Fractional Uncertainty in Physics: Definition, Formula, and Worked Examples
Of the three forms of uncertainty used in CIE 9702 — absolute, fractional, and percentage — fractional uncertainty is the one students understand least precisely. They know the formula exists somewhere between absolute and percentage, they vaguely recall it is a decimal rather than a percentage, and they apply it inconsistently in multi-step calculations.
The March 2025 CIE Physics examiner report states explicitly that “many candidates incorrectly calculated an answer using a fractional uncertainty in Physics” — meaning the error is not isolated or occasional but widespread and repeatable.
This guide defines fractional uncertainty precisely, explains the formula and where it sits in relation to absolute and percentage uncertainty, works through multiple exam-style examples at increasing complexity, and covers every mistake the CIE examiner identifies in students who handle this topic incorrectly.
What Is Fractional Uncertainty in Physics? The Precise Definition
Fractional uncertainty in Physics is the absolute uncertainty in a measurement expressed as a fraction of the measured value. It is dimensionless — it has no unit — because both the absolute uncertainty and the measured value share the same unit, which cancels in the division.
Fractional uncertainty = Absolute uncertainty / Measured value
In symbolic notation:
ΔQ / Q
Where:
- ΔQ is the absolute uncertainty (in the same unit as the measurement)
- Q is the measured value (in the same unit as the absolute uncertainty)
- ΔQ / Q is the fractional uncertainty (dimensionless, no unit)
Example: A student measures a resistance R = 47 Ω with an absolute uncertainty ΔR = ±2 Ω.
Fractional uncertainty = ΔR / R = 2 / 47 = 0.043
This tells you that the uncertainty is approximately 4.3% of the measured value — which connects directly to the percentage uncertainty form.
The relationship between all three forms is fixed and exact:
| Form | Expression | Example (R = 47 ± 2 Ω) |
| Absolute uncertainty | ΔQ | ±2 Ω |
| Fractional uncertainty | ΔQ / Q | 0.043 |
| Percentage uncertainty | (ΔQ / Q) × 100% | 4.3% |
Percentage uncertainty is simply fractional uncertainty multiplied by 100. They carry identical information in different formats. CIE 9702 mark schemes accept both forms in combination rule calculations — but they must not be mixed within the same calculation without converting.
Why Fractional Uncertainty in Physics Is Used
Fractional uncertainty in Physics solves the problem that absolute uncertainty alone cannot. An absolute uncertainty of ±2 Ω means something very different depending on the size of the resistance being measured.
For R = 5 Ω: fractional uncertainty = 2/5 = 0.40 → 40% — the measurement is highly uncertain. For R = 500 Ω: fractional uncertainty = 2/500 = 0.004 → 0.4% — the measurement is very reliable.
The absolute uncertainty is the same in both cases. The fractional uncertainty reveals the actual quality of each measurement relative to its magnitude. This is why fractional uncertainty is the correct tool for comparing the reliability of different measurements and for identifying which measurement dominates the total uncertainty in a calculated result.
For AS Level Physics and A Level Physics students, this concept matters most in Paper 3 and Paper 5 questions that ask: “Identify the measurement that contributes most to the uncertainty in the final result.” The answer is always the measurement with the highest fractional uncertainty in Physics — or equivalently, the highest percentage uncertainty.
Calculating Fractional Uncertainty From Different Sources
Fractional uncertainty can be calculated from different types of measurement, depending on how the data is collected.
From a Single Reading on an Analogue Instrument
For a single reading from an analogue instrument, the absolute uncertainty is typically ±half the smallest scale division.
Example: A ruler has divisions every 1 mm. A single measurement gives L = 84 mm. Absolute uncertainty: ΔL = ±0.5 mm Fractional uncertainty: ΔL / L = 0.5 / 84 = 0.0060
From a Measurement Using Two Readings
When a measurement requires two readings (e.g. measuring a length with a ruler where both start and end positions are recorded), both readings contribute uncertainty and the absolute uncertainty doubles.
Example: Initial reading: 10.0 mm (±0.5 mm), Final reading: 94.0 mm (±0.5 mm) Length L = 84.0 mm Absolute uncertainty: ΔL = 0.5 + 0.5 = ±1.0 mm Fractional uncertainty: ΔL / L = 1.0 / 84.0 = 0.012
This is double the previous example — an important distinction. Using ±0.5 mm for a two-reading measurement is one of the most consistently penalised errors in Paper 3.
From Repeated Measurements
When a measurement is repeated multiple times, the absolute uncertainty is taken as ±half the range of the readings.
Example: Five timing measurements (s): 1.42, 1.45, 1.38, 1.44, 1.41 Mean t = 1.42 s Range = 1.45 − 1.38 = 0.07 s Absolute uncertainty: Δt = ±0.07/2 = ±0.035 s Fractional uncertainty: Δt / t = 0.035 / 1.42 = 0.025
The free topical past paper workbooks at Quality Notes include structured questions on all three methods of calculating fractional uncertainty in Physics, drawn from multiple years of CIE 9702 past papers.
Combination Rules Using Fractional Uncertainty
This is where fractional uncertainty does its most important work in CIE 9702 — in multi-variable calculations where uncertainties from several measured quantities must be combined to find the total uncertainty in a derived result.
Rule 1: Addition and Subtraction — Use Absolute Uncertainty, NOT Fractional
When quantities are added or subtracted, add their absolute uncertainties. Do not use fractional uncertainty here.
If z = x + y or z = x − y: Δz = Δx + Δy
This is the most common rule-switching error in CIE 9702 uncertainty questions.
Rule 2: Multiplication and Division — Add Fractional Uncertainties
When quantities are multiplied or divided, add their fractional uncertainties.
If z = x × y or z = x / y: Δz/z = Δx/x + Δy/y
This is entirely equivalent to adding percentage uncertainties — both forms give the same result. The choice between fractional and percentage in multiplication/division questions is a matter of preference, not correctness, provided you are consistent.
Example: Speed v = distance / time d = (2.40 ± 0.05) m → Δd/d = 0.05/2.40 = 0.021 t = (1.80 ± 0.10) s → Δt/t = 0.10/1.80 = 0.056 v = 2.40/1.80 = 1.333 m s⁻¹ Δv/v = 0.021 + 0.056 = 0.077 Δv = 0.077 × 1.333 = 0.103 ≈ 0.1 m s⁻¹ (1 sig fig) Result: v = (1.3 ± 0.1) m s⁻¹
Rule 3: Powers — Multiply Fractional Uncertainty by the Power
When a quantity is raised to a power n, multiply its fractional uncertainty by n.
If z = xⁿ: Δz/z = n × (Δx/x)
Example: Period of a pendulum: T = 2π√(l/g) → l = gT²/4π² Length l = (0.850 ± 0.005) m → Δl/l = 0.005/0.850 = 0.0059 Period T = (1.847 ± 0.020) s → ΔT/T = 0.020/1.847 = 0.0108
Using l = gT²/4π² — T appears squared, l appears to the first power: Δg/g = Δl/l + 2(ΔT/T) = 0.0059 + 2(0.0108) = 0.0059 + 0.0216 = 0.0275 g = 4π² × 0.850 / 1.847² = 9.81 m s⁻² Δg = 0.0275 × 9.81 = 0.270 ≈ 0.3 m s⁻² (1 sig fig) Result: g = (9.8 ± 0.3) m s⁻²
Rule 4: Constants and Exact Numbers — Zero Contribution
Mathematical constants (π, 4, 2) and exact conversion factors carry no uncertainty and contribute zero to the combined fractional uncertainty.
Converting Between Fractional and Absolute Uncertainty in the Final Answer
After combining fractional uncertainties through multiplication, division, or power rules, the result is a fractional uncertainty in the derived quantity. To express the final answer correctly, this must be converted back to an absolute uncertainty.
Step: Multiply the fractional uncertainty of the result by the calculated value of the result.
Absolute uncertainty in result = (fractional uncertainty) × (calculated value)
This final step is where many students stop too early — they calculate the fractional uncertainty correctly but forget to convert it back to an absolute uncertainty before writing the final answer. A result expressed as “g = 9.81 ± 0.0275” is not a correct final answer — the 0.0275 is the fractional uncertainty, not the absolute uncertainty in m s⁻². The absolute uncertainty is 0.0275 × 9.81 = 0.27 ≈ 0.3 m s⁻².
Multi-Step Worked Example: Resistivity Calculation
A student measures the resistance R, length L, and diameter d of a wire to calculate resistivity ρ using:
ρ = RA / L = R × π(d/2)² / L = Rπd² / 4L
Measurements:
- R = (15.4 ± 0.5) Ω
- d = (0.42 ± 0.01) mm = (4.2 × 10⁻⁴ ± 1 × 10⁻⁵) m
- L = (1.250 ± 0.005) m
Step 1: Calculate fractional uncertainty of each measurement. ΔR/R = 0.5/15.4 = 0.0325 Δd/d = 0.01/0.42 = 0.0238 ΔL/L = 0.005/1.250 = 0.0040
Step 2: Apply combination rules. ρ = Rπd²/4L → d is squared, R and L appear to first power: Δρ/ρ = ΔR/R + 2(Δd/d) + ΔL/L Δρ/ρ = 0.0325 + 2(0.0238) + 0.0040 Δρ/ρ = 0.0325 + 0.0476 + 0.0040 = 0.0841
Step 3: Calculate ρ. A = π(2.1 × 10⁻⁴)² = 1.385 × 10⁻⁷ m² ρ = 15.4 × 1.385 × 10⁻⁷ / 1.250 = 1.71 × 10⁻⁶ Ω m
Step 4: Calculate absolute uncertainty. Δρ = 0.0841 × 1.71 × 10⁻⁶ = 1.44 × 10⁻⁷ ≈ 0.1 × 10⁻⁶ Ω m (1 sig fig)
Final answer: ρ = (1.7 ± 0.1) × 10⁻⁶ Ω m
This example demonstrates the complete five-step process — individual fractional uncertainties, combination with the power rule applied to d², calculation of the result, and conversion back to absolute uncertainty — all formatted correctly for CIE 9702 mark schemes.
Fractional Uncertainty in Physics in Graph Analysis: Paper 5
The March 2025 CIE examiner report flagged a specific error in Paper 5 analysis questions: “Some candidates determined the uncertainty in the y-intercept by assuming that the fractional uncertainty in the gradient was the same as the fractional uncertainty in the y-intercept.” This is incorrect — gradient and intercept uncertainties must each be determined separately from the worst acceptable line, not transferred between each other.
In Paper 5 graph analysis, fractional uncertainty in the gradient is calculated as:
Fractional uncertainty in gradient = |gradient of best-fit line − gradient of worst acceptable line| / gradient of best-fit line
This is then converted to a percentage uncertainty if required. The y-intercept uncertainty is calculated separately using the same worst acceptable line — the y-value where that line crosses the y-axis compared to where the best-fit line crosses it.
Each uncertainty is independent. Assuming one equals the other — as many candidates did in March 2025 — loses the marks allocated for each calculation separately.
For structured support on every graph-based skill including fractional uncertainty in gradients, recorded lessons at Quality Notes provide worked Paper 5 examples with the exact steps the mark scheme rewards.
Common Exam Mistakes on Fractional Uncertainty
Leaving the answer as a fractional uncertainty instead of converting to absolute. After calculating Δz/z = 0.084, students must convert: Δz = 0.084 × z. Writing only 0.084 as the uncertainty in the final answer is dimensionally wrong and scores zero on the presentation mark.
Using fractional uncertainty in Physics when absolute uncertainty is required. When adding or subtracting measurements, absolute uncertainties are added — not fractional uncertainties. The March 2025 CIE examiner report specifically noted this error as widespread: “many candidates incorrectly calculated an answer using a fractional uncertainty” in a context requiring absolute uncertainty.
Not applying the power factor to squared or cubed quantities. When d appears as d² in a formula, its fractional uncertainty must be multiplied by 2 before adding to the total. Forgetting this factor is the most common error in resistivity and volume uncertainty calculations and is flagged in examiner reports across multiple recent sessions.
Using single-reading absolute uncertainty for a two-reading measurement. A length measured as the difference between two ruler readings has absolute uncertainty ±(0.5 + 0.5) = ±1 mm, not ±0.5 mm. Using the single-reading uncertainty halves the absolute uncertainty and consequently halves the fractional uncertainty in Physics — producing an answer that understates the actual measurement uncertainty.
Mixing fractional and percentage uncertainty in the same calculation. Fractional uncertainty (0.043) and percentage uncertainty (4.3%) differ by a factor of 100. Adding 0.043 to 4.3% without converting produces an answer wrong by almost that factor. Always work in one form consistently throughout the same calculation.
For personalised support identifying specific patterns in your uncertainty errors and building a targeted revision plan, students counselling at Quality Notes is available to help. For comprehensive chapter-by-chapter notes covering measurement skills and all other CIE 9702 topics, books and revision notes at Quality Notes are written specifically around the current syllabus.
People Also Ask About Fractional Uncertainty in Physics
What is fractional uncertainty in physics?
Fractional uncertainty is the absolute uncertainty in a measurement divided by the measured value: ΔQ/Q. It is dimensionless — it has no unit — because the units of the numerator and denominator cancel. It expresses the uncertainty as a proportion of the measurement and is used in combination rules for multiplication, division, and powers.
What is the difference between fractional and percentage uncertainty?
Fractional uncertainty is ΔQ/Q (a decimal, e.g. 0.043). Percentage uncertainty is (ΔQ/Q) × 100% (a percentage, e.g. 4.3%). They carry identical information — percentage uncertainty is simply fractional uncertainty multiplied by 100. Both are accepted in CIE 9702 combination rule calculations, but they must not be mixed within the same calculation without converting.
When do you use fractional uncertainty in Physics?
Fractional uncertainty is used when combining uncertainties in quantities that are multiplied, divided, or raised to powers. When quantities are added or subtracted, absolute uncertainties are used instead. Applying fractional uncertainty to addition or subtraction, or absolute uncertainty to multiplication, is the most common rule error in CIE 9702.
How do you calculate fractional uncertainty in Physics from repeated measurements?
Take the range of the repeated measurements (maximum minus minimum) and divide by 2 to get the absolute uncertainty. Then divide the absolute uncertainty by the mean value to get the fractional uncertainty: (x_max − x_min) / (2 × mean).
How does fractional uncertainty change with powers?
When a quantity Q is raised to a power n, its fractional uncertainty is multiplied by n. For Q², the fractional uncertainty doubles. For Q^(1/2) (a square root), it halves. For Q³, it triples. This power rule applies regardless of whether the power is a whole number or a fraction.
How do you express a final answer after combining fractional uncertainties?
After combining fractional uncertainties through multiplication, division, and power rules, multiply the total fractional uncertainty in Physics by the calculated value to obtain the absolute uncertainty in the result. Round the absolute uncertainty to 1 significant figure, then round the calculated value to match the same decimal place. Express the result as (value ± absolute uncertainty) with correct SI units.
Conclusion
Fractional uncertainty in Physics sits at the centre of the CIE 9702 uncertainty framework. It converts absolute measurement doubt into a proportional form that can be meaningfully combined across different quantities with different units and magnitudes. It is the natural language of multiplication and division uncertainty — and the tool that connects individual measurements to the reliability of any calculated result.
Students who understand fractional uncertainty in Physics precisely — not as a formula to recall but as a concept with a clear physical meaning — handle every uncertainty question with the same systematic approach: identify each source, calculate its fractional form, combine using the appropriate rule, convert back to absolute, and present in the format the mark scheme rewards.
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