Is there a power X^n?}
    F -->|Yes| G[Multiply % by the power<br/>%Z = n × %X]
    C --> H[Quote result with absolute uncertainty<br/>rounded to 1 s.f.]
    E --> H
    G --> H

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Worked Example 1 — Adding Two Lengths

A student measures two pieces of string and adds them end-to-end.

• L₁ = 245 ± 2 mm
• L₂ = 130 ± 1 mm

Calculate the total length and its uncertainty.

Solution:

L = L₁ + L₂ = 245 + 130 = 375 mm

ΔL = ΔL₁ + ΔL₂ = 2 + 1 = 3 mm

L = (375 ± 3) mm

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Worked Example 2 — Subtraction Can Magnify Relative Uncertainty

Two masses are measured with a digital balance:

• m₁ = 52.34 ± 0.01 g
• m₂ = 52.18 ± 0.01 g

Calculate m₁ − m₂ and its percentage uncertainty.

Solution:

m₁ − m₂ = 52.34 − 52.18 = 0.16 g

Δ(m₁ − m₂) = 0.01 + 0.01 = 0.02 g (absolute uncertainties add)

Percentage uncertainty = (0.02 / 0.16) × 100 = 12.5%

Δ = (0.16 ± 0.02) g ≈ ±13%

Note the enormous percentage uncertainty, compared to just 0.02% on each individual mass. This is the subtraction catastrophe: subtracting two nearly equal numbers throws away precision. You should avoid experimental designs that depend on such subtractions whenever possible.

Exam Tip: OCR sometimes asks you to "explain why the method gives a large uncertainty". If the method involves subtracting two similar values, say so, and state that "the absolute uncertainty does not change but the value becomes small, inflating the percentage uncertainty".

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Worked Example 3 — Multiplication for Area

A rectangular plate has width w = 30.0 ± 0.5 mm and length l = 80.0 ± 0.5 mm. Calculate the area and its uncertainty.

Solution:

A = w × l = 30.0 × 80.0 = 2400 mm²

Percentage uncertainties:

• %w = (0.5 / 30.0) × 100 = 1.67%
• %l = (0.5 / 80.0) × 100 = 0.625%

Add:

%A = 1.67 + 0.625 = 2.29%

Absolute uncertainty:

ΔA = (2.29 / 100) × 2400 = 55 mm²

Quoted result: A = (2400 ± 60) mm² or A = (2.4 ± 0.1) × 10³ mm² to 1 s.f. on the uncertainty.

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Worked Example 4 — Division for Density

A metal block is weighed and measured.

• mass m = (50.00 ± 0.05) g
• volume V = (18.5 ± 0.3) cm³

Calculate the density and its absolute uncertainty in SI units (kg m⁻³).

Solution:

ρ = m/V = 50.00 / 18.5 = 2.7027 g cm⁻³

Convert to SI: g cm⁻³ = 10³ kg m⁻³, so ρ = 2703 kg m⁻³ (approximately).

Percentage uncertainties:

• %m = (0.05/50.00) × 100 = 0.10%
• %V = (0.3/18.5) × 100 = 1.62%
• %ρ = 0.10 + 1.62 = 1.72%

Absolute uncertainty:

Δρ = (1.72/100) × 2703 = 47 kg m⁻³ ≈ 50 kg m⁻³ (1 s.f.)

Quoted result: ρ = (2700 ± 50) kg m⁻³.

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Worked Example 5 — Using Powers (Kinetic Energy)

A student measures the mass and speed of a trolley:

• m = (1.20 ± 0.01) kg
• v = (2.50 ± 0.05) m s⁻¹

Calculate the kinetic energy and its uncertainty.

Solution:

E_k = ½ m v² = 0.5 × 1.20 × (2.50)² = 0.5 × 1.20 × 6.25 = 3.75 J

Percentage uncertainties:

• %m = (0.01/1.20) × 100 = 0.833%
• %v = (0.05/2.50) × 100 = 2.00%

v is squared, so its percentage uncertainty is doubled:

%(v²) = 2 × 2.00 = 4.00%